Financial Risk Paper Collection

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Financial Risk The articles in this collection analyze financial risk from a multitude of angles and disciplines, including accounting, economics, and mathematics. Topics covered by articles in this collection include the influence of mood on the willingness to take financial risks, risk models, risk economics, financial risk following the global financial crisis, and risk and longevity in investments and hedges. Simply click on the article titles below to read them online for free*! *free access is available until 31/12/2014 ●

A cautionary note on natural hedging of longevity risk ❍

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A closed-form approximation for valuing European basket warrants under credit risk and interest rate risk ❍

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Quantitative Finance Yung-Ming Shiu, Pai-Lung Chou & Jen-Wen Sheu

A generalized pricing framework addressing correlated mortality and interest risks: a change of probability measure approach ❍

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North American Actuarial Journal Nan Zhu & Daniel Bauer

Stochastics: An International Journal of Probability and Stochastic Processes iaoming Liu, Rogemar Mamon & Huan Gao

An industrial organization theory of risk sharing

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A simulation model for calculating solvency capital requirements for non-life insurance risk ❍

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North American Actuarial Journal M. Martin Boyer & Charles M. Nyce

Scandinavian Actuarial Journal Jonas Alm

A threshold based approach to merge data in financial risk management Journal of Applied Statistics ❍ Silvia Figini, Paolo Giudici & Pierpaolo Uberti Open Access ❍

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Asset allocation with risk factors ❍

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Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks ❍

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Scandinavian Actuarial Journal Haizhong Yang, Wei Gao & Jinzhu Li

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks ❍

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Quantitative Finance Letters Kris Boudt & Benedict Peeters

Scandinavian Actuarial Journal Yang Yang & Dimitrios G. Konstantinides

Competition and risk in Japanese banking

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The European Journal of Finance Hong Liu & John O.S. Wilson

Credit risk modelling and sustainable agriculture: Asset evaluation and rural carbon revenue Journal of Sustainable Finance & Investment ❍ Emmanuel Olatunbosun Benjamin Open Access ❍

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Credit valuation adjustment and wrong way risk ❍

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Developing equity release markets: Risk analysis for reverse mortgages and home reversions ❍

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North American Actuarial Journal Daniel H. Alai, Hua Chen, Daniel Cho, Katja Hanewald & Michael Sherris

Downside risk management of a defined benefit plan considering longevity basis risk ❍

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Quantitative Finance Letters Umberto Cherubini

North American Actuarial Journal Yijia Lin, Ken Seng Tan, Ruilin Tian & Jifeng Yu

Estimating the risk–return profile of new venture investments using a risk-neutral framework and ‘thick’ models ❍

The European

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Evaluating lotteries, risks, and risk‐mitigation programs ❍

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Quantitative Finance Halis Sak & Wolfgang Hörmann

First passage time for compound Poisson processes with diffusion: Ruin theoretical and financial applications ❍

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Jian Zhou

Fast simulations in credit risk ❍

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The European Accounting Review Miles B. Gietzmann & Angela K. Pettinicchio

Extreme risk spillover among international REIT markets ❍

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Quantitative Finance Begoña Font & Alfredo Juan Grau

External auditor reassessment of client business risk following the issuance of a comment letter by the SEC ❍

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Journal of Risk Research Mei Wang & Paul S. Fischbeck

Exchange rate and inflation risk premia in the EMU ❍

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Journal of Finance Beat Reber

Scandinavian Actuarial Journal David Landriault & Tianxiang Shi

Gaussian risk models with financial constraints

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GJR-GARCH model in value-at-risk of financial holdings ❍

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Applied Financial Economics Sun-Joong Yoon & Suk Joon Byun

Internal loss data collection implementation: evidence from a large UK financial institution ❍

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Journal of Risk Research Eun-sung Kim

Implied risk aversion and volatility risk premiums ❍

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Journal of Risk Research Patrick Parnaby

How did enterprise risk management first appear in the Korean public sector? ❍

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Applied Financial Economics Y. C. Su, H. C. Huang & Y. J. Lin

Health and finance: Exploring the parallels between health care delivery and professional financial planning ❍

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Scandinavian Actuarial Journal Krzysztof D•bicki, Enkelejd Hashorva & Lanpeng Ji

Journal of Risk Research Cormac Bryce, Robert Webb & Jennifer Adams

Keeping some skin in the game: How to start a capital market in longevity risk transfers ❍

North American

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Actuarial Journal Enrico Biffis & David Blake

Learning to organise risk management in organisations: What future for enterprise risk management? Journal of Risk Research ❍ Frank Schiller & George Prpich Open Access ❍

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Maimonides risk parity ❍

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Managing the invisible: Identifying value-maximizing combinations of risk and capital ❍

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Quantitative Finance Bernd Scherer

Measuring basis risk in longevity hedges ❍

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North American Actuarial Journal William H. Panning

Market risks in asset management companies ❍

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Quantitative Finance Letters Philip Z. Maymin & Zakhar G. Maymin

North American Actuarial Journal Johnny Siu-Hang Li PhD, FSA & Mary R. Hardy PhD, FIA, FSA, CERA

Measuring operational risk in financial institutions ❍

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Applied Financial Economics Séverine Plunus, Georges Hübner & Jean-Philippe Peters

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Misclassifications in financial risk tolerance ❍

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Modeling of commercial real estate credit risks ❍

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Applied Financial Economics Su-Lien Lu, Kuo-Jung Lee & Chia-Chang Yu

Multivariate models for operational risk ❍

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Journal of Sustainable Finance & Investment Mehdi Beyhaghi & James P. Hawley

Momentum strategy and credit risk ❍

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Quantitative Finance Yong Kim

Modern portfolio theory and risk management: Assumptions and unintended consequences ❍

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Journal of Risk Research Caterina Lucarelli, Pierpaolo Uberti & Gianni Brighetti

Quantitative Finance Klaus Böcker & Claudia Klüppelberg

Numerical analysis for Spread option pricing model of markets with finite liquidity: First-order feedback model ❍

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International Journal of Computer Mathematics A. Shidfar, Kh. Paryab, A.R. Yazdanian & Traian A. Pirvu

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On dependence of volatility on return for stochastic volatility models ❍

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On risk charges and shadow account options in pension funds ❍

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The European Journal of Finance Georges Dionne & Thouraya Triki

Optimal investment of an insurer with regime-switching and risk constraint ❍

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International Journal of Computer Mathematics Karina Gibert & Dante Conti

On risk management determinants: What really matters? ❍

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Scandinavian Actuarial Journal Peter Løchte Jørgensen & Nadine Gatzert

On the understanding of profiles by means of postprocessing techniques: an application to financial assets ❍

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Stochastics: An International Journal of Probability and Stochastic Processes Mikhail Martynov & Olga Rozanova

Scandinavian Actuarial Journal Jingzhen Liu, Ka-Fai Cedric Yiu & Tak Kuen Siu

Optimal investment under dynamic risk constraints and

partial information ❍

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Optimal risk classification with an application to substandard annuities ❍

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Stochastics: An International Journal of Probability and Stochastic Processes Masahiko Egami & Kazutoshi Yamazaki

Real earnings management uncertainty and corporate credit risk ❍

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The European Accounting Review George Batta, Ricardo Sucre Heredia & Marc Weidenmier

Precautionary measures for credit risk management in jump models ❍

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North American Actuarial Journal Nadine Gatzert, Gudrun Schmitt-Hoermann & Hato Schmeiser

Political connections and accounting quality under high expropriation risk ❍

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Quantitative Finance Wolfgang Putschögl & Jörn Sass

The European Accounting Review Tsung-Kang Chen, Yijie Tseng & Yu-Ting Hsieh

Risk aversion vs. individualism: What drives risk taking in household finance?

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Risk perception and management in smallholder dairy farming in Tigray, Northern Ethiopia ❍

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Quantitative Finance Rama Cont, Romain Deguest & Giacomo Scandolo

Robust risk measurement and model risk ❍

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Journal of Risk Research Lijana Baublyte, Martin Mullins & John Garvey

Robustness and sensitivity analysis of risk measurement procedures ❍

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Accounting and Business Research Stephen G. Ryan

Risk selection in the London political risk insurance market: The role of tacit knowledge, trust and heuristics ❍

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Journal of Risk Research Kinfe Gebreegziabher & Tewodros Tadesse

Risk reporting quality: Implications of academic research for financial reporting policy ❍

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The European Journal of Finance Wolfgang Breuer, Michael Riesener & Astrid Juliane Salzmann

Quantitative Finance Paul Glasserman & Xingbo Xu

Specificity of reinforcement for risk behaviors of the Balloon Analog Risk Task using math models of performance

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Stochastic modelling of mortality and financial markets ❍

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Journal of Risk Research Nicole Prause & Steven Lawyer

Scandinavian Actuarial Journal Helena Aro & Teemu Pennanen

The adoption and design of enterprise risk management practices: An empirical study The European Accounting Review ❍ Leen Paape & Roland F. Speklé Open Access ❍

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The case for convex risk measures and scenario-dependent correlation matrices to replace VaR, C-VaR and covariance simulations for safer risk control of portfolios ❍

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The effectiveness of using a basis hedging strategy to mitigate the financial consequences of weather-related risks ❍

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Quantitative Finance Letters William T. Ziemba

North American Actuarial Journal Linda L. Golden PhD, Charles C. Yang PhD & Hong Zou PhD

The first five years of the EU Impact Assessment system: A risk economics perspective on gaps between rationale and practice ❍

Journal of Risk

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The impact of banking and sovereign debt crisis risk in the eurozone on the euro/US dollar exchange rate ❍

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China Journal of Accounting Studies Tianshu Zhang & Jun Huang

The UK's prudential borrowing framework: A retrograde step in managing risk? ❍

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Journal of Risk Research John E. Grable & Michael J. Roszkowski

The risk premium of audit fee: Evidence from the 2008 Financial Crisis ❍

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Applied Mathematical Finance Wolfgang Karl Härdle & Brenda López Cabrera

The influence of mood on the willingness to take financial risks ❍

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Applied Financial Economics Stefan Eichler

The implied market price of weather risk ❍

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Research Jacopo Torriti & Ragnar Löfstedt

Journal of Risk Research John Hood, Darinka Asenova, Stephen Bailey & Melina Manochin

Towards a new framework to account for environmental risk in sovereign credit risk analysis

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Transfer pricing as a tax compliance risk ❍

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The European Journal of Finance Yi-Mien Lin, Chin-Fang Chao & Chih-Liang Liu

Up and down credit risk ❍

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Accounting and Business Research Sven P. Jost, Michael Pfaffermayr & Hannes Winner

Transparency, idiosyncratic risk, and convertible bonds ❍

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Journal of Sustainable Finance & Investment Margot Hill Clarvis, Martin Halle, Ivo Mulder & Masaru Yarime

Quantitative Finance Tomasz R. Bielecki, Stéphane Crépey & Monique Jeanblanc

Why do state-owned enterprises over-invest? Government intervention or managerial entrenchment ❍

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China Journal of Accounting Studies Jun Bai & Lishuai Lian

Published on 13 June 2014. Last updated on 8 July 2014. Copyright © 2014 Informa UK Limited, an Informa Group company.

North American Actuarial Journal, 18(1), 104–115, 2014 C Society of Actuaries Copyright  ISSN: 1092-0277 print / 2325-0453 online DOI: 10.1080/10920277.2013.876911

A Cautionary Note on Natural Hedging of Longevity Risk Nan Zhu1 and Daniel Bauer2 1

Department of Mathematics, Illinois State University, Normal, Illinois Department of Risk Management and Insurance, Georgia State University, Atlanta, Georgia

2

In this article, we examine the so-called natural hedging approach for life insurers to internally manage their longevity risk exposure by adjusting their insurance portfolio. In particular, unlike the existing literature, we also consider a nonparametric mortality forecasting model that avoids the assumption that all mortality rates are driven by the same factor(s). Our primary finding is that higher order variations in mortality rates may considerably affect the performance of natural hedging. More precisely, although results based on a parametric single factor model—in line with the existing literature—imply that almost all longevity risk can be hedged, results are far less encouraging for the nonparametric mortality model. Our finding is supported by robustness tests based on alternative mortality models.

1. INTRODUCTION Longevity risk, the risk that policyholders will live longer than expected, has recently attracted increasing attention from both academia and insurance practitioners. Different ways have been suggested on how to manage this risk, for example, by transferring it to the financial market via mortality-linked securities (see, e.g., Blake et al. 2006). One approach that is particularly appealing at first glance since it can be arranged from within the insurer is “natural hedging,” that is, adjusting the insurance portfolio to minimize the overall exposure to systematic mortality risk (longevity risk). Cox and Lin (2007) first formally introduce this concept of mortality risk management for life insurers. They find that, empirically, companies selling both life and annuity policies generally charge cheaper prices for annuities than companies with only a single business line. Since then a number of studies have occurred in the insurance literature showing “that natural hedging can significantly lower the sensitivity of an insurance portfolio with respect to mortality risk” (Gatzert and Wesker 2012, p. 398; see also Bayraktar and Young 2007; Wetzel and Zwiesler 2008; Tsai et al. 2010; Wang et al. 2010). However, these contributions arrive at their positive appraisal of the natural hedging approach within model-based frameworks; that is, their conclusions rely on conventional mortality models such as the Lee-Carter model (Lee and Carter 1992) or the CairnsBlake-Dowd (CBD) model (Cairns et al. 2006b). While these popular models allow for a high degree of numerical tractability and serve well for many purposes, they come with the assumption that all mortality rates are driven by the same low-dimensional stochastic factors. Therefore, these models cannot fully capture disparate shifts in mortality rates at different ages, which could have a substantial impact on the actual effectiveness of natural hedging. To analyze the impact of the mortality forecasting model on the effectiveness of natural hedging, in this article, we compare results under several assumptions for the future evolution of mortality in the context of a stylized life insurer. In particular, aside from considering deterministic mortality rates and a simple-factor model as in previous studies, we also use a nonparametric forecasting model that arises as a by-product of the mortality modeling approach presented in Zhu and Bauer (2013). The advantage of a nonparametric model is that we do not make functional assumptions on the mortality model, especially the potentially critical factor structure indicated above.1 Our results reveal that the efficiency of natural hedging is considerably reduced when relying on the nonparametric model—which underscores the problem when relying on model-based analyses for risk management decisions more generally. We perform various robustness tests for this finding. In particular, we consider a setting without financial risk, and we repeat the calculations for alternative mortality models. Although these analyses reveal additional insights, the primary result is robust Address correspondence to Nan Zhu, Department of Mathematics, Illinois State University, 100 North University Street, Normal, IL 61790. E-mail: [email protected] 1Similar

arguments can be found in other insurance-related studies: e.g., Li and Ng (2010) use a nonparametric framework to price mortality-linked securities.

104

A CAUTIONARY NOTE ON NATURAL HEDGING OF LONGEVITY RISK

105

to these modifications: Natural hedging only marginally reduces the exposure of the company to systematic mortality risk. This meager performance can be viewed as further evidence endorsing market-based solutions for managing longevity risk. The remainder of the article is structured as follows: Section 2 briefly introduces the considered mortality forecasting models. Section 3 discusses the calculation of the economic capital for a stylized life insurance company, and Section 4 revisits the natural hedging approach within our economic capital framework. Section 5 conducts the robustness tests. Section 6 concludes. 2. MORTALITY FORECASTING MODELS We commence by introducing the mortality forecasting models that will be primarily used in this article. In particular, we consider two representative models within the forward-mortality framework developed in Zhu and Bauer (2013): a parametric single-factor model and a nonparametric model for the annualized mortality innovations. Employing two models from the same framework facilitates the interpretation of similarities and differences within certain applications. Moreover, as is detailed in Zhu and Bauer (2011), the use of conventional spot mortality models (Cairns et al. 2006a) will typically require so-called nested simulations in the numerical realizations within our Economic Capital framework, which in turn will considerably increase the computational difficulty of the optimization procedures described below. Since we are primarily interested in how the assumption of a low-dimensional factor structure—rather than the choice of any specific mortality forecasting model—affects the performance of the natural hedging approach in model-based analyses, we believe that our model choice serves well as a representative example to draw more general conclusions. Nevertheless, in Section 5 we conduct robustness tests of our results based on alternative factor and nonparametric models that are used in existing literature. Underlying the approach is a time series of generation life tables for some population for years t1 , t2 , . . . , tN . More precisely, in each of these tables labeled by its year tj , we are given forward-looking survival probabilities τ px (tj ) for ages x = 0, 1, 2, . . . , 100 and terms τ = 0, 1, 2, . . . , 101 − x, where τ px (t) denotes the probability for an x-year old to survive for τ periods until time t + τ .2 Mathematically this is equivalent to τ px (t) 1{ϒx−t >t}

   = EP 1{ϒx−t >t+τ }  Ft ∨ {ϒx−t > t}

for an (x − t) > 0 year old at time 0, where ϒx0 denotes the (random) time of death or future lifetime of an x0 -year-old at time 0. In particular, τ px (t) will account for projected mortality improvements over the future period [t, t + τ ). Now, for each year tj , 1 ≤ j < N, and for each term/age combination (τ, x) with 1 ≤ x ≤ 100 and 0 ≤ τ ≤ 100 − x, we define    τ +1+tj +1 −tj px−tj +1 +tj (tj ) τ +1 px (tj +1 ) , 1 ≤ j < N. (1) F (tj , tj +1 , (τ, x)) = − log τ px (tj +1 ) τ +tj +1 −tj px−tj +1 +tj (tj ) Hence, F (tj , tj +1 , (τ, x)) measures the log-change of the one-year marginal survival probability for an individual aged x at time tj +1 over the period [tj +1 + τ, tj +1 + τ + 1) from projection at time tj +1 relative to time tj . Further, we define the = 5,050, vector F¯ (tj , tj +1 ) = vec(F (tj , tj +1 , (τ, x)), 1 ≤ x ≤ 100, 0 ≤ τ ≤ 100 − x), with dim(F¯ (tj , tj +1 )) = 100×101 2 j = 1, 2, . . . , N − 1. Proposition 2.1 in Zhu and Bauer (2013) shows that under the assumption that the mortality age/term structure is driven by a time-homogeneous diffusion and with equidistant evaluation dates, that is, tj +1 − tj ≡ , F¯ (tj , tj +1 ), j = 1, . . . , N − 1, are independent and identically distributed (iid). Therefore, in this case a nonparametric mortality forecasting methodology is immediately given by bootstrapping the observations F¯ (tj , tj +1 ), j = 1, . . . , N − 1 (Efron, 1979). More precisely, with equation (1), we can generate simulations for the generation life tables at time tN+1 , {τ px (tN+1 )}, by sampling (with replacement) F¯ (tN , tN+1 ) from {F¯ (tj , tj +1 ), j = 1, . . . , N − 1} in combination with the known generation life tables at time tN , {τ px (tN )}. This serves as the algorithm for generating our nonparametric mortality forecasts. A related approach that we consider in the robustness tests (Section 5) relies on the additional assumption that F¯ (tj , tj +1 ) are iid Gaussian random vectors. In this case we can directly sample from a Normal distribution with the mean and the covariance matrix estimated from the sample. To introduce corresponding factor models, it is possible to simply perform a factor analysis of the iid sample {F¯ (tj , tj +1 ), j = 1, . . . , N − 1}, which shows that for population mortality data, the first factor typically captures the vast part of the systematic variation in mortality forecasts. However, as is detailed in Zhu and Bauer (2013), factor models developed this way are not necessarily self-consistent; that is, expected values derived from simulations of future survival probabilities do not necessarily align with the forecasts engrained in the current generation life table at time tN . 2In

particular, in this article, we use a maximal age of 101 though generalizations are possible.

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To obtain self-consistent models, it is convenient to introduce the so-called forward force of mortality (Cairns et al. 2006a), μt (τ, x) = −

∂ log {τ px (t)} , ∂τ

so that we have   p (t) = exp − τ x

τ 0

 μt (s, x) ds .

(2)

Time-homogeneous (forward) mortality models can then be represented by an infinite-dimensional stochastic differential equation of the form dμt = (A μt + α) dt + σ dWt , μ0 (·, ·) > 0,

(3)

∂ , and (Wt ) is a d-dimensional Brownian where α and σ are sufficiently regular, function-valued stochastic processes, A = ∂τ∂ − ∂x motion. Bauer et al. (2012a) show that for self-consistent models, we have the drift condition

α(τ, x) = σ (τ, x) ×



τ

σ ′ (s, x) ds,

(4)

0

and for time-homogeneous, Gaussian models (where α and σ are deterministic) to allow for a factor structure, a necessary and sufficient condition is σ (τ, x) = C(x + τ ) × exp{Mτ } × N, for some matrices M, N, and a vector-valued function C(·). By aligning this semiparametric form with the first factor derived in a factor analysis described above, Zhu and Bauer (2013) propose the following specification for the volatility structure in a single-factor model:

σ (τ, x) = k + c ed(x+τ ) (a + τ ) e−bτ .

(5)

Together with equations (2), (3), and (4), equation (5) presents the parametric factor mortality forecasting model employed in what follows. We refer to Zhu and Bauer (2013) for further details, particularly on how to obtain maximum-likelihood estimates for the parameters k, c, d, a, and b. 3. ECONOMIC CAPITAL FOR A STYLIZED INSURER In this section we employ the mortality forecasting approaches outlined in the previous section to calculate the Economic Capital (EC) of a stylized life insurance company. We start by introducing the framework for the EC calculations akin to Zhu and Bauer (2011). Subsequently we describe the data used in the estimation of the underlying models and resulting parameters. In addition to calculating the EC for a base case company with fixed investments, we derive an optimal static hedge for the financial risk by adjusting the asset weights. 3.1. EC Framework Consider a (stylized) newly founded life insurance company selling traditional life insurance products only to a fixed population. More specifically, assume that the insurer’s portfolio of policies consists of nterm x,i i-year term-life policies with face value Bterm for i-year endowment policies with face value B for x-year old individuals, and nann x-year old individuals, nend end x single-premium x,i life annuities with an annual benefit of Bann paid in arrears for x-year old individuals, x ∈ X , i ∈ I. Furthermore, assume that the benefits/premiums are calculated by the Equivalence Principle based on the concurrent generation table and the concurrent yield curve without the considerations of expenses or profits. In particular, we assume that the insurer is risk-neutral with respect to mortality risk; that is, the valuation measure Q for insurance liabilities is the product measure of the risk-neutral measure for financial and the physical measure for demographic events. Under these assumptions, the insurer’s Available Capital at time zero, AC0 , defined as the difference of the market value of assets and liabilities, simply amounts to its initial equity capital E. The available capital at time 1, AC1 , on the other hand, equals

A CAUTIONARY NOTE ON NATURAL HEDGING OF LONGEVITY RISK

107

to the difference in the value of the insurer’s assets and liabilities at time 1, denoted by A1 and V1 , respectively. More specifically, we have ⎛ ⎞

A 1 (0)

Ax:i (0) x:n ax (0) nann nterm + Bend nend ⎠ × R1 , A1 = ⎝E + Bann x + Bterm ¨ x:i (0) x,i ¨ x:i (0) x,i a a x∈X x∈X ,i∈I x∈X ,i∈I

ann

ann term a¨ x+1 (1) nx − Dx (0, 1) + Bterm V1 = Bann Dx,i (0, 1) + Bend Dend x,i (0, 1) x∈X

+Bterm

x∈X ,i∈I

x∈X ,i∈I

+Bend



A



x∈X ,i∈I

A 1 (0)

1

x+1:i−1

(1) −

Ax+1:i−1 (1) −

x:i

a¨ x:i (0)

x∈X ,i∈I





term a¨ x+1:i−1 (1) × nterm x,i − Dx,i (0, 1)



Ax:i (0) end a¨ x+1:i−1 (1) × nend x,i − Dx,i (0, 1) . a¨ x:i (0)

Here R1 is the total return on the insurer’s asset portfolio. Dcon x,i (0, 1) is the number of deaths between time 0 and time 1 in the cohort of x-year-old policyholders with policies of term i and of type con ∈ {ann, term, end}. And a¨ x (t), Ax:i (t), etc. denote the present values of the contracts corresponding to the actuarial symbols at time t, which are calculated based on the yield curve and the generation table at time t. For instance, a¨ x (t) =

∞

τ px (t) p(t, τ ),

τ =0

where τ px (t) is the time-t (forward) survival probability as defined in Section 2, and p(t, τ ) denotes the time t price of a zero-coupon bond that matures in τ periods (at time t + τ ). The EC calculated within a one-year mark-to-market approach of the insurer can then be derived as (Bauer et al. 2012b) EC = ρ (AC0 − p(0, 1) AC1 ),    L

where L denotes the one-period loss and ρ(·) is a monetary risk measure. For example, if the EC is defined based on Value-at-Risk (VaR) such as the Solvency Capital Requirement (SCR) within the Solvency II framework, we have EC = SCR = VaRα (L) = arg min{P(L > x) ≤ 1 − α}, x

(6)

where α is a given threshold (99.5% in Solvency II). If the EC is defined based on the Conditional Tail Expectation (CTE), on the other hand, we obtain EC = CTEα = E [L|L ≥ VaRα (L)] .

(7)

In this article, we define the economic capital based on VaR (equation [6]), and choose α = 95%. 3.2. Data and Implementation For estimating the mortality models in this article, we rely on female U.S. population mortality data for the years 1933–2007 as available from the Human Mortality Database.3 More precisely, we use ages ranging between 0 and 100 years to compile 46 consecutive generation life tables (years t1 = 1963, t2 = 1964, . . . , t46 = 2008) based on Lee-Carter mean projections, with each generated independently from the mortality experience of the previous 30 years.4 That is, the first table (year t1 = 1963) uses mortality data from years 1933–1962, the second table (t2 ) uses years 1934–1963, and so forth. 3Human Mortality Database. University of California, Berkeley (U.S.), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de. 4More precisely, for the estimation of the Lee-Carter parameters, instead of the original approach we use the modified weighted-least-squares algorithm (Wilmoth 1993) and further adjust κt by fitting a Poisson regression model to the annual number of deaths at each age (Booth et al. 2002).

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TABLE 1 Estimated Parameters of Factor Mortality Forecasting Model (5) k 2.3413 × 10−6

c

d

a

b

3.3722 × 10−8

0.1041

3.1210

0.0169

TABLE 2 Estimated Parameters of Capital Market Model μ 0.0866

σA

ρ

κ

γ

σr

λ

r0 (06/2008)

0.1372

−0.0078

0.0913

0.0123

0.0088

−0.7910

0.0188

Having obtained these generation tables {τ px (tj )}, j = 1, . . . , N = 46, we derive the time series of F¯ (tj , tj +1 ), j = 1, 2, . . . , 45, which serve as the underlying sample for our nonparametric forecasting methodology and as the basis for the maximum likelihood parameter estimates of our mortality factor model.5 In particular, time tN = 2008 corresponds to time 0, whereas time tN+1 = 2009 corresponds to time 1 in our EC framework. Table 1 displays the parameter estimates of the parametric factor model (5).6 For the asset side, we assume that the insurer only invests in 5-, 10-, and 20-year U.S. government bonds as well as an equity index (S&P 500) S = (St )t≥0 . For the evolution of the assets, we assume a generalized Black-Scholes model with stochastic interest rates (Vasicek model), that is, under P, 

, S0 > 0, 1 − ρ 2 σA dB(2) dSt = St μ dt + ρ σA dB(1) t + t drt = κ (γ − rt ) dt +

σr dB(1) t ,

(8)

r0 > 0,

where μ, σA , κ, γ , σr > 0, and ρ ∈ [−1, 1], and (Bt(1) ) and (Bt(2) ) are independent Brownian motions that are independent of (Wt ). Moreover, we assume that the market price of interest rate risk is constant and denote it by λ; that is, we replace μ by rt and γ by γ − (λσr )/κ for the dynamics under the risk-neutral measure Q. We estimate the parameters based on U.S. data from June 1988 to June 2008 using a Kalman filter. In particular, we use monthly data of the S&P 500 index,7 Treasury bills (three months), and government bonds with maturities of one, three, five, and ten years.8 The parameter estimates are displayed in Table 2. Based on time 1 realizations of the asset process, S1 , and the instantaneous risk-free rate, r1 , we have R1 = ω1

S1 p(1, 4) p(1, 9) p(1, 19) + ω2 + ω3 + ω4 , S0 p(0, 5) p(0, 10) p(0, 20)

where ωi , i = 1, . . . , 4, are the company’s proportions of assets invested in each category. A procedure to generate realizations of S1 , r1 , and p(t, τ ) with the use of Monte Carlo simulations is outlined in Zaglauer and Bauer (2008). 3.3. Results Table 3 displays the portfolio of policies for our stylized insurer. For simplicity and without loss of generality, we assume that the company holds an equal number of term/endowment/annuity contracts for different age/term combinations and that the face values coincide; of course, generalizations are possible. The initial capital level is set to E = $20,000,000. The insurer’s assets and liabilities at time 0, A0 and V0 , are calculated at $1,124,603,545 and $1,104,603,545, respectively. 5Of course, the underlying sample of 45 realizations is rather small for generating a large bootstrap sample, which limits the scope of the approach for certain applications (such as estimating VaR for high confidence levels, which is of practical interest). We come back to this point in our robustness tests (Section 5). 6A Principal Component Analysis indicates that 85% of the total variation in the F ¯ (tj , tj +1 ), j = 1, 2, . . . , 45, is explained by the leading factor for our dataset. Generally, the percentage of total variation explained is slightly larger for female data in comparison to male data (Zhu and Bauer 2013), suggesting that for female populations a single-factor model is more appropriate. 7Yahoo! Finance, http://finance.yahoo.com. 8Federal Reserve Economic Data (FRED), http://research.stlouisfed.org/fred2/.

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TABLE 3 Portfolio of Policies for Stylized Life Insurer term/end/ann

x

i

30 35 40 45

20 15 10 5

2, 500 2, 500 2, 500 2, 500

$100,000 $100,000 $100,000 $100,000

40 45 50

20 15 10

5, 000 5, 000 5, 000

$50,000 $50,000 $50,000

60 70

(40) (30)

2, 500 2, 500

$18,000 $18,000

nx,i

Bterm/end/ann

Term life

Endowment

Annuities

We consider three different approaches to modeling mortality risk: (1) a deterministic evolution of mortality given by the life table at time 0 (2008), {τ px (0)}; (2) the parametric factor model (5); and, (3), the nonparametric mortality model also introduced in Section 2. Within each approach, we use 50,000 simulations of the assets and liabilities to generate realizations of the loss (L) at time 1, where in addition to financial and systematic mortality risk, we also consider unsystematic mortality risk by sampling the number of deaths within each cohort. Finally, we can calculate the EC via the resulting empirical distribution functions and the given risk measure ρ. In particular, for VaR we rely on the empirical quantile. Table 4 displays the results for two assumptions regarding the insurer’s investments. For the results in the first row of Table 4, we assume that the company does not optimize its asset allocation but invests a fixed 30% in the equity index (see, e.g., ACLI 2011), and the rest in government bonds to match the duration of its liabilities (at 10.2560). Without stochastic mortality, we find EC levels of around $60,000,000, which suggests that the current capital position of $20,000,000 is not sufficient; that is, the firm is undercapitalized. Surprisingly, including systematic mortality risk appears to have little influence on the results in this case: The EC increases by only $787,832 (1.30%) or $2,004,332 (3.30%) when introducing mortality risk via the factor mortality model or nonparametric mortality model, respectively. However, this changes dramatically when we allow the insurer to pursue a more refined allocation strategy to better manage the financial risk exposure. In the second row of Table 4, we display the results when the insurer optimally chooses (static) asset weights in order to minimize the EC. The corresponding portfolio weights are displayed in Table 5. We find that although the EC level decreases vastly under all three mortality assumptions so that the company is solvent according to the 95% VaR capital requirement (EC ≤ AC0 ), the relative impact of systematic mortality risk now is highly significant. More precisely, the (minimized) EC increases by 208.31% (to $9,871,987) and 213.86% (to $10,049,401) if systematic mortality risk is considered via the factor model and nonparametric model, respectively. This underscores an important point in the debate about the economic relevance of mortality and longevity risk: Although financial risk indices may be more volatile and thus may dominate systematic mortality risk, there exist conventional methods and (financial) instruments to hedge against financial risk. Of course, naturally the question arises if we can use a similar approach to protect against systematic mortality risk, either by expanding the scope of securities considered on the asset side toward mortality-linked securities or by adjusting the composition of the insurance portfolio on the liability side. The former approach has been considered in a number of papers (see, e.g., Cairns et al. 2013, Li and Luo 2012; and references therein), but a liquid market of corresponding instruments is only slowly emerging. The latter approach—which is commonly referred to as natural hedging (Cox and Lin 2007) and which is in the focus of this TABLE 4 Economic Capital for Different Investment Strategies

95% VaR (no hedging) 95% VaR (financial hedging)

Deterministic Mortality

Factor Model

Nonparametric Model

$60,797,835 $3,201,921

$61,585,667 $9,871,987

$62,802,167 $10,049,401

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TABLE 5 Financial Hedging: Optimal Weights

Stock 5-year bond 10-year bond 20-year bond

Deterministic Mortality

Factor Model

Nonparametric Model

0.2% 2.5% 87.3% 10.0%

1.5% 0.1% 88.0% 10.4%

0.9% 0.5% 90.8% 7.8%

article—has also received attention in the insurance literature and is reported to perform well (Wetzel and Zwiesler 2008; Tsai et al. 2010; Wang et al. 2010; Gatzert and Wesker 2012). Before we explore this approach in more detail in the next section, it is helpful to emphasize that the results for the two mortality models—the nonparametric model and the parametric factor model—are very similar across both cases. This may not be surprising because these models originate from the same framework. Essentially, one can interpret the factor model as a parsimonious approximation of the nonparametric model that nevertheless captures the majority of the “important” variation—with resulting statistical advantages, for example, in view of its estimation. However, there are also pitfalls for its application in the context of analyzing the performance of hedges as we will see in the next section. 4. NATURAL HEDGING OF LONGEVITY RISK Akin to the previous section, we consider the possibility of reducing the risk exposure by adjusting the portfolio weights. However, although there we adjusted asset weights in order to minimize the exposure to financial risk, here we focus on adjusting the composition of the liability portfolio in order to protect against mortality/longevity risk. More specifically, we fix the number of endowment and annuity contracts in the life insurer’s portfolio to the same values as in the previous section (see Table 3) and vary the number of term-life policies nterm , where, for simplicity, we assume nterm ≡ nterm x,i is constant across age/term combinations (x, i). For each nterm , the EC is then calculated analogously to the previous section under the assumption that the insurer hedges against financial risk; that is, we determine the “optimal” asset allocation separately for each nterm . Finally, we determine the ∗ optimal number of term-life policies, nterm that minimizes the EC for the life insurer.9 We start by considering the factor mortality model and compare it to the case without stochastic mortality risk. Figure 1 shows the EC as a function of the number of term-life policies in the insurer’s portfolio nterm (we will refer to this as an “EC curve”). We first find that in the case of no systematic mortality risk (deterministic mortality), EC increases in the number of term policies. The reason is twofold: On the one hand, an increase leads to higher premiums and, thus, assets, which increases asset risk. On the other hand, although there is no systematic mortality risk, the number of deaths in each cohort is a random variable due to nonsystematic mortality risk, which clearly increases in the number of policies. In contrast, under the stochastic factor mortality model, the EC is a convex function of nterm , which initially decreases and then increases sharply; that is, it is U-shaped. The optimal number ∗ of policies, nterm , is approximately 60,000 and the corresponding minimal EC is $4,165,973, which is only slightly larger than the corresponding EC level under deterministic mortality ($4,128,345). Therefore, in line with other papers on natural hedging, it appears that an appropriately composed insurance portfolio can serve well for hedging against systematic mortality/longevity risk. However, when repeating the same exercise based on the nonparametric forecasting model, the situation changes considerably. As is also depicted in Figure 1, in this case we can observe an only very mild effect of natural hedging when nterm is relatively small, ∗ and it is far less pronounced compared with the factor model. In particular, at the optimal term-insurance exposure nterm = 60,000, under the factor model, the capital level is at $13,872,739 for the nonparametric model, which is far greater than the corresponding capital level under deterministic mortality ($4,128,345). The intuition for this result is as follows: As indicated at the end of Section 3.3, the two models behave quantitatively alike in “normal” circumstances and particularly yield similar capital levels for the initial portfolio. This is not surprising since the rationale behind the single factor model—akin to other single-factor models such as the Lee-Carter model—is that the majority of the variation across ages and terms can be explained by the leading factor (85% for our dataset). Essentially, the residuals for 9Note that we implicitly assume that the insurer can place arbitrarily many term-life insurance policies in the market place at the same price, which may be ∗ unrealistic for large nterm . Moreover, we assume that underwriting profits and losses can be transferred between different lines of business and that there are no other technical limitations when pursuing natural hedging. However, such limitations would only cast further doubt on the natural hedging approach, so we refrain from a detailed discussion of these aspects.

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2.5

x 10

deterministic mortality factor mortality model non-parametric mortality model

Optimal economic capital

2

1.5

1

0.5

0

1

2

3

4

5

6

7

8

Exposure of term-life

9

10 4

x 10

FIGURE 1. Optimal Longevity Hedging. (color figure available online)

lower ages are small in absolute terms and mostly unsystematic, whereas the residuals for higher ages (beyond 50) are relatively large in absolute terms and mostly systematic. The latter are responsible for the high proportions of the variation explained in absolute terms. However, under the natural hedging approach, the large exposure in the term-life lines leads to a considerable rescaling of the profile of the residuals across terms and ages, so that this similarity breaks down. In particular, the residuals for lower-age groups become increasingly important, which in turn are considerably influenced by higher-order factors including but also beyond the second factor—some of which do not carry a systematic shape at all. Thus, for the analysis of the effectiveness of natural hedging, the consideration of higher order/nonsystematic variation indeed might be important. Again, we would like to emphasize that this is not general criticism of these models. For many applications, such as forecasting mortality rates, abstracting from these small and unsystematic variations is expedient. We solely challenge the reliance on lowdimensional factor models for the analysis of hedging performance. Indeed, our results indicate that natural hedging may not be as effective as asserted in the existing literature.

5. ROBUSTNESS OF THE RESULTS Of course, the question may arise to what extent the results on the performance of natural hedging are driven by the details of our setup. Thus, in this section we repeat the EC calculations under modified assumptions. In particular, we examine the impact of financial risk on the results, we consider modifications of our mortality models, and we derive EC curves for alternative mortality models. 5.1. The Impact of Financial Risk To analyze the role played by financial risk in the results, we recalculate the EC levels for different term-life exposures under a deterministic evolution of the asset side—so that the results are solely driven by systematic and unsystematic mortality risk. More precisely, in our asset model (equation [8]), we set both volatility terms σA and σr to zero, and we use the risk-neutral drift parameters throughout. In particular, the equity index S is now risk-less and returns the risk-free rate. Figure 2(a) displays the corresponding optimal EC curves.

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2.5

7

x 10

2.5

x 10

deterministic mortality factor mortality model non-parametric mortality model

deterministic mortality factor mortality model non-parametric mortality model 2

Optimal economic capital

Optimal economic capital

2

1.5

1

0.5

0

1.5

1

0.5

0

1

2

3

4

5

6

7

8

Exposure of term-life

(a) Optimal EC without financial risk

9

10

0

0

1

2

3

4

4

5

6

7

8

Exposure of term-life

x 10

9

10 4

x 10

(b) Optimal EC with financial risk (Q-measure)

FIGURE 2. The Impact of Financial Risk. (color figure available online)

We find that the EC levels under each of the mortality models are similar to the case with financial risk (Fig. 1). For large values of nterm , the EC here even exceeds the corresponding values in the case with financial risk, which appears surprising at first glance. The reason for this observation is the change of probability measure in the financial setting and the associated risk premia paid over the first year. For comparison, we also plot the EC curves for all mortality assumptions when relying on the Q-measure throughout in Figure 2(b). We find that the (hypothetical) EC, as expected, is always greater than without the consideration of financial risk, though the difference is not very pronounced. This indicates that the static hedging procedure eliminates most of the financial risk or, in other words, that financial risk does not contribute too much to the total EC. 5.2. Modifications of the Mortality Models As indicated in note 5, the relatively small size of the sample underlying our nonparametric forecasting approach may be problematic for certain applications such as estimating VaR for a high confidence level. To analyze the impact of the small sample size on our results, we follow the approach also described in Section 2 that relies on the additional assumption that F¯ (tN , tN+1 ) is Gaussian distributed. Then, rather than sampling F¯ (tN , tN+1 ) from the empirical realizations, we generate random vectors with the mean vector and the covariance matrix estimated from the underlying sample. Figure 3(a) shows the resulting EC curve in comparison to the deterministic mortality case. We find that the results are very similar to the nonparametric model underlying Figure 1. In particular, there is only a rather mild effect of natural hedging when nterm is small, and the economic capital levels considerably exceed those for the deterministic mortality case. As also was indicated in Section 2, the competing model used in the calculations in Sections 3 and 4, while originating from a factor analysis, presents a self-consistent, parametric approximation. In particular, the entire term structure is driven by only a handful parameters in this case, so that it is not immediately clear what aspects of the model are responsible for the results. Thus, as an intermediary step, we also provide results based on a (high-dimensional) single-factor model. In particular, instead of directly relying on the leading principal component, we estimate a one-factor model following the approach from Bai and Li (2012), which allows for heteroscedasticity in the error term so that some variation will also be picked up for lower ages. More precisely, this approach posits a factor form F¯ (tj , tj +1 ) = α + β ∗ λtj + δtj , j = 1, . . . , N − 1, 2 where α and β are 5050 × 1 vectors, E[δt ] = 0 and E[δt × δt′ ] = error = diag{σ12 , . . . , σ5050 }, which is estimated via maximum likelihood. Here, we employ the leading factor from the Principal Component Analysis as the starting value in the numerical optimization of the log-likelihood, and the resulting factors overall are very similar.

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2.5

7

x 10

2.5

x 10

deterministic mortality stochastic mortality

deterministic mortality stochastic mortality 2

Optimal economic capital

Optimal economic capital

2

1.5

1

0.5

1.5

1

0.5

0

1

2

3

4

5

6

7

Exposure of term-life

(a) Gaussian nonparametric model

8

9

10 4

x 10

0

1

2

3

4

5

6

7

Exposure of term-life

8

9

10 4

x 10

(b) One-factor nonparametric model

FIGURE 3. Modifications of the Mortality Models. (color figure available online)

Figure 3(b) provides the EC curve based on this factor model. We find that the effect of natural hedging is far less pronounced than for the parametric factor model, and the Economic Capitals are considerably higher than for the deterministic mortality case throughout. However, we also observe that in contrast to the nonparametric approach in Figure 1, the EC curve is “flat” in the sense that the increased exposure to term-life insurance has only a little effect on the economic capital level. This indicates that the factor loadings for the lower age range are very small, or, in other words, that much of the variation is driven by higher order variations. For the parametric factor model, on the other hand, the parametric form is fit across all terms and ages, which appears to yield a more significant relationship between low and high ages in the first factor. Thus, the parametric nature of the model also seems to be an important driver for the positive appraisal of the natural hedging approach, at least in our setting. 5.3. Alternative Mortality Models As a final robustness check, we repeat the calculations for alternative mortality models that do not fall within our framework. We start by providing results for the (stochastic) Lee-Carter model as another model where all variation is driven by a single factor. More precisely, we use the Lee-Carter parameters estimated at time tN —which also serve for generating the corresponding generation life table that we use for the calculation of time 0 premiums—and simulations of the κtN+1 to generate life tables at time 1 with each based on the median projection starting from a simulated value of κtN+1 . Figure 4(a) presents the resulting EC curve. We make two primary observations. On the one hand, the EC curve exhibits a U-shape similar to the parametric factor model in Figure 1, that is, natural hedging again is found to be highly effective under this model. In particular, the optimal exposure to term life policies nterm∗ again is around 60,000 with a corresponding minimum capital of $4,150,010. This finding is not surprising since it was exactly this positive appraisal of the natural hedging approach in previous contributions that serves as the primary motivation of this article. On the other hand, we observe that the magnitude of EC is considerably lower than in the models considered in Sections 3 and 4. Again, this finding is not surprising since it is exactly the underestimation of the risk in long-term mortality trends that serves as the motivation for the underlying approach in Zhu and Bauer (2013). As a second nonparametric modeling approach, we implement the model proposed by Li and Ng (2010), which relies on mx,t bootstrapping one-year mortality reduction factors rx,tj = mx,tj +1 for j = 1, . . . , N − 1 and mx,t being the central death rate for j age x in year t. More precisely, as proposed in Li and Ng (2010), we use a block bootstrap method with a block size of two to capture the serial dependency in the data. For consistency with the other mortality models in this article, we use 30 years of the historical data (1978–2007) and ages x ranging from 0 to 100. Figure 4(b) shows the resulting EC curves, where, of course, the deterministic curve is calculated based on a generation table compiled also using this model. The capital levels are lower than for the approaches considered in Sections 3 and 4 though larger than for the (stochastic) Lee-Carter model, which is due to differences in the model structures. However, the observations regarding natural hedging are in

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16

6

x 10

16

x 10

14

12

12

Optimal economic capital

Optimal economic capital

deterministic mortality stochastic model 14

10

8

10

8

6

6

4

4

2

0

1

2

3

4

5

6

Exposure of term-life

(a) Lee-Carter model

7

8

9

10 4

x 10

deterministic mortality stochastic mortality

2

0

1

2

3

4

5

6

7

8

Exposure of term-life

9

10 4

x 10

(b) One-factor nonparametric model

FIGURE 4. Alternative Mortality Models. (color figure available online)

line with our results from Section 4. More precisely, the effect is not very pronounced under the nonparametric model—we see only a very mild U-shape—and the EC level increases considerably for higher values of nterm . 6. CONCLUSION In this article, we analyze the effectiveness of natural hedging in the context of a stylized life insurer. Our primary finding is that higher order variations in mortality rates may considerably affect the performance of natural hedging. More precisely, although results based on a parametric single-factor model imply that almost all longevity risk can be hedged by appropriately adjusting the insurance portfolio (in line with the existing literature), the results are far less encouraging when including higher order variations via a nonparametric mortality forecasting model. Of course, this is not a general endorsement of these more complicated models. Simple (or parsimonious) models may have many benefits in view of their tractability, their statistical properties, or their forecasting power. We solely show that relying on “simple” models for analyzing the performance of hedges may be misleading since they contain assumptions on the dependence across ages that are not necessarily supported by the data. At a broader level, we believe our results call for more caution toward model-based results in the actuarial literature in general. ACKNOWLEDGMENTS A previous version entitled “A Natural Examination of the Natural Hedging Approach” was presented at the Eighth International Longevity Risk and Capital Markets Solutions Conference. We are grateful for valuable comments from the conference participants. Moreover, we thank two anonymous referees whose suggestions greatly improved the article. All remaining errors are our own. REFERENCES ACLI (American Council of Life Insurers). 2011. Life Insurers Fact Book 2011. https://www.acli.com/Tools/Industry Facts/Life Insurers Fact Book/Documents/ 2011 Fact Book.pdf Bai, J., and K. Li, 2012. Statistical Analysis of Factor Models of High Dimension. Annals of Statistics 40: 436–465. Bauer, D., F. E. Benth, and R. Kiesel, 2012a. Modeling the Forward Surface of Mortality. SIAM Journal on Financial Mathematics 3: 639–666. Bauer, D., A. Reuss, and D. Singer, 2012b. On the Calculation of the Solvency Capital Requirement Based on Nested Simulations. ASTIN Bulletin 42: 453–499. Bayraktar, E., and V. R. Young, 2007. Hedging Life Insurance with Pure Endowments. Insurance: Mathematics and Economics 40: 435–444. Blake, D., A. J. G. Cairns, and K. Dowd, 2006. Living with Mortality: Longevity Bonds and other Mortality-Linked Securities. British Actuarial Journal 12: 153–197. Booth, H., J. Maindonald, and L. Smith, 2002. Applying Lee-Carter under Conditions of Variable Mortality Decline. Population Studies 56: 325–336. Cairns, A. J. G., D. Blake, and K. Dowd, 2006a. Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk. ASTIN Bulletin 36: 79–120.

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Cairns, A.J.G., D. Blake, and K. Dowd, 2006b. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance 73: 687–718. Cairns, A. J. G., D. Blake, K. Dowd, and G. D. Coughlan, 2014. Longevity Hedge Effectiveness: A Decomposition. Quantitative Finance 14: 217–235. Cox, S. H., and Y. Lin, 2007. Natural Hedging of Life and Annuity Mortality Risks. North American Actuarial Journal 11: 1–15. Efron, B., 1979. Bootstrap Methods: Another Look at the Jackknife. Annals of Statistics 7: 1–26. Gatzert, N., and H. Wesker, 2012. The Impact of Natural Hedging on a Life Insurer’s Risk Situation. Journal of Risk Finance 13: 396–423. Lee, R. D., and L. R. Carter, 1992. Modeling and Forecasting U. S. Mortality. Journal of the American Statistical Association 87: 659–675. Li, J. S.-H., and A. C.-Y. Ng, 2010. Canonical Valuation of Mortality-Linked Securities. Journal of Risk and Insurance 78: 853–884. Li, J. S.-H., and A. Luo, 2012. Key q-Duration: A Framework for Hedging Longevity Risk. ASTIN Bulletin 42: 413–452. Tsai, J. T., J. L. Wang, and L. Y. Tzeng, 2010. On the Optimal Product Mix in Life Insurance Companies Using Conditional Value at Risk. Insurance: Mathematics and Economics, 46: 235–241. Wang, J. L., H. C. Huang, S. S. Yang, and J. T. Tsai, 2010. An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach. Journal of Risk and Insurance 77: 473–497. Wetzel, C., and H. J. Zwiesler, 2008. Das Vorhersagerisiko der Sterblichkeitsentwicklung: Kann es durch eine geeignete Portfoliozusammensetzung minimiert werden? Bl¨atter DGVFM 29: 73–107. Wilmoth, J., 1993. Computational Methods for Fitting and Extrapolating the Lee-Carter Model of Mortality Change. Technical, University of California, Berkeley. Zaglauer, K., and D. Bauer, 2008. Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment. Insurance: Mathematics and Economics 43: 29–40. Zhu, N., and D. Bauer, 2011. Applications of Gaussian Forward Mortality Factor Models in Life Insurance Practice. Geneva Papers on Risk and Insurance—Issues and Practice 36: 567–594. Zhu, N., and D. Bauer, 2013. Modeling and Forecasting Mortality Projections. Working paper, Georgia State University.

Discussions on this article can be submitted until October 1, 2014. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Quantitative Finance, 2013 Vol. 13, No. 8, 1211–1223, http://dx.doi.org/10.1080/14697688.2012.741693

A closed-form approximation for valuing European basket warrants under credit risk and interest rate risk YUNG-MING SHIU*†, PAI-LUNG CHOU‡ and JEN-WEN SHEUx †Department of Risk Management and Insurance, Risk and Insurance Research Center, National Chengchi University, 64, Sec. 2, Zhi-Nan Road, Wen-Shan District, Taipei, Taiwan ‡Department of Risk Management and Insurance, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan xDepartment of Finance, Fortune Institute of Technology, Kaohsiung, Taiwan (Received 20 January 2011; in final form 8 October 2012) Over the past few years, many financial institutions have actively traded basket warrants in the over-the-counter market. Prior research has proposed an approach to valuing single-stock options subject to credit. However, this approach cannot be applied directly to the case of basket warrants. Using the martingale method, we propose a closed-form approximation for valuing European basket warrants using a continuous-time model, with credit risk and interest rate risk considered simultaneously. Finally, several numerical examples are utilized to demonstrate the characteristics of basket warrants under credit risk. Keywords: Derivatives pricing; Derivatives securities; Stochastic interest rates; Credit risk JEL Classification: G1, G13

1. Introduction With the liberalization of global financial markets and the instability of the world economy, many derivatives have been developed to meet the increasing needs of investors. Among these derivative securities, basket warrants have gradually become more popular over the past decade. In essence, basket warrants are actually basket options. These options have a basket of two or more underlying assets whose prices determine basket warrants’ payoffs. However, basket warrants and ordinary options are different in terms of issuing institutions. Basket warrants are normally issued by financial institutions such as investment banks. Investors thus face the credit risk of issuers. As the global financial markets are rapidly changing, investors are also concerned with interest rate risk. The purpose of this paper is to value European basket warrants, with credit risk and interest rate risk considered simultaneously. Since securities companies which issue basket warrants may default on their obligations, investors should take into account the creditworthiness of issuing organizations when purchasing warrants. Johnson and Stulz (1987) was one of

the first studies to examine the pricing of these options with default risk, also known as vulnerable options. Hull and White (1995), Jarrow and Turnbull (1995), Klein (1996), and Hung and Liu (2005) also indicate the importance of taking into account counterparty risk when pricing options traded on the over-the-counter (OTC) market due to the unavailability of a clearing mechanism. When valuing vulnerable options, Klein (1996) does not consider the interest rate risks, while Johnson and Stulz (1987) assume a fixed interest rate. Both Gentle (1993) and Milevsky and Posner (1998) also assume fixed interest rates when valuing basket options.{ Based on the assumption of fixed interest rates, the value of warrants would be underestimated if interest rates are highly volatile before maturity. Hull and White (1995) and Jarrow and Turnbull (1995) assume independence between the assets of the option writer and the underlying asset of the option, especially when the option writer is a large and well diversified financial institution (Hull and White 1995). However, Klein (1996) argues that the option writer may still default due to the volatile changes in the value of the underlying asset even if the option writer has undertaken hedging. Johnson and Stulz

*Corresponding author. Email: [email protected] {It is noted that Gentle (1993) and Milevsky and Posner (1998) do not consider default risk when pricing basket options. Ó 2013 Taylor & Francis

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Y.-M. Shiu et al.

(1987) further suggest that the option writer may default, possibly due to decreases in the value of the assets of the writer and/or the growth of the value of the option. If the value of the option grows to a larger extent than that of the assets of the writer, it is likely that default may occur. Prior studies also discuss the correlations among the assets of the option writer, the assets underlying the options, and the interest rate. Merton (1974) was one of the first to study the link between the value of the assets of the option writer and the default event, considering the relation between the interest rate and the assets underlying the options. Extending Merton’s (1974) work, Longstaff and Schwartz (1995) simultaneously consider both default risk and interest rate risk. They further point out that the changes in the value of the assets of the firm and in the interest rates have a significant impact on credit spread when pricing risky bonds. However, the interest rate risk is generally not taken into account in the literature when valuing vulnerable options. In this paper, we argue the importance of considering both risks and simultaneously model these two risks when pricing basket warrants for the following reason. In the past few years, warrant writers have been exposed to bankruptcy risk due to the Financial Crisis of 2007–2010. They are also adversely affected by interest rate volatility. Furthermore, prices of financial assets are significantly interrelated. Unlike prior studies, we therefore simultaneously take into account credit risk, interest rate risk, and correlations between assets when valuing basket warrants. Our valuation formula provides flexibility in pricing warrants with underlying assets including bonds, stocks and other types of securities. It is actually the general closed-form solution of the models of Black and Scholes (1973), Smith (1976), Gentle (1993), Hull and White (1995), Klein (1996), and Klein and Inglis (1999). The remainder of the paper is organized as follows. The following section develops a theoretical model for valuing European basket warrants subject to interest rate risk and to financial distress on the part of the warrant writer. Next, we derive a closed-form approximation valuation formula for vulnerable basket call and put warrants. In the penultimate section, we utilize several numerical examples to show the properties of our pricing formula. Finally, the last section concludes this paper.

2. The model This article’s framework for valuing basket warrants can be considered an extension of the framework for pricing vulnerable options proposed by Klein and Inglis (1999). In their paper, Klein and Inglis (1999) value calls/puts on a single asset, while in this study we price basket warrants. The underlying of basket warrants is a basket of assets. Specifically, the basket of assets is actually a portfolio of n kinds of different tradable stocks. We assume that the warrants are traded under a continuous-time frame and that markets are perfect and frictionless, i.e. transaction costs and taxes are ignored. Suppose that the market value of each of these n stocks follows Geometric Brownian Motion (GBM). The dynamics of the market value of a particular stock are then stated as follows:

dSit ¼ lSi dt þ rSi dWSi ; Sit

i ¼ 1; . . . ; n;

ð1Þ

where lSi and rSi are the instantaneous expected return on stock i underlying the warrant and the instantaneous standard deviation of the return (both assumed to be constants), respectively, Sit is the price of the ith stock at time t, WSi is a standard Wiener process, i ¼ 1; . . . ; n, and \WSi ; WSj[t ¼ qij t, for every i; j ¼ 1; . . . ; n. Credit risk, also known as default risk, is defined as the risk that warrant writers, such as investment banks or securities firms, will be unable to make the required payments when their debt obligations fall due. The bankrupt event occurs when the writer’s value of assets at the expiration date (VT) are smaller than its value of debts (D). If the warrant writer is bankrupt, its payments to investors depend on the value of its assets on the warrant’s expiration date. If the value of the assets of the warrant writer VT falls below the fixed threshold value D⁄ on the warrant maturity date T, then default occurs. The value of D⁄ is allowed to be less than the value of D, the outstanding liabilities of the writer. As in Klein and Inglis (1999), D is simplified as the value of the zero-coupon bonds issued by the writer. We assume that D is the same as D⁄ in the process of valuation to allow for capital forbearance of the warrant writer. Once the writer goes bankrupt, the warrant holder can only claim ð1  aÞVT =D, where 0  a  1 is the costs associated with the financial distress when the writer becomes bankrupt. Suppose that the value of the assets of the warrant writer V follows a GBM. The dynamic process of V is as follows: dV ¼ lV dt þ rV dWV ; V

ð2Þ

where lV and rV are the instantaneous expected return on the assets of the writer and the instantaneous standard deviation of the return (both assumed to be constants), respectively, and WV is a standard Wiener process. As for the interest rate risk, let Pðt; T Þ represent the price of a zero-coupon bond at time t paying one dollar at time T, where T represents the expiration date of the warrants. Therefore, dPðt; T Þ ¼ lP dt þ rP ðT  tÞ dWP ; Pðt; T Þ

ð3Þ

where lP and rP denote the instantaneous expected return on the zero-coupon bond and the instantaneous standard deviation of the return, respectively, and WP follows a standard Wiener process. Let B(t) denote a money market account which corresponds to the future value of the wealth accumulated from an investment of $1 at an interest rate of r(t). Its dynamic process is dBðtÞ ¼ rðtÞBðtÞ dt, where r(t) is the instantaneous interest rate at time t. Under the risk-neutral probability measure Q, the dynamics of the zero coupon bond price are dPðt; T Þ ¼ rðtÞ dt þ rP ðT  tÞ dWPQ : Pðt; T Þ

ð4Þ

1213

Valuing European basket warrants As shown in equations (3) and (4), the instantaneous expected return on the zero-coupon bond under the probability measure P is replaced by the instantaneous interest rate r(t) under the risk-neutral measure Q. In addition, the market value of the underlying stock, the value of the assets of the warrant writer, and the price of the zero-coupon bond are all assumed to be correlated with each other under the probability measure P. The instantaneous correlations between WV and WP , between WS and WP , and between WV and WS are qVP , qSP , and qVS , respectively. Our framework described in this section can be applied to the models proposed by Vasicek (1977), Hull and White (1990), and Heath et al. (1992). It is also worth noting that the valuation formula for the vulnerable European basket warrants is the general closed-form solution of the models of Black and Scholes (1973), Smith (1976), Gentle (1993), Hull and White (1995), Klein (1996), and Klein and Inglis (1999). In the ‘Numerical examples’ section below, we use the Cox, Ingersoll and Ross (1985) model (CIR model), which is a special case of the Hull and White (1990) model. 3. Valuation of vulnerable basket warrants The warrant writer is considered to be bankrupt if the value of its asset VT falls below the value D on the maturity date. If the writer defaults on its obligations at maturity, the claims of the warrant purchasers would not be completely satisfied. Thus, we use the maturity date T as a reference point in time. Let CTB be the payoff of a European basket call warrant at maturity date T. CTB is defined as CTB ¼ maxfBKT  K; 0g; P where BKT ¼ ni¼1 wi SiT , i ¼ 1; . . . ; n: BKT represents the weighted average value of n kinds of different stocks at time T, K is the strike price of the basketPcall warrant, wi represents the weight of the ith stock and ni¼1 wi ¼ 1. Let FiS ðt; T Þ ¼ Sit =Pðt; T Þ; and FiS ðt; T Þ is the forward price at time t of the ith asset for the settlement date T. BKT ¼

n X i¼1

n  X wi SiT ¼ wi FiS ðt; T Þ i¼1

i

i¼1

where the modified weight is

K ; S i¼1 wi Fi ðt; T Þ

i¼1

S^iT ¼

K  ¼ Pn

Xi ¼ 1;

SiT : S Fi ðt; T Þ

If the money market account B(t) is used as the numeraire, the discounted price of the asset is Q -martingale. If P(t,T), the price of a zero-coupon bond with a maturity date T at time t, is used as the numeraire, the discounted price of the asset will be QT -martingale. Since PðT ; T Þ ¼ BðT Þ ¼ 1, CTB , the payoff function of a vulnerable European basket call warrant with a maturity date T, can be given as follows: 8 < ðBKT  KÞ; T CTB ¼ ðBKT  KÞð1aÞV ; D : 0;

if BKT  K[ 0 and VT  D; if BKT  K[ 0 and VT\ D; if BKT  K  0;

where BKT represents the weighted average value of n kinds of different stocks at time T, K is the strike price of the basket call warrant, and VT and D are the warrant writer’s assets and liabilities, respectively. When VT 0, define τ (b) = inf {t ≥ 0 : Rt = b}. From Kendall (1957), it is known that the distribution of the first passage time τ (b) satisfies the following identity:

∗ Corresponding

author. E-mail: [email protected] © 2012 Taylor & Francis

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∞ x

Pr (τ (b) ≤ t)

db = b



t 0

Pr (Rs > x)

ds , s

(1)

for t, x > 0 (see also Borovkov and Burq (2001) and references therein for further discussion on the distribution of this first passage time). In a ruin theoretical context, Eq. (1) leads to the determination of the distribution of various first hitting times of interest. Among them, we mention the distribution of the time to ruin in the dual risk model with exponential interinnovation times and with or without a diffusion component (see, e.g. Avanzi et al. (2007) and Avanzi and Gerber (2008)), as well as the distribution of the time to reach a given surplus level in the classical compound Poisson risk model (see, e.g. Gerber (1990)). We propose to extend (1) by further incorporating one particular property of this first passage, namely the number of negative jumps before τ (b). For this to be of interest, we shall restrict the class of spectrally negative Lévy processes to those which takes the form of a compound Poisson process perturbed by a Brownian motion with drift, that is Rt = ct + σ Wt − Mt ,

(2)

with c  = 0, σ > 0, W = {Wt , t ≥ 0} a standard Brownian motion, and M = {Mt , t ≥ 0} an independent (of W ) compound Poisson process. More precisely, we define the process M as

Mt =

 0,

Nt i=1

Xi ,

Nt > 0, Nt = 0,

where {Nt , t ≥ 0} is a Poisson process with rate λ > 0 and is defined via the sequence of ∞ independent and identically distributed (iid) interclaim time random variables (r.v.’s) {Ti }i=1 ∞ with density k(t) = λe−λt . {X i }i=1 also form a sequence of iid r.v.’s with density p, and Laplace  ∞ −sx ∞ transform  p (s) = 0 e p (x) d x. We assume that the r.v.’s {X i }i=1 are also independent of the Poisson process {Nt , t ≥ 0}. For spectrally negative Lévy processes not of the form (2), it is well known that these processes have infinitely many jumps in every interval (i.e. infinite intensity of infinitely small activity). Thus, the present analysis is not applicable to this class of spectrally negative Lévy processes. Our main objective is to generalize (1) for the class of compound Poisson processes with diffusion by jointly analyzing the number of jumps before the first passage time. Our approach makes use of Lagrange’s expansion theorem and establishes an interesting connection with Kendall’s identity. In the process, an alternative proof of Kendall’s identity which is both relatively straightforward and only involves simple algebraic manipulations is provided for spectrally negative Lévy processes of the form (2). An implied result of the generalization is the joint "density" of the first passage time and the number of negative jumps until this first passage time. In Section 4, we point out that our result can be directly used to find the finite-time ruin probability in a dual risk model with diffusion (see, e.g. Avanzi and Gerber (2008)). It is worth mentioning that this dual risk model can also be applied in a fluid flow context (see, e.g. Avanzi and Gerber 2008). Finally, we show that our main result can be used to price pathdependent options on an insurer’s stock price. Numerical examples are provided for illustrations.

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2. Kendall’s identity: revisited 2.1. Preamble In this section, the joint Laplace transform of the first passage time to level b and the number of jumps until this first passage time is derived. Consider the process   γ Nt e−δt+s Rt

(3)

t≥0

for γ ∈ (0, 1] and δ > 0. Under the condition



γ E e−δT1 es cT1 +σ WT1 −X 1 = 1,

(4)

it is easy to verify that (3) is a martingale. Let ξ be the unique positive solution of (4). When s = ξ, the process (3) is a bounded martingale before the first passage time to level b. By the optional sampling theorem, we have E γ Nτ (b) e−δτ (b) 1 (τ (b) < ∞) = e−ξ b ,

(5)

where 1 (A) =



1, 0,

A is true, otherwise.

Routine calculations yield

 

E e−δT1 es cT1 +σ WT1 −X 1 = =

∞ 0

 p (s) λe−(λ+δ−cs)t E esσ Wt dt  λ

λ + δ − cs − 21 σ 2 s 2

Using partial fractions, λ λ + δ − cs −

σ2 2 2 s

=

 p (s) .

−2λ σ2

(s − ρ1 ) (s − ρ2 )

where

ρi =

− σ2c2

±



2c σ2

2

+ 8 λ+δ σ2

 2  c 2 λ+δ c +2 2 =− 2 ± 2 σ σ σ   1  2 2 = 2 −c ± c + 2σ (λ + δ) . σ

,

(6)

(7)

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In the sequel, we assume (without loss of generality) that ρ1 = −

  1  2 + 2σ 2 (λ + δ) , c c − σ2

(8)

ρ2 = −

  1  c + c2 + 2σ 2 (λ + δ) . 2 σ

(9)

and

It is worth pointing out that ρ1 > 0 and ρ2 < 0 for c  = 0. Finally, substituting (6) and (7) into (4) yields s − ρ1 +

2λγ 1  p (s) = 0, σ 2 s − ρ2

(10)

an identity particularly relevant to the use of Lagrange’s expansion theorem in the next section. 2.2. Main result In this section, a generalization of Kendall’s identity is proposed for the compound Poisson process with diffusion. Theorem 1 For a compound Poisson process with diffusion, we have 

∞ x

 db Pr τ (b) ≤ t, Nτ (b) = n = b



t 0

Pr (Rs > x, Ns = n)

ds , s

(11)

for t, x > 0 and n = 0, 1, ... Proof

We employ a transform-based approach to derive (11). Indeed, using (5), ∞ 

γn

n=0

=





∞ 0

∞ 0

e−αx





∞

e−αx

x

0

∞

e−δt



∞ x



 db Pr τ (b) ≤ t, Nτ (b) = n dtd x b

 e−ξ b db d x. δ b

(12)

From the representation (10), the use of Lagrange’s expansion theorem (see, e.g. Cohen (1982)) allows to re-write e−ξ b as

 

  ∞  γ n −2λ n d n−1  p (x) n  −xb e (−b)  n! x − ρ2 σ2 d x n−1 x=ρ1 n=1    ∞ −x(b+y) ∗n

n ∞  n n−1  dy p e (y) d γ −2λ  0 = e−ρ1 b + (−b) n−1  n 2  n! σ dx (x − ρ2 ) n=1 x=ρ1    ∞ j −ρ1 y ∗n ∞ n−1 n 2λ n   − b) dy y e p (y − j − 2)! γ (2n b −ρ1 b (13) =e + b n! σ 2 (n − j − 1)! j! (ρ1 − ρ2 )2n−1− j

e−ξ b = e−ρ1 b +

n=1

j=0

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D. Landriault & T. Shi

From (8) and (9), we know that 2  2 c + 2σ 2 (λ + δ) σ2 c  = 2 ρ1 + 2 , σ

ρ1 − ρ2 =

which implies that (13) becomes   ∞ n−1   c −(2n−1− j) γn  ζ j,n (2n − j − 2)! ρ1 + 2 b n! σ  ∞ n=1  j=0   e−ρ1 y y j p ∗n (y − b) dy , ×

e−ξ b = e−ρ1 b +

(14)

b

where ζ j,n = Given that



1 (n − j − 1)! j!

c −n = ρ1 + 2 σ



∞

e

0

−ρ1 x

for n = 1, 2, ..., it follows that e−ξ b = e−ρ1 b +

∞  

∞

n=1 b

λ 2σ 2



n

2 j+1 . −

c

x n−1 e σ 2 (n − 1)!

x



d x,

 γ n e−ρ1 y ϕn (y |b ) dy,

(15)

where ϕn (y |b )

n−1

b  = ζ j,n n! j=0

2b = n! (n − 1)!



b

y

x j (y − x)2(n−1)− j e

λ 2σ 2

n 

y

b

−

c (y−x) σ2

p ∗n (x − b) d x

(y + x)n−1 (y − x)n−1 e

−

c (y−x) σ2

p ∗n (x − b) d x.

(16)

From (8), it is also clear that 1

e−ρ1 b = e σ 2

(c−|c|)b

e

|c| σ2



  2 1− 1+ 2σ2 (λ+δ) b c

.

(17)

Given that an inverse Gaussian r.v. with parameters κ and µ (κ, µ > 0) and density q (t) = has Laplace transform

 κ  1 − κ(t−µ)2 2 e 2µ2 t , 2πt 3  q (s) = e

κ µ



  2 1− 1+ 2µκ s

x > 0,

,

(18)

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Scandinavian Actuarial Journal

(17) can be rewritten as e

−ρ1 b

= =





∞ 0 ∞

0

e



−δt



e−δt

√ √

e−λt 2πσ 2 t 3 e−λt

1 (c−|c|)b σ2

be

−

1 2σ 2 t

e

−

(ct−b)2

1 2σ 2 t



(|c|t−b)2



dt

dt.

(19)

 γ n e−δt f n (t |b ) dt,

(20)

2πσ 2 t 3

be

Substituting (19) into (15) yields e−ξ b =

∞  

n=0 0

∞

where f n (t |b ) =

⎧ ⎨ √ e−λt ⎩

2π σ 2 t 3 −λt √e 2π σ 2 t 3

1

−

(ct−b)2

n = 0,

be 2σ 2 t ,  ∞ − 1 (ct−y)2 2σ 2 t ϕn (y |b ) dy, b ye

n = 1, 2, . . .

(21)

Interchanging the order of integration, (21) becomes e−λt 2b f n (t |b ) = √ 2 3 n! − 1)! (n 2πσ t for n = 1, 2, ... where



λ 2σ 2

n 

∞

b

∞

−

p ∗n (x − b) In (x, t) d x,

c

(y−x) −

1

(22)

(ct−y)2

y (y + x)n−1 (y − x)n−1 e σ 2 dy e 2σ 2 t x 

   ∞ n−1 − 1 y 2 −x 2 − 1 (ct−x)2 = e 2σ 2 t y y2 − x 2 e 2σ 2 t dy.

In (x, t) =

x

Using integration by parts, it is not difficult to show that n     ∞  2 y − x2 y − 12 y 2 −x 2  − 12 (ct−x)2 2σ t e In (x, t) = e 2σ t dy 2n σ 2t x =

1 In+1 (x, t) , 2nσ 2 t

(23)

for n = 1, 2, ... Through a recursive use of (23) together with its starting point  ∞

 − 1 y 2 −x 2 − 12 (ct−x)2 2σ t I1 (x, t) = e ye 2σ 2 t dy x

=

one concludes that

e

−

1 2σ 2 t

(ct−x)2



∞

e

−

2 0   − 12 (ct−x)2 2 = σ t e 2σ t ,

In (x, t) =

1 2σ 2 t

w

dw

(2σ 2 t)n − 1 (ct−x)2 , (n − 1)! e 2σ 2 t 2

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D. Landriault & T. Shi

for n = 1, 2, ... Substituting into (22) yields

f n (t |b ) =

Note that

⎧  ⎪ ⎪ ⎨ b e−λt √

1

2

− (ct−b) 1 e 2σ 2 t 2π σ 2 t

t

⎪ (λt)n e−λt  ∞ ∗n ⎪ ⎩ bt b p (x − b) n! f n (t |b ) =





,

√ 1 e 2π σ 2 t

− 12 2σ t

(ct−x)2



n = 0,

.

(24)

d x, n = 1, 2, . . .

b (λt)n e−λt f Rt |Nt (b |n ) , t n!

(25)

for n = 0, 1, 2, . . . Substituting (25) and (20) into (12) followed by some simple manipulations, one arrives at 

 db Pr τ (b) ≤ t, Nτ (b) = n e e γ dtd x b x 0 0 n=0   ∞  ∞ −δt   ∞ ∞  e b (λt)n e−λt db −αx n e γ = f Rt |Nt (b |n ) dt d x δ t n! b 0 x 0 n=0     ∞ ∞ e−δt ∞  dt (λt)n e−λt e−αx γn = Pr (Rt > x |Nt = n ) dx δ n! t 0 0 n=0  ∞ −δt  ∞ ∞  dt e −αx n e γ = Pr (Rt > x, Nt = n) d x. δ t 0 0

∞ 

n



∞

−αx



∞

−δt



∞

n=0

Using integration by parts, one concludes that ∞ 



∞



∞

dt e−δt Pr (Rt > x, Nt = n) d x δ t 0 0 n=0     ∞ t ∞ ∞  ds −δt −αx n Pr (Rs > x, Ns = n) e e γ = dtd x. s 0 0 0 γ

n

e

−αx

n=0

By the uniqueness property of Laplace transforms and probability generating functions, the result follows.  Remark 2 Given that Rs (s > 0) has a density at x ∈ R, it is immediate from Theorem 1 that the (defective) joint "density" of the first passage time τ (b) and the jumps until the first passage time Nτ (b) at (t, n) is given by f n (t |b ). Remark 3 As expected, Theorem 1 still holds when σ = 0 (note that the proof has to be modified accordingly). We point out that the marginal distribution of the first passage time τ (b) was discussed by, e.g., Gerber and Shiu (1998, Eq. 5.15) together with its connection with a particular version of the ballot theorem. In our context, Theorem 1 provides a generalization of this result: the first passage time τ (b) is a mixed random variable with a mass point at b/c of Pr(τ (b) = λ b/c, Nτ (b) = 0) = e− c b . When at least one claim occurs before the first passage time, the joint

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"density" of τ (b) , Nτ(b) at (t, n) is given by κn (t |b ) =

b (λt)n e−λt ∗n p (ct − b) , t n!

for t > b/c and n = 1, 2, ... In the following section, we consider a large class of distributions which leads to a mathematically tractable expression for f n (t |b ).

3. Mixed Erlang distributed jumps In this section, we assume that the Laplace transform of the negative jumps is of the form  p (s) = C

where

C (z) =

β β +s

∞ 



,

(26)

cjz j,

j=1

 with c j ≥ 0 for j = 1, 2, ... and ∞ j=1 c j = 1. The reader is referred to (Tijms, 1994, p.163) for a proof that any continuous and positive random variable can be approximated arbitrary accurately by a mixed Erlang density and to Willmot and Woo (2007) and Willmot and Lin (2011) for an extensive analysis of this class of distributions. Under this distributional assumption, (24) becomes b (λt)n e−λt f n (t |b ) = t n! b = t

=

(λt)n e−λt n!



0

⎧

∞ ∞ ⎨

1

⎩

j=1 ∞ 

√ 2πσ 2 t

b (λt)n e−λt e t n!

⎫ j x j−1 e−βx ⎬ 1 β − 1 (ct−b−x)2 c∗n dx e 2σ 2 t √ j ( j − 1)! ⎭ 2πσ 2 t

j=1

c∗n j

βj ( j − 1)!



2 −β ct−b− βσ2 t ∞ 

√ 2πσ 2 t

c∗n j

j=1



∞

0

x j−1 e−βx e

βj ( j − 1)!

∞ 0

1 2σ 2 t

x j−1 e

−

(ct−b−x)2

1 2σ 2 t



dx



 2 x− c−βσ 2 t−b

d x,

(27)

where c∗n j are defined via the transform relationship (C (z))n =



−

∞  j=1

j c∗n j z .

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D. Landriault & T. Shi

Simple modifications of the integrand in (27) results in

f n (t |b ) =

b t

(λt)n e−λt

e

n!



2 −β z t + βσ2 t ∞ 

√

2πσ 2 t

j−1 

j c∗n j β

k=0

j=1

(z t ) j−1−k αk (z t ) , ( j − 1 − k)!k! (28)

where



αk (z) =

∞

xke

−

1 2σ 2 t

x2

d x,

−z

and   z t = c − βσ 2 t − b. For k odd (say k = 2i + 1), we have α2i+1 (z) =



=

1 2

=

∞

|z|



x 2i+1 e ∞

z2

yi e

−

−



1 (i!) 2σ 2 t 2

1 2σ 2 t

1 2σ 2 t

y

x2

dx

dy

i i+1 



z2 2σ 2 t

l

e

−

z2 2σ 2 t

For k even (say k = 2i), using integration by parts, one finds that 2



α2i (z) = σ t (−z)

1 2 2i−1 − 2σ 2 t z

e

+ (2i − 1)



∞ −z

y

(29)

.

l!

l=0

1 2 2(i−1) − 2σ 2 t y

e



dy .

By repeating this argument, one finds

α2i (z) = =

  i  i  ∞  γi  2 i− j+1 − 1 z2 − 1 y2 σ t e 2σ 2 t dy + γi σ 2 t (−z)2 j−1 e 2σ 2 t γj −z j=1

   i  i+ 1 √  −z γi  2 i− j+1 2 − 1 z2 σ t 2π 1 −  + γi σ 2 t , √ (−z)2 j−1 e 2σ 2 t γj σ t j=1

(30) where  is the cumulative distribution function of a Normal random variable with mean 0 and variance 1, and γi =

i &

k=1

(2k − 1) .

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4. Applications 4.1. First passage times in dual risk model with diffusion and in fluid flow model The dual risk process with diffusion U = {Ut , t ≥ 0} is defined as (31)

Ut = u − ct + σ Wt + Mt = u − Rt ,

where u is the initial surplus level, and c is the non-negative expense rate (see, e.g. (Grandell, 1991, p.8) and Avanzi and Gerber (2008)). As pointed out by, e.g., Avanzi and Gerber (2008), the dual risk model is well suited to model the cash flow dynamics of a portfolio of life annuities, or of companies specializing in inventions and discoveries. Let ς = inf {t ≥ 0 : Ut < 0} be the time to ruin for this surplus process. By a reflective argument, one can easily conclude that the time to ruin ς with an initial surplus u in the dual risk process U corresponds to the first passage time to level u, namely τ (u), of the process {Rt , t ≥ 0}. Therefore, one can directly use the main result of Section 2 to obtain finite-time ruin probabilities in the dual risk process with no more than a given number of claims in the interim. Many researchers have analyzed the ruin probability in the dual risk process (see, e.g. Cramér (1955, Section 5.13) and Mazza and Rulliére (2004)). A traditional way to obtain the distribution of the time to ruin is through the numerical inversion of its Laplace transform (which is known under various distributional assumptions). The explicit expression (24) can be considered as an alternative to calculate finite-time ruin probabilities. More interestingly, it allows to break the contribution to the finite-time ruin probability by the number of claims until ruin, which by itself is of interest. Moreover, the resulting approximative quantity provides an insightful ruin related quantity. The dual risk process (31) can also be considered as a second-order fluid flow queue, where u is the initial fluid in the system and Mt is the non-decreasing fluid arrivals into the queue. We assume a linear service rate c and use the Brownian motion to represent the traffic noise. A more general definition of a second-order fluid queue can be found in Kulkarni (1997), Rabehasaina and Sericola (2004), as well as references therein. Fluid flow models are widely used in engineering to analyze the behavior of telecommunication flow, whereby the fluid represents the signals temporarily stored in a buffer. The first passage time to level 0 is the duration of a busy period of the buffer. Similarly, Eq. (24) can be directly used to obtain the joint (defective) distribution of the busy period and the number of signal arrivals. In the following example, we will illustrate the usage of Eq. (24) to obtain the finite ruin probabilities in a dual risk model. Example 1 We assume that the jump size distribution follows a mixture of 3 Erlangs with Laplace transform  p (s) = 0.3 ·

0.05 + 0.5 · 0.05 + s

0.05 0.05 + s

2

+ 0.2 ·

0.05 0.05 + s

3

,

s ≥ 0.

378

D. Landriault & T. Shi Table 1.

Finite-time ruin probability with no more than n jumps (T = 10).

u\n

0

1

2

3

4

5

6

7

8

9

5 10 25 50

0.8607 0.7408 0.4724 0.1138

0.9107 0.8235 0.5765 0.1151

0.9176 0.8349 0.5882 0.1151

0.9183 0.8359 0.5888 0.1151

0.9183 0.8359 0.5889 0.1151

0.9183 0.8359 0.5889 0.1151

0.9183 0.8359 0.5889 0.1151

0.9183 0.8359 0.5889 0.1151

0.9183 0.8359 0.5889 0.1151

0.9183 0.8359 0.5889 0.1151

Table 2.

Finite-time ruin probability with no more than n jumps (T = 50).

u\n

0

1

2

3

4

5

6

7

8

9

5 10 25 50

0.8607 0.7408 0.4724 0.2232

0.9164 0.8368 0.6253 0.3676

0.9334 0.8690 0.6916 0.4536

0.9421 0.8860 0.7288 0.5089

0.9476 0.8966 0.7528 0.5462

0.9512 0.9036 0.7686 0.5706

0.9536 0.9081 0.7784 0.5849

0.9549 0.9107 0.7837 0.5920

0.9556 0.9119 0.7861 0.5948

0.9558 0.9124 0.7870 0.5957

Also, let λ = 0.15, σ = 0.2, and the expense rate c = 5. Tables 1 and 2 present the values

 of finite-time ruin probabilities with no more than n jumps (i.e. Pr ς ≤ T, Nς ≤ n ) for a time horizon of T = 10 and T = 50, respectively. As expected, longer is the time horizon, more jumps are required to observe the convergence of the joint cumulative distribution function of the time to ruin and the number of claims at ruin to the finite-time ruin probability (as n → ∞). Theoretically, we know that

 lim Pr ς ≤ T, Nς ≤ n = Pr (ς ≤ T ) .

n→∞

4.2. Pricing path-dependent exotic options In recent years, jump-diffusion processes have been widely used to model financial assets. In general, researchers are using two-sided jumps to represent random gains and losses of a company (see, e.g. Kou and Wang (2004)). However, many authors have argued of the relevance of using one-sided jump-diffusion processes to model the stock price of insurance companies, where the one-sided negative jumps represent the impact of catastrophic losses (see, e.g. Gerber and Landry (1998), Cox (2004) and Lin and Wang (2009)). Rather than investigating catastrophe-linked securities (in which the catastrophe loss is the exercise trigger, as in, e.g. Cox (2004) and Lin and Wang (2009)), we are interested here in analyzing general path-dependent options issued on the insurer’s stock. Specifically speaking, we will use the main result of Section 2 to price the up-and-in call option of an insurer. Other up-and-in, up-and-out call (put) options can be obtained in a similar fashion. An up-and-in call is a regular call option that will be activated only if the price of the underlying asset rises above a certain price level (see, e.g. Hull (2008)). The underlying asset process of an insurance company S = {St , t ≥ 0} is assumed to satisfy St = S0 exp (ct + σ Wt − Mt ) ,

379

Scandinavian Actuarial Journal

where S0 is the initial stock price. The process is further assumed to be under the risk-neutral probability measure Q with a continuous risk-free rate r > 0. To ensure that the discounted   stock price process e−r t St , t ≥ 0 is a martingale under Q, c is assumed to be given by c=r−

σ2 − λ( p (1) − 1). 2

(32)

Under the risk-neutral probability measure, the price of an up-and-in call option with time to maturity T, strike price K , barrier level H (H > S0 ) and up to n negative jumps is given by Cn (T, H, K ) = E

Q



e

−r T

+

(ST − K ) 1

max St ≥ H, N T ≤ n

0≤t≤T



= n − K e−r T n , where n = E and

Q



e

−r T

ST 1 ST ≥ K , max St ≥ H, N T ≤ n 0≤t≤T



,



n = P Q ST ≥ K , max St ≥ H, N T ≤ n . 0≤t≤T

(33)

(34)

Naturally, the price of an up-and-in call option is the limit of Cn (T, H, K ) as n → ∞, i.e. C(T, H, K ) = lim Cn (T, H, K ). n→∞

Let b = ln SH0 . By conditioning on the first passage time of the process R to level b and on the number of jumps, (34) becomes n = P Q (ST ≥ K , τ (b) ≤ T, N T ≤ n) n  T  P Q ( ST ≥ K , N T ≤ n| τ (b) = t, Nτ (b) = k) f k (t |b ) dt. = k=0 0

Noting that Rτ (b) = b and Sτ (b) = H , and making use of the strong Markov property of the process R, one deduces

 P Q ST ≥ K , N T ≤ n| τ (b) = t, Nτ (b) = k n    P Q ( Sτ (b) e RT −b ≥ K , N T −τ (b) = j − k  τ (b) = t) = j=k

= =

n  j=k

P Q (RT −t ≥ ln

n  

∞

j=k ln

K H

K , N T −t = j − k) H

g j−k (y; T − t) dy,

380

D. Landriault & T. Shi Table 3.

Prices of the up-and-in call option with no more than n jumps (λ = 0.01).

K \n

0

1

2

3

4

5

6

B-S Price

80 90 100 110 120 130

16.7702 12.9167 9.1955 5.8394 3.2213 1.6269

16.9191 13.0303 9.2752 5.8891 3.2481 1.6400

16.9198 13.0308 9.2756 5.8893 3.2482 1.6401

16.9198 13.0308 9.2756 5.8893 3.2482 1.6401

16.9198 13.0308 9.2756 5.8893 3.2482 1.6401

16.9198 13.0308 9.2756 5.8893 3.2482 1.6401

16.9198 13.0308 9.2756 5.8893 3.2482 1.6401

16.9182 13.0296 9.2746 5.8885 3.2477 1.6396

where gn (y; t) is the density of Rt with n jumps, namely

gn (y; t) =

Therefore,

⎧  ⎪ −λt √ ⎪ e ⎨

1

2

− (ct−y) 1 e 2σ 2 t 2π σ 2 t

⎪ (λt)n e−λt  ∞ ∗n ⎪ ⎩ 0 p (x) n! n =

n  n  

k=0 j=k 0





√ 1 e 2π σ 2 t

T

f k (t |b )



n = 0,

,

∞ ln

1 2σ 2 t

−

K H

(ct−y−x)2



d x,

n = 1, 2, . . .

g j−k (y; T − t) dydt.

(35)

Using the same change of numeraire arguments as in Kou and Wang (2004), an expression  such that for n can be obtained. Indeed, define a new probability measure Q  St dQ = exp = e−r t dQ S0

  σ2 − − λ( p (1) − 1) t + σ Wt − Mt . 2

 measure, Rt is a jump-diffusion process Under the Q 





Rt = c Q t + σ WtQ − MtQ ,







Q Q where c Q = c + σ 2 , and {Wt , t ≥ 0} defined as Wt = Wt − σ t is a standard Brownian   motion, and {MtQ , t ≥ 0} is a compound Poisson process with Poisson arrival rate λ Q = λ p (1)  e−x Q and secondary distribution having density p (x) =  p(x) for x ≥ 0. p (1)  measure is given by Furthermore, (34) under the Q

   n = S0 E Q 1 ST ≥ K , max St ≥ H, N T ≤ n 0≤t≤T

 (ST ≥ K , τ (b) ≤ T, N T ≤ n) = S0 Q 

= S0 nQ , 







where nQ is the value of n replacing c, λ and p(x) by c Q , λ Q and p Q (x) respectively. Example 2 Assume S0 = 100, H = 120, T = 1, r = 0.05, σ = 0.2, and the jump sizes have 2 50 50 . We point out that this example is identical Laplace transform  p (s) = 0.9 50+s + 0.1 50+s to Kou and Wang (2004, Section 4.3) except for the jump size density. Indeed, Kou and Wang

381

Scandinavian Actuarial Journal Table 4.

Prices of the up-and-in call option with no more than n jumps (λ = 3).

K \n

0

1

3

5

7

9

11

12

80 90 100 110 120 130

1.1824 0.9338 0.6906 0.4642 0.2774 0.1530

4.3804 3.4380 2.5192 1.6717 0.9822 0.5322

12.5362 9.7395 7.0325 4.5727 2.6175 1.3803

16.4717 12.7252 9.1159 5.8652 3.3135 1.7243

17.3013 13.3417 9.5341 6.1150 3.4418 1.7847

17.3973 13.4115 9.5800 6.1413 3.4547 1.7904

17.4043 13.4164 9.5831 6.1430 3.4555 1.7908

17.4045 13.4166 9.5833 6.1431 3.4555 1.7908

(2004) assume a jump-diffusion process which allows for both positive and negative exponential jumps, whereas only one-sided jumps are considered in this paper. However, for comparative purposes, the mean of the jump size was preserved. Tables 3 and 4 contain the price for up-and-in call options with no more than n negative jumps when λ = 0.01 and λ = 3, respectively. Note that the drift c of the Brownian motion (as defined in (32)) is 0.0302 (0.0946) when λ = 0.01 (λ = 3). Remark that the last column of Table 3 gives the price of up-and-in call options under the Black-Scholes model (λ = 0). We observe that the values of Cn (T, H, K ) for λ = 0.01 and n relatively large are very close to the Black-Scholes price of these up-and-in call options, as anticipated. Also, as the Poisson arrival rate λ of the jump-diffusion processes gets larger, the speed of convergence (in n) of Cn (T, H, K ) to C(T, H, K ) gets slower. Indeed, for a jumpdiffusion process with a large value of λ, more jumps are expected (on average) within a given time horizon. When the strike price is no less than the barrier level, the up-and-in call option becomes a regular call option. This can be formally proven as highlighted in the following remark. Remark 4 When K ≥ H , (35) is consistent with the expression for regular call options. Indeed, for y > 0, by using Laplace transform arguments, it can be shown that the convolution of g j (y; t) and f k (t |b ) satisfies j  

k=0 0

T

g j−k (y; T − t) f k (t |b ) dt = g j (y + b; T ),

which, substituted into (35), yields n =

n  

∞

j=0 ln

K H

g j (y + b; T )dy =

n  

∞

j=0 ln

K S0

g j (y; T )dy = P Q (ST ≥ K , N T ≤ n).

Acknowledgements Financial support for David Landriault from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Support from the Society of Actuaries is also gratefully acknowledged by Tianxiang Shi.

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D. Landriault & T. Shi

References Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Communications in Statistics. Stochastic Models 11 (1), 21–49. Avanzi, B. & Gerber, H. U. (2008). Optimal dividends in the dual model with diffusion. ASTIN Bulletin 38 (2), 653–667. Avanzi, B., Gerber, H. U. & Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance: Mathematics and Economics 41 (1), 111–123. Borovkov, K. & Burq, Z. (2001). Kendall’s identity for the first crossing time revisited. Electronic Communications in Probability 6, 91–94. Cohen, J. W. (1982). The single server queue. 2nd ed. North-Holland Publication Company: Amsterdam. Cox, S. H., Fairchild, J. R. & Pedersen, H. W. (2004). Valuation of structured risk management products. Insurance: Mathematics and Economics 34 (2), 259–272. Cramér, H. (1955). Collective risk theory: a survey of the theory from the point of view of the theory of stochastic processes. Stockholm: Ab Nordiska Bokhandeln. Gerber, H. U. (1990). When does the surplus reach a given target? Insurance: Mathematics and Economics 9, 115–119. Gerber, H. U. & Landry, B. (1998). On the discount penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Mathematics and Economics 22, 263–276. Gerber, H. U. & Shiu, E. S. W. (1998). On the time value of ruin. North American Actuarial Journal 2 (1), 48–78. Grandell, J. (1991). Aspects of risk theory. Springer series in statistics. Probability and its applications. New York: Springer-Verlag. Hull, J. C. (2008). Options, futures, and other derivatives. 7th ed. London: Pearson Prentice Hall. Kendall, D. G. (1957). Some problems in the theory of dams. Journal of the Royal Statistical Society: Series B 19, 207–212. Kou, S. G. & Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management Science 50 (9), 1178–1192. Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queuing, Models and Applications in Science and Engineering. Boca Raton, FL: CRC Press. p. 321–338. Lin, X. S., & Wang, T. (2009). Pricing perpetual American catastrophe put options: a penalty function approach. Insurance: Mathematics and Economics 44, 287–295. Mazza, C. & Rulliére, D. (2004). A link between wave governed random motions and ruin processes. Insurance: Mathematics and Economics 35, 205–222. Rabehasaina, L. & Sericola, B. (2004). A second order Markov modulated fluid queue with linear service rate. Journal of Applied Probability 41 (3), 758–777. Tijms, H. C. (1994). Stochastic models: an algorithmic approach. Chichester: John Wiley & Sons, Ltd. Willmot, G. E. & Lin, X. S. (2011). Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry 27 (1), 2–16. Willmot, G. E. & Woo, J. K. (2007). On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal 11 (2), 99–115.

Scandinavian Actuarial Journal, 2013 http://dx.doi.org/10.1080/03461238.2013.850442

Gaussian risk models with financial constraints KRZYSZTOF DEBICKI†, ¸ ENKELEJD HASHORVA‡ and LANPENG JI∗ ‡ †Mathematical Institute, University of Wrocław, Wrocław, Poland ‡Department of Actuarial Science, University of Lausanne, Lausanne, Switzerland (Accepted September 2013) In this paper, we investigate Gaussian risk models which include financial elements, such as inflation and interest rates. For some general models for inflation and interest rates, we obtain an asymptotic expansion of the finite-time ruin probability for Gaussian risk models. Furthermore, we derive an approximation of the conditional ruin time by an exponential random variable as the initial capital tends to infinity. Keywords: finite-time ruin probability; conditional ruin time; exponential approximation; Gaussian risk process; inflation; interest AMS Subject Classifications: Primary: 91B30; Secondary: 60G15; 60G70

1. Introduction A central topic in the actuarial literature, inspired by the early contributions of Lundberg (1903) and Cramér (1930), is the computation of the ruin probability over both finite-time and infinitetime horizon; see e.g. Rolski et al. (1999), Mikosch (2004), Asmussen & Albrecher (2010), and the references therein. As mentioned in Mikosch (2004), calculation of the ruin probability is considered as the ‘jewel’ of the actuarial mathematics. In fact, exact formulas for both finite-time and infinite-time ruin probability are known only for few special models. Therefore, asymptotic methods have been developed to derive expansions of the ruin probability as the initial capital/reserve increases to infinity. Following Chapter 11.4 in Rolski et al. (1999), the risk reserve process of an insurance company can be modeled by a stochastic process {U˜ (t), t ≥ 0} given as U˜ (t) = u + ct −



t

Z (s) ds,

0

t ≥ 0,

(1.1)

where u ≥ 0 is the initial reserve, c > 0 is the rate of premium received by the insurance company, and {Z (t), t ≥ 0} is a centered Gaussian process with almost surely continuous sample paths; the process {Z (t), t ≥ 0} is frequently referred to as the loss rate of the insurance company. ∗ Corresponding

author. E-mail: [email protected] © 2013 Taylor & Francis

K. D¸ebicki et al.

2

Under the assumption that {Z (t), t ≥ 0} is stationary the asymptotics of the infinite-time ruin probability of the process (1.1) defined by 

ψ∞ (u) = P

inf

t∈[0,∞)

 U˜ (t) < 0 ,

u≥0

has been investigated in Hüsler & Piterbarg (2004), D¸ebicki (2002), Dieker (2005) and Kobelkov (2005); see also Hüsler & Piterbarg (1999) and Hashorva et al. (2013). Therein the exact speed of convergence to 0 of ψ∞ (u) as u → ∞ was dealt with. In order to account for the financial nature of the risks, and thus for the time-value of the money, as well as other important economic factors, in this paper we shall consider a more general risk process which includes inflation/deflation effects and interest rates (cf. Chapter 11.4 in Rolski et al. (1999)). Essentially, in case of inflation, a monetary unit at time 0 has the value e−δ1 (t) at time t, where δ1 (t), t ≥ 0 is a positive function with δ1 (0) = 0. In case of interest, a monetary unit invested at time 0 has the value eδ2 (t) at time t, where δ2 (t), t ≥ 0 is another positive function with δ2 (0) = 0. Assuming first that the premium rate and the loss rate have to be adjusted for inflation, we arrive at the following risk reserve process: u+c



t 0

eδ1 (s) ds −



0

t

eδ1 (s) Z (s)ds,

t ≥ 0.

Since the insurance company invests the surplus, and thus accounting for investment effects, the resulting risk reserve process is    t  t U (t) = eδ2 (t) u + c eδ1 (s)−δ2 (s) ds − eδ1 (s)−δ2 (s) Z (s)ds , 0

0

t ≥ 0.

(1.2)

We shall refer to {U (t), t ≥ 0} as the risk reserve process in an economic environment; see Chapter 11.4 in Rolski et al. (1999) for a detailed discussion on the effects of financial factors on the risk reserve processes. In the case that δ1 (t) = 0, δ2 (t) = δt, t ≥ 0, with δ > 0, the random process {U (t), t ≥ 0} reduces to a risk reserve process with constant force of interest. For a class of stationary Gaussian processes {Z (t), t ≥ 0} with twice differentiable covariance function, the exact asymptotics of the infinite-time ruin probability for the risk reserve process with constant force of interest was obtained in He & Hu (2007). Since therein the authors considered only smooth Gaussian process, the method of proof relied on the well-known Rice method; see e.g. Piterbarg (1996). Let T be any positive constant. The principal goal of this contribution is the derivation of the exact asymptotics of the finite-time ruin probability of the risk reserve process U given by ψT (u) := P = P



inf U (t) < 0

t∈[0,T ]



sup



t∈[0,T ]

0

t

e



−δ(s)

Z (s) ds − c



0

t

e

−δ(s)

ds



>u



(1.3)

3

Scandinavian Actuarial Journal

as u → ∞, where {Z (t), t ≥ 0} is a general centered Gaussian process with almost surely continuous sample paths and δ(t) = δ2 (t) − δ1 (t), t ≥ 0 is some measurable real-valued function satisfying δ(0) = 0. Note in passing that δ(t) > 0 means that the interest contributes more to the risk reserve process than the inflation at time t, and vice versa. In Theorem 2.1 below we shall show that ψT (u) has asymptotically, as u → ∞, (nonstandard) normal distribution. This emphasizes the qualitative difference between asymptotics in finite- and infinite-time horizon scenario; see He & Hu (2007). A related, interesting and vastly analyzed quantity is the time of ruin which in our model is defined as τ (u) = inf {t ≥ 0 : U (t) < 0}, u ≥ 0. (1.4)

Using that P {τ (u) < T } = P inf t∈[0,T ] U (t) < 0 , investigation of distributional properties of the time of ruin under the condition that ruin occurs in a certain time period has attracted substantial attention; see e.g. the seminal contribution Segerdahl (1955) and the monographs Embrechts et al. (1997) and Asmussen & Albrecher (2010). Recent results for infinite-time Gaussian and Lévy risk models are derived in Hüsler (2006), Hüsler & Piterbarg (2008), Hüsler & Zhang (2008), Griffin & Maller (2012), Griffin (2013) and Hashorva & Ji (2014). In Theorem 2.4 we derive a novel result, which shows that as u → ∞, the sequence of random variables {ξu , u > 0}, defined (on the same probability space) by  d  ξu = u 2 (T − τ (u))(τ (u) < T )

(1.5) d

converges in distribution to an exponential random variable (here = stands for the equality of the distribution functions). This, somewhat surprising result, contrasts with the infinite-time case analyzed by Hüsler & Piterbarg (2008) and Hashorva & Ji (2014), where the limiting random variable is normally distributed. Organization of the paper: The main results concerning the finite-time ruin probability and the approximation of ξu are displayed in Section 2, whereas the proofs are relegated to Section 3. We conclude this contribution with a short Appendix.

2. Main results Let the loss rate of the insurance company {Z (t), t ≥ 0} be modeled by a centered Gaussian process with almost surely continuous sample paths and covariance function Cov (Z (s)), Z (t) = R(s, t). As mentioned in the Introduction we shall require that δ(0) = 0. For notational simplicity we shall define below Y (t) :=



t 0

e−δ(s) Z (s)ds,

σ 2 (t) := Var (Y (t)),

δ (t) :=



t 0

e−δ(s) ds,

t ∈ [0, T ].

(2.6)

K. D¸ebicki et al.

4

In what follows, let σ ′ (t) be the derivative of σ (t) and let  denote the survival function of a N (0, 1) random variable. We are interested in the asymptotic behavior of (1.3) as the initial reserve u tends to infinity, i.e. we shall investigate the asymptotics of ψT (u) = P



sup



t∈[0,T ]

  Y (t) − cδ (t) > u

as u → ∞. In our first result below we derive an asymptotic expansion of ψT (u) in terms of c, σ (T ), δ (T ).

Theorem 2.1 Let {Z (t), t ≥ 0} be a centered Gaussian process with almost surely continuous sample paths and covariance function R(s, t), s, t ≥ 0. Further let δ(t), t ≥ 0, be some measurable function with δ(0) = 0. If σ (t) attains its maximum over [0, T ] at the unique point t = T and σ ′ (T ) > 0, then

ψT (u) = P N > (u + c δ (T ))/σ (T ) (1 + o(1))

(2.7)

holds as u → ∞, with N a N (0, 1) random variable.

The following result is an immediate consequence of Theorem 2.1. Corollary 2.2 Let {Z (t), t ≥ 0} and δ(t), t ≥ 0 be given as in Theorem 2.1. If R(s, t) > 0 for any s, t ∈ [0, T ], then (2.7) is satisfied. Remark 2.3

t (a) It follows from the proof of Theorem 2.1 that (2.7) still holds if Y (t) := 0 e−δ1 (s) Z (s)ds

t −δ (s) and δ (t) := 0 e 2 ds in (2.6). (b) In the asymptotic behavior of ψT (u) the positive constant σ ′ (T ) does not appear. It appears however explicitly in the the approximation of the conditional ruin time as shown in our second theorem below. Along with the analysis of the ruin probability in risk theory an important theoretical topic is the behavior of the ruin time. For infinite-time horizon results in this direction are well known; see e.g. Asmussen & Albrecher (2010), Hüsler & Piterbarg (2008) and Hashorva & Ji (2014) for the normal approximation of the conditional distribution of the ruin time τ (u) given that τ (u) < ∞. In our second result below we show that (appropriately rescaled) ruin time τ (u) conditioned that τ (u) < T is asymptotically, as u → ∞, exponentially distributed with parameter σ ′ (T )/(σ (T ))3 . Theorem 2.4

Under the conditions of Theorem 2.1, we have

 lim P u 2 (T − τ (u)) ≤

u→∞

    σ ′ (T )  x τ (u) < T = 1 − exp − 3 x , x ≥ 0. σ (T )

(2.8)

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Scandinavian Actuarial Journal

Note in passing that (2.8) means the convergence in distribution d

ξu → ξ,

(2.9)

u → ∞,

where ξ is exponentially distributed such that eT := E (ξ ) =

σ 3 (T ) > 0. σ ′ (T )

We present next three illustrating examples. Example 2.5 Let {Z (t), t ≥ 0} be an Ornstein-Uhlenbeck process with parameter λ > 0, i.e. Z is a stationary process with covariance function R(s, t) = exp(−λ|s − t|). If δ(t) = δt, t ≥ 0 with δ ∈ (0, λ), then δ (t) =

 1 1 − e−δt , δ

σ 2 (t) =

    2 1 1 − e−(λ+δ)t , t ∈ [0, T ]. 1 − e−2δt − 2 2 (λ − δ)δ λ −δ

Therefore, from Corollary 2.2, we obtain that      2 1 1 1 − e−2δT − 2 1 − e−(λ+δ)T ψT (u) = √ 2 (λ − δ)δ λ − δ u 2π ⎛ ⎞   (u + c/δ 1 − e−δT )2 × exp ⎝−      ⎠ (1 + o(1)) 2 1 −(λ+δ)T 1 − e−2δT − λ2 −δ 2 (λ−δ)δ 2 1−e

as u → ∞. Furthermore, in view of Theorem 2.4 the convergence in (2.9) holds with eT = Example 2.6

((λ + δ)(1 − e−2δT ) − 2δ(1 − e−(λ+δ)T ))2 . (λ − δ)(λ + δ)2 δ 2 e−δT (e−δT − e−λT )

Let {Z (t), t ≥ 0} be a Slepian process, i.e. Z (t) = B(t + 1) − B(t),

t ≥ 0,

with B a standard Brownian motion. For this model we have R(s, t) = max(1 − |s − t|, 0). If further δ(t) = δt, t ≥ 0 with δ = 0, then δ (t) =

 1 1 − e−δt , δ

σ 2 (t) =

1 1 2 2t δ+1 − 3 − 2 e−δt + 2 e−δt + 3 e−2δt , t ∈ [0, 1]. δ2 δ δ δ δ

Consequently, Corollary 2.2 implies, as u → ∞ ψ1 (u) =



 2    δ δu + c 1 − e−δ δ − 1 + (δ + 1)e−2δ −1 u exp − (1 + o(1)). 2πδ 3 2(δ − 1 + (δ + 1)e−2δ )

K. D¸ebicki et al.

6

Further by Theorem 2.4 the convergence in (2.9) holds with e1 =

(δ − 1 + (δ + 1)e−2δ )2 . δ 4 e−δ − δ 4 (δ + 1)e−2δ

Example 2.7 Let {Z (t), t ≥ 0} be a standard Brownian motion and assume that δ(t) = t 2 /2, t ≥ 0. Since R(s, t) = min(s, t) we obtain δ (t) =

√

2π(1/2 − (t)),

√ √ √ √ √ σ 2 (t) = ( 2 − 1) π − 2 2π(t) + 2 π( 2t), t ∈ [0, T ].

Applying once again Corollary 2.2 we obtain  √ √ √ √ √ ( 2 − 1) π − 2 2π(T ) + 2 π( 2T ) −1 ψT (u) = u 2π ⎛ ⎞   2 √ u + 2πc(1/2 − (T )) ⎜ ⎟  ⎠ (1 + o(1)) × exp ⎝−  √ √ √ √ √ 2 ( 2 − 1) π − 2 2π(T ) + 2 π( 2T )

as u → ∞. Finally, by Theorem 2.4 the convergence in distribution in (2.9) holds with 2 √ √ √ √ √ ( 2 − 1) π − 2 2π(T ) + 2 π( 2T ) , eT = √ √ 2π(ϕ(T ) − ϕ( 2T )) where ϕ = − ′ is the density function of N (0, 1) random variable.

3. Proofs Before presenting proofs of Theorems 2.1 and 2.4, we introduce some notation. Let ˜ δ(t) gu (t) = u+c σ (t) and define Y (t) gu (T ) , σ X2 u (t) := Var (X u (t)) , σ (t) gu (t)     X u (s) X u (t) Y (s) Y (t) r X u (s, t) := Cov , , . = Cov σ X u (s) σ X u (t) σ (s) σ (t) X u (t) :=

Then, we can reformulate (1.3) for all large u as ψT (u) = P =P



sup t∈[0,T ]





Y (t) gu (T ) σ (t) gu (t)



> gu (T ) 

sup X u (t) > gu (T ) ,

t∈[0,T ]



u ≥ 0.

(3.10)

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Proof of Theorem 2.1 that

7

We shall derive first a lower bound for ψT (u). It follows from (3.10) 

 Y (T ) > gu (T ) σ (T ) = (gu (T ))   (u + c δ (T ))2 σ (T ) (1 + o(1)) = √ u −1 exp − 2σ 2 (T ) 2π

ψT (u) ≥ P

(3.11)

as u → ∞. Next, we derive the upper bound. Since R(s, t) = R(t, s) for any s, t ∈ [0, T ], we have σ 2 (t) := Var (Y (t)) = 2

 t 0

0

w

e−δ(v)−δ(w) R(v, w) dvdw.

(3.12)

Further, since by the assumption the function σ (t) attains its unique maximum over [0, T ] at t = T and that σ ′ (T ) > 0, there exists some θ1 ∈ (0, T ) such that σ (t) is strictly increasing on [θ1 , T ] and inf

t∈[θ1 ,T ]

σ ′ (t) > 0

(3.13)

implying that for u sufficiently large σ X′ u (t) =

σ ′ (t) u + c δ (T )) δ (T ) ceδ(t) σ (t)(u + c − >0 σ (T ) u + c δ (t) (u + c δ (t))2 σ (T )

for all t ∈ [θ1 , T ]. Hence, for sufficiently large u, σ X u (t) is strictly increasing on [θ1 , T ]. Furthermore, since gu (T ) gu (t)   (σ (T ) − σ (t)) u + c δ (t) − cσ (t)( δ (T ) − δ (t))   = , σ (T ) u + cδ (t)

1 − σ X u (t) = 1 −

then by the definitions of δ (t) and σ (t) for any ε1 > 0 there exist some constants K > 0 and θ2 ∈ (0, T ) such that δ (T ) − δ (t) ≤ K (T − t),

(1 − ε1 )σ ′ (T )(T − t) ≤ σ (T ) − σ (t) ≤ (1 + ε1 )σ ′ (T )(T − t) are valid for all t ∈ [θ2 , T ]. Therefore, we conclude that for u sufficiently large (1 − ε1 )2

σ ′ (T ) σ ′ (T ) (T − t) ≤ 1 − σ X u (t) ≤ (1 + ε1 ) (T − t) σ (T ) σ (T )

(3.14)

K. D¸ebicki et al.

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for t ∈ [θ2 , T ]. For any s < t we have 

 Y (s) Y (t) , σ (s) σ (t) Var (Y (t) − Y (s)) − (σ (t) − σ (s))2 = 2σ (s)σ (t) Var (Y (t) − Y (s)) ≤ 2σ (s)σ (t)

t t R(v, w)e−δ(w)−δ(v) dwdv = s s . 2σ (s)σ (t)

1 − r X u (s, t) = 1 − Cov

The above implies that for sufficiently large u and s, t ∈ [θ2 , T ] 1 − r X u (s, t) ≤ C(t − s)2 ,

(3.15)

−δ(w)−δ(v)

where C = maxw,v∈[θ2 ,T ] |R(v,w)|e . Consequently, in the light of (3.14) and (3.15), for 2σ 2 (w) any ε > 0 sufficiently small, we have for some θ0 ∈ (max(θ1 , θ2 ), T ) σ X u (t) ≤

1 ′

(T ) 1 + (1 − ε)(1 − ε1 )2 σσ (T ) (T − t)

and r X u (s, t) ≥ e−(1+ε)C(t−s)

2

for all s, t ∈ [θ0 , T ]. Next, define a centered Gaussian process {Yε (t), t ≥ 0} as Yε (t) =

ξε (t) ′

(T ) 1 + (1 − ε)(1 − ε1 )2 σσ (T ) (T − t)

,

where {ξε (t), t ≥ 0} is a centered stationary Gaussian process with covariance function 2 Cov (ξε (t), ξε (s)) = e−(1+ε)C(t−s) . In view of Slepian Lemma (cf. Adler & Taylor (2007) or Berman (1992)) we obtain

P



sup

t∈[θ0 ,T ]

≤P ≤P





 Y (t) − c δ (t) > u sup

t∈[θ0 ,T ]







=P



sup X u (t) > gu (T ) t∈[θ0 ,T ]

X u (t)/σ X u (t) ′

(T ) 1 + (1 − ε)(1 − ε1 )2 σσ (T ) (T − t) 







> gu (T )

sup Yε (t) > gu (T )

t∈[θ0 ,T ]

= (gu (T ))(1 + o(1))

(3.16)

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as u → ∞, where the last asymptotic equivalence follows from (iii) of Theorem 4.1 in Appendix. Moreover since for u sufficiently large there exists some λ ∈ (0, 1) such that sup σ X u (t) ≤

t∈[0,θ0 ]

and P



(1 + λ)σ (t) (1 + λ)σ (θ0 ) ≤ < 1 σ (T ) σ (T ) t∈[0,θ0 ] sup

sup X u (t) > a t∈[0,θ0 ]





2Y (t) ≤ P sup >a t∈[0,θ0 ] σ (T )



≤

1 2

for some positive number a, we get from Borell inequality (e.g. Piterbarg (1996)) that, for u sufficiently large       P sup Y (t) − cδ (t) > u = P sup X u (t) > gu (T ) t∈[0,θ0 ]

t∈[0,θ0 ]

≤ 2



(gu (T ) − a)σ (T ) (1 + λ)σ (θ0 )



= o((gu (T ))) (3.17)

as u → ∞. Combining (3.16) and (3.17), we conclude that    ψT (u) ≤ P

sup X u (t) > gu (T ) + P

t∈[0,θ0 ]

sup X u (t) > gu (T )

t∈[θ0 ,T ]



= (gu (T ))(1 + o(1))   (u + c δ (T ))2 σ (T ) (1 + o(1)) = √ u −1 exp − 2σ 2 (T ) 2π

as u → ∞, which together with (3.11) establishes the proof.



Proof of Corollary 2.2 Since R(s, t) > 0 for any s, t ∈ [0, T ] it follows from (3.12) that σ (t) attains its unique maximum over [0, T ] at t = T and σ ′ (T ) > 0. Therefore, the claim follows immediately from Theorem 2.1.  Proof of Theorem 2.4 In the following, we shall use the same notation as in the proof of Theorem 2.1. First note that for any x > 0   P τ (u) < T − xu −2

2 P u (T − τ (u)) > x|τ (u) < T = . P {τ (u) < T } X u (t) := With Tu := T − xu −2 and

Y (t) gu (Tu ) σ (t) gu (t)

the above can be re-written as

X u (t) > gu (Tu ) P supt∈[0,Tu ]

. P u (T − τ (u)) > x|τ (u) < T = P supt∈[0,T ] X u (t) > gu (T ) 

2



As in the proof of Theorem 2.1 we have   P sup X u (t) > gu (Tu ) ≥ (gu (Tu )). t∈[0,Tu ]

K. D¸ebicki et al.

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In order to derive the upper bound we use a time change such that P





X u (t) > gu (Tu ) = P sup

t∈[0,Tu ]





X u (Tu t) > gu (Tu ) . sup

t∈[0,1]

Similar argumentations as in (3.14) and (3.15) yield that, for some θ0 ∈ (0, 1) σ X u (Tu t) ≤

1 1+

σ ′ (Tu ) 2σ (Tu ) Tu (1 − t)

and 2

−2C Tu (t−s) r X u (Tu s, Tu t) ≥ e

2

hold for all s, t ∈ [θ0 , 1] and all u sufficiently large. Consequently, in view of (iii) in Theorem 4.1 and similar argumentations as in the proof of Theorem 2.1 we conclude that P



 sup X u (t) > gu (Tu ) ≤ (gu (Tu ))(1 + o(1))

t∈[0,Tu ]

as u → ∞. Hence

  (gu (Tu )) P u 2 (T − τ (u)) > x|τ (u) < T = (1 + o(1)) (gu (T ))  2  g (T ) − gu2 (Tu ) = exp u (1 + o(1)), 2

u → ∞.

(3.18)

After some standard algebra, it follows that gu2 (T ) − gu2 (Tu ) =

(u + c δ (T ))2 (u + c δ (Tu ))2 2σ ′ (T ) − = − x(1 + o(1)) σ 2 (T ) σ 2 (Tu ) σ 3 (T )

(3.19)

as u → ∞. Consequently, by (3.18)

    σ ′ (T ) lim P u 2 (T − τ (u)) > x|τ (u) < T = exp − 3 x , u→∞ σ (T )

which completes the proof.



4. Appendix We give below an extension of Theorem D.3 in Piterbarg (1996) suitable for a family of Gaussian processes which is in particular useful for the proof of our main results. We first introduce two well-known constants appearing in the asymptotic theory of Gaussian processes. Let {Bα (t), t ≥ 0} be a standard fractional Brownian motion with Hurst index α/2 ∈ (0, 1] which is a centered Gaussian process with covariance function

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Cov(Bα (t), Bα (s)) =

1 α (t + s α − | t − s |α ), 2

s, t ≥ 0.

The Pickands constant is defined by    √  1 α ∈ (0, ∞) α ∈ (0, 2] 2Bα (t) − t Hα = lim E exp sup T →∞ T t∈[0,T ] and the Piterbarg constant is given by Pαb



  √  α ∈ (0, ∞), α ∈ (0, 2], = lim E exp sup 2Bα (t) − (1 + b)t T →∞

b > 0.

t∈[0,T ]

See, for instance, Piterbarg (1996) and D¸ebicki & Mandjes (2003) for properties of the above two constants. Assume, in what follows, that θ and T are two positive constants satisfying θ < T . Let {ηu (t), t ≥ 0} be a family of Gaussian processes satisfying the following three assumptions: A1 : The variance function ση2u (t) of ηu attains its maximum over [θ, T ] at the unique point t = T for any u large enough, and further there exist two positive constants A, β and a function A(u) satisfying limu→∞ A(u) = A such that σηu (t) has the following expansion around T for all u large enough σηu (t) = 1 − A(u)(T − t)β (1 + o(1)),

t ↑ T.

A2 : There exist two constants α ∈ (0, 2], B > 0 and a function B(u) satisfying limu→∞ B(u) = B such that the correlation function rηu (s, t) of ηu has the following expansion around T for all u large enough rηu (s, t) = 1 − B(u)|t − s|α (1 + o(1)),

min(s, t) ↑ T.

A3 : For some positive constants Q and γ , and all u large enough E (ηu (s) − ηu (t))2 ≤ Q|t − s|γ for any s, t ∈ [θ, T ]. Let {ηu (t), t ≥ 0} be a family of Gaussian processes satisfying Assumptions

Theorem 4.1 A1–A3.

(i) If β > α, then P



sup ηu (t) > u

t∈[θ,T ]

= √

1

Bα 1

2π A β

Hα Ŵ





 2  2 1 u − β2 −1 α exp − +1 u (1 + o(1)), β 2

as u → ∞.

K. D¸ebicki et al.

12

(ii) For β = α we have P



sup ηu (t) > u

t∈[θ,T ]



 2 A 1 u = √ PαB u −1 exp − (1 + o(1)), 2 2π

as u → ∞.

(iii) If β < α, then P



sup ηu (t) > u

t∈[θ,T ]

Proof of Theorem 4.1 large enough



 2 u 1 = √ u −1 exp − (1 + o(1)), 2 2π

as u → ∞.

Since from assumptions A1–A2 we have that, for any ε > 0 and u

(A − ε)(T − t)β (1 + o(1)) ≤ 1 − σηu (t) ≤ (A + ε)(T − t)β (1 + o(1)),

t↑T

and (B − ε)|t − s|α (1 + o(1)) ≤ 1 − rηu (s, t) ≤ (B + ε)|t − s|α (1 + o(1)),

min(s, t) ↑ T

Theorem D.3 in Piterbarg (1996) gives tight asymptotic upper and lower bounds, and thus the claims follow by letting ε → 0. 

Acknowledgements We are thankful to the referee for several suggestions which improved our manuscript. K. D¸ebicki was partially supported by NCN Grant No 2011/01/B/ST1/01521 (2011-2013). All the authors kindly acknowledge partial support by the Swiss National Science Foundation Grant 200021-1401633/1 and by the project RARE -318984, a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

References Adler, R. J. & Taylor, J. E. (2007). Random fields and geometry. New York: Springer. Asmussen, S. & Albrecher, H. (2010). Ruin probabilities, 2nd edn. New Jersey: World Scientific. Berman, M. S. (1992). Sojourns and extremes of stochastic processes. Boston: Wadsworth & Brooks/ Cole. Cramér, H. (1930). On the mathematical theory of risk. Stockholm: Skandia Jubilee Volume. Reprinted in: Martin-Löf, A. (Ed.) Cramér, H. (1994). Collected works. Berlin: Springer. Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stochastic Processes and their Applications 115, 207–248. D¸ebicki, K. (2002). Ruin probability for Gaussian integrated processes. Stochastic Processes and their Applications 98, 151–174. D¸ebicki, K. & Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. Journal of Applied Probability 40, 704–720. Embrechts, P., Klüpelberg, C. & Mikosch, T. (1997). Modeling extremal events for finance and insurance. Berlin: Springer. Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. The Annals of Applied Probability 23, 1506–1543.

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Griffin, P. S. & Maller, R. A. (2012). Path decomposition of ruinous behaviour for a general Lévy insurance risk process. The Annals of Applied Probability 22, 1411–1449. Hashorva, E. & Ji, L. (2014). Approximation of passage times of γ -reflected processes with fractional Brownian Motion as input. Journal of Applied Probability 51(3). In press. Hashorva, E., Ji, L. & Piterbarg, V. I. (2013). On the supremum of γ -reflected processes with fractional Brownian Motion as input. Stochastic Processes and their Applications 123, 4111–4127. He, X. & Hu, Y. (2007). Ruin probability for the integrated Gaussian process with force of interest. Journal of Applied Probability 44, 685–694. Hüsler, J. Exremes and ruin of Gaussian processes. In International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo, June 28–30, 2006. Hüsler, J. & Piterbarg, V. I. (1999). Extremes of a certain class of Gaussian processes. Stochastic Processes and their Applications 83, 257–271. Hüsler, J. & Piterbarg, V. I. (2004). On the ruin probability for physical fractional Brownian motion. Stochastic Processes and their Applications 113, 315–332. Hüsler, J. & Piterbarg, V. I. (2008). A limit theorem for the time of ruin in Gaussian ruin problem. Stochastic Processes and their Applications 118, 2014–2021. Hüsler, J. & Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statistics and Probability Letters 78, 1230–1235. Kobelkov, S. (2005). The ruin problem for the stationary Gaussian process. Theory of Probability and Its Applications 49, 155–163. Lundberg, F. (1903). Approximerad framställning av sannolikhetsfunktionen. Uppsala: Aterförsäkring av kollektivrisker. Akad. Afhandling. Almqvist och Wiksell. Mikosch, T. (2004). Non-Life insurance mathematics. An introduction with stochastic processes. Berlin: Springer. Piterbarg, V. I. (1996). Asymptotic methods in the theory of Gaussian processes and fields. Vol. 148, Translations of mathematical monographs. Providence, RI: AMS. Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. L. (1999). Stochastic processes for insurance and finance. Chichester: John Wiley and Sons. Segerdahl, C. O. (1955). When does ruin occur in the collective theory of risk? Scandinavian Actuarial Journal 1–2, 22–36.

Applied Financial Economics, 2011, 21, 1819–1829

GJR-GARCH model in value-at-risk of financial holdings Y. C. Sua, H. C. Huangb,* and Y. J. Lina a

Department of Finance, National Taiwan University, 50, Taipei, Taiwan Department of Finance, Chung Yuan Christian University, Chung Li, Taiwan

b

In this study, we introduce an asymmetric Generalized Autoregressive Conditional Heteroscedastic (GARCH) model, Glosten, Jagannathan and Runkle-GARCH (GJR-GARCH), in Value-at-Risk (VaR) to examine whether or not GJR-GARCH is a good method to evaluate the market risk of financial holdings. Because of lacking the actual daily Profit and Loss (P&L) data, portfolios A and B, representing FuBon and Cathay financial holdings are simulated. We take 400 observations as sample group to do the backward test and use the rest of the observations to forecast the change of VaR. We find GJR-GARCH works very well in VaR forecasting. Nonetheless, it also performs very well under the symmetric GARCH-in-Mean (GARCH-M) model, suggesting no leverage effect exists. Further, a 5-day moving window is opened to update parameter estimates. Comparing the results under different models, we find that the model is more accurate by updating parameter estimates. It is a trade-off between violations and capital charges. Keywords: market risk; value-at-risk; GARCH; GJR GARCH; financial holdings JEL Classification: G2; G21

I. Introduction Value-at-Risk (VaR) can determine market risk capital requirements for large banks and has been set through the Market Risk Amendment to the Basle Accord. It has become a standard measure of financial market risk (Jorion, 2006).1 There is plenty

of literature on VaR including statistical descriptions and examinations under the following models (Giot and Laurent, 2004; Li and Lin; 2004; Angelidis and Benos; 2006; Kuester et al., 2006; Alexander and Sheedy, 2008). First, historical simulation models utilize empirical quantiles by the available past data.2 Second, fully parametric models describe the entire

*Corresponding author. E-mail: [email protected] Nevertheless, Yamai and Yoshiba (2005) propose the ‘tail risk’ to document that VaR models are useless for measuring market risk when the events are in the ‘tails’ of distributions. Wong (2008) use a risk measure that takes into account the extreme losses beyond VaR, and expected shortfall proposed by Artzner et al. (1997, 1999) to remedy the shortcoming of VaR. 2 Pe´rignon and Smith (2010) find that historical simulation is the most popular VaR method in the world, as 73% of banks that disclose their VaR method report using historical simulation. Moreover, Pe´rignon et al. (2008) find that commercial banks exhibit a systematic excess of conservatism when setting their VaR, which contradicts the common wisdom that banks intentionally understate their market risk to reduce their market risk capital charges. 1

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online ß 2011 Taylor & Francis http://www.tandfonline.com DOI: http://dx.doi.org/10.1080/09603107.2011.595677

1819

1820 distributions of return volatility. Third, extreme value theory parametrically models only the tails of the return distribution. Fourth, quantile regression theory directly models a specific quantile rather than the whole return distribution. Because trading data is highly confidential,3 most of studies compare VaR modelling approaches and implementation procedures using illustrative portfolios (Hendricks, 1996; Marshall and Siegel, 1997; Pritsker, 1997). Berkowitz and O’Brien (2002) release the first detailed analysis of the performance models actually in use. They indicate that Generalized Autoregressive Conditional Heteroscedasticity (GARCH) in VaR outperform the six banks’ actual VaRs. Su et al. (2010) adopt some actual data of two active financial holdings and simulate two portfolios. After comparing VaR under five different models, they find that GARCH still outperforms other models. Su and Chiang (2004) try to put the leverage effect4 in VaR under the GARCH model. They introduce Exponential GARCH-in-Mean (EGARCH-M) to compare with GARCH-in-Mean (GARCH-M) and find that the asymmetric GARCH model is better in VaR forecasting than the symmetric GARCH model. According to the extant studies, there is an excellent performance in VaR forecasting5 under the GARCH model6 and the leverage effect exists. Since there are other asymmetric GARCH models that can also show the leverage effect, if we use the same data formation, do these results still hold under the different models? GARCH models vary in conditional variance equations. Does the form of mean equation affect the accuracy of VaR forecasting? As more and more parameters are considered in modelling, the model should be more complete and provide more information. Is a model with more parameters relatively good at forecasting? Speaking of providing more information, there is no more update in parameters after backward test.7 Su et al. (2010) 3

Y. C. Su et al. and Su and Chiang (2004) both use 400 observations to do backward testing and compare the rest 216 Profit and Loss (P&L) observations for forward test. Will it be more accurate by updating parameters? In order to answer those questions, we introduce the Glosten, Jagannathan and Runkle-GARCH-M (GJR-GARCH-M) model. Three forms of time series mean equations, which are Autoregressive Moving Average (ARMA)(1, 1), Autoregressive (AR)(1) and Moving Average (MA)(1) are adopted. Further, we introduce new information in backward testing by opening a 5-day moving window under GJR-GARCH-M. Therefore, we can compare the results under different models. The purposes of this article are as follows. First, GJR-GARCH model is introduced in VaR to examine whether or not it is a better method to evaluate the market risk of financial holdings. Second, we compare the results under different models to explore the most fitted one for each portfolio. Third, we update estimates by opening a 5-day moving window to improve the forward testing results. The conclusions are as follows. First, a strong GARCH effect exists in portfolios A and B. The coefficient of conditional variance is significant and large under all models. Second, GJR-GARCH-M and GARCH-M work very well in VaR forecasting. Third, new information does not always improve the model accuracy. When we use ARMA(1, 1) in mean equation, there is an overfitting problem. On the contrary, updating parameter estimates improves the accuracy of VaR modelling for portfolio A. It increases the number of violations and the aggregate violation for portfolio B at 95% confidence level. This is a trade-off between violations and capital charges.8 Fourth, the evidence of the leverage effect is not pronounced. The remainder of this article proceeds as follows. In Section II, we simulate two portfolios and generate our data. We illustrate all the models in Section III.

In response to the lack of ready-to-use data, Pe´rignon et al. (2008) develop a data extraction technique to extract the data from the graph included in the banks’ annual reports. Therefore, the daily VaR and P&L data are not anonymous. 4 The leverage effect means that the negative shock generally has a greater impact on stock return volatility than the positive shock does. 5 While many models of the accuracy of VaR are available in the literature, little is known on the accuracy of disclosed VaRs. Pe´rignon and Smith (2010) contribute to fill this gap. They find that although there is an overall upward trend in the quantity of information released to the public, the quality of VaR disclosure shows no sign of improvement over time. 6 Since financial returns exhibit three widely reported stylized facts, which are volatility clustering, substantial kurtosis and mild skewness of the returns, ‘standard’ methods, based on the assumption of iid-ness and normality, tend not to suffice, which has led to various alternatives for VaR prediction (for example, GARCH model). 7 Back testing a VaR model represents assessing its ex post performance using a sample of actual historical VaR and P&L data. 8 Cuoco and Liu (2006) document the financial institution chooses the VaR by trading off the cost of higher capital requirements in the current period resulting from a higher reported VaR against the benefit of a lower probability of higher capital requirements in the future as a result of a loss exceeding the reported VaR. Moreover, the minimum capital requirement is then equal to the sum of a charge to cover credit risk and a charge to cover general market risk.

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GJR-GARCH model in value-at-risk of financial holdings Table 1. Summary for operational income and net P&L for subsidiaries (TWD, in thousands) Panel A: Fu Bon Financial Holding Company Holding company Operational income Net P&L

79 539 359 9 427 427

Bank 13 211 721 3 983 650

Security 7 486 837 3 618 926

Property insurance 30 939 247 2 078 926

Panel B: Cathay Financial Holding Company Cathay Bank UWCCB Bank

Property insurance

Life insurance

Operational income Net P&L

8 108 382 162 558

315 348 275 6 459 991

7 563 820 873 107

25 719 751 10 287 215

Life insurance

Investment trust

27 401 422 407 850

500 433 153 805

Notes: This table summarizes related operational incomes and net P&L of the two portfolios. Panel A presents the portfolio A, which is a simulated sample of Fu Bon Financial Holding Company and Panel B shows the portfolio B, which is simulated from Cathay Financial Holding Company. We source those data from the third quarterly financial reports of individual subsidiary under the two financial holding companies in 2002.

Our empirical results are discussed in Section IV. Section V concludes.

II. Data The trading related data of financial holding companies in Taiwan are highly confidential and just part of them are public accessible. In order to solve the problem and to compare with the results of other methods, we form two simulated portfolios from framework of two financial holding companies as Su et al. (2010) conduct. The portfolio A is a simulated sample of Fu Bon Financial Holding Company and the portfolio B is simulated from Cathay Financial Holding Company. The related operational incomes and net P&L of the two portfolios are summarized in Table 1. Those data are sourced from the third quarterly financial reports of individual subsidiary under the two financial holding companies in 2002. The two companies are different in business scope and trading activities. Because there is lack of related trading data in public, we cannot just take their actual positions to fit the models. Therefore, we make some assumptions to simulate portfolios A and B. The two portfolios consist of only three asset classes, which are Foreign Exchange (FX), stocks and government bonds. Three asset classes are exactly the same trading instruments. The only difference between the two portfolios is the allocation of the three asset classes that are determined by raw data and our assumptions. We illustrate the assumptions of Su et al. (2010) to form the two portfolios. Thus, we can get daily P&L. The correlation coefficient of daily P&L in two portfolios is about 0.9571.

Assumptions by Su et al. (2010) According to Su et al. (2010), the assumptions are as follows. First, for banks, we assume that only 20% out of their operational incomes derive from the treasury department, which the main trading activities come from, and their main trading instruments are FX related. For simplicity, we choose the vanilla FX transaction, which means that no forward rate arrangement or option products have been included in the list. All currency pair will be booked in at their initial trading pair but will be converted to Taiwan Dollar (TWD) base, for accounting purpose, on calculation period. Second, we assume for securities, majority of the revenue comes from the brokerage. For a full license securities firm in Taiwan, they also could trade their own portfolio and government bonds. Hence we assume 80% of their revenue from brokerage, 15% from equity trading and 5% from the government bonds. Third, for property and life insurance, we assume 85% of the revenue is from the insurance related brokerage. 15% of the revenue is from the trading of their own portfolios and the spilt for instruments are 5% of government bonds and 10% of stocks. Fourth, for investment trust, we assume 50% of the revenue is from trading of the stocks. Fifth, all bonds are denoted in Taiwan Government bonds domicile in TWD. All securities are from Taiwan Stock Exchange or GreTai Securities Market. No derivatives for bonds and securities have been included in the portfolio for simplicity. Portfolio formation Based on raw data and the assumptions of Su et al. (2010), we can simulate two dummy portfolios, A and B, which represent Fu Bon and Cathay. Both of them

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Y. C. Su et al.

Table 2. Size and allocation categorized by investment asset classes (TWD, in thousands) Portfolio A

TWD ($)

Percentage

Panel A: Portfolio A FX Government bond Stock Sum Portfolio B

7 370 520 600 3 291 360 300 7 207 263 950 17 869 144 850 TWD ($)

41.25 18.45 40.33 100 Percentage

6 656 714 200 16 172 832 850 32 345 665 700 55 175 212 750 Portfolio A

12.07 29.31 58.62 100 Portfolio B

Panel B: Portfolio B FX Government bond Stock Sum

Panel C: Percentage of investment asset allocation for portfolios A and B FX 41.25% 12.07% Government bond 18.45% 29.31% Stock 40.33% 58.62% Sum 100% 100% Notes: This table presents the size and allocation of portfolios A and B, which represent Fu Bon and Cathay. Both of them contain three types of asset classes that are FX, stocks and government bonds.

contain three types of asset classes that are FX, stocks and government bonds. Size and allocation of portfolios A and B are shown in Table 2. The distribution of the two portfolios is different. Portfolio A nearly equally holds foreign exchange and stocks (42% and 40%), and comparatively fewer government bonds (18%). Portfolio B focuses on stock holdings (59%), government bonds (29%) in second and has a few foreign exchange exposures (12%). They are also different in size. The size of Portfolio B is almost three times as big as that of portfolio A. In order to eliminate the size effect, we presume that they are at the same size of total asset dollar volume, meaning that the size of the portfolio will not impact the performance result of VaR forecasting tools adopted. Then, we keep their distributions only and get a summary table for asset investment allocation of both portfolios A and B in Panel C of Table 2. Further, we assume that the investing instruments of the three asset classes are also identical except the proposition. We simulate the investing instruments of the two portfolios. As Su et al. (2010) conduct, 9

we select 74 stocks from Taiwan Stock Exchange list with a well spread over all the industry categories, eight Taiwan government bonds in cash base and the top five most commonly traded currency pairs in Taiwan, which are USD/TWD, USD/YEN, USD/ ECU, AUD/USD and CAD/USD. The positions for both portfolios can be established because we have their asset allocation and the simulated investing instruments. We have simulated the trading activities of both portfolios that have three asset classes contained exactly the same instruments and the investment dollar amounts of the individual instrument according to the proposition assigned and assumptions. Nonetheless, there are some exclusive conditions. First, market risk comes from the activities of trading and the price change of the underlying instruments. Thus, all brokerage incomes related items do not require a capital reserve for these activities. Second, we also exclude all the revenue from the retail mortgage business lines.9 Holding period and daily P&L The holding period of the simulated positions for portfolios A and B are both from 28 November 2000 to 15 April 2003. There are 617 working days during this period. We can obtain daily P&L by marking the investing instruments to the market everyday. The quantity of the investing instruments is the same and there is no position movement. Thus, we can have 617 observations in this sample period. We depart these observations into two groups in order to form our models and test model fitness. One group is used as sample data, called Sample Group. We take 400 observations for backward testing. The other one is Testing Group. The rest 216 observations are used for forward testing. We loose one observation in order to estimate the parameters. Finally, we can compare our forward testing results with Testing Group and explore whether or not GJR-GARCH model can capture daily P&L change more accurately. Table 3 presented the statistics summaries of the daily P&L returns for both portfolios. Portfolio A performs better because it has a greater mean than portfolio B and a lower SD. The magnitude of market risk is bigger for portfolio B because its 95th and 99th percentile is more downward. Not only the daily P&L distributions of the two portfolios but also those of the two Sample Groups are not in a

The reasons are as follows. First, the majority of the retail loans are prefixed at a ‘floated’ foundation in Taiwan. That means banks are allowed to adjust the rates whenever the primary rate are adjusted. Therefore, banks are always protected by a spread and there is less price volatility. Second, retail funding is more related in credit risk management area based in Basle II amendments.

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GJR-GARCH model in value-at-risk of financial holdings Table 3. Daily P&L returns statistics summary Obs.

Mean

SD

95th percentile

Panel A: Portfolio A and B (28 November 2000–15 April 2003) Portfolio A 617 0.008849% 0.672722% 1.046497% Portfolio B 617 0.002237% 0.958983% 1.571410% Panel B: Sample Groups (28 November 2000–1 March 2003) Sample Group of Portfolio A 400 0.008476% 0.718468% 1.117172% Sample Group of Portfolio B 400 0.005928% 0.993236% 1.721987%

99th percentile

Skewness

Excess kurtosis

1.558714%

0.003136

1.823645

2.127068%

0.182604

0.501733

1.604896%

0.017258

1.745985

2.191302%

0.185259

0.214208

Notes: The holding period of the simulated positions for portfolios A and B are both from 28 November 2000 to 15 April 2003. There are 617 working days during this period. We can obtain daily P&L by marking the investing instruments to the market everyday. The quantity of the investing instruments is the same and there is no position movement. Then, we can have 617 observations in this period. We depart these observations into two groups in order to form our models and test model fitness. One group is used as sample data, called Sample Group. We take 400 observations for backward testing. The other one is Testing Group. The rest 216 observations are used for forward testing. We loose one observation in order to estimate the parameters. Finally, we can compare our forward testing results with Testing Group and see if GJR-GARCH model can capture daily P&L change more accurately. There are statistics summaries of the of daily P&L returns for both portfolios in this table.

normal distribution. Portfolio A has a slightly negative skewness; tail to the left, and portfolio B has a positive skewness; tail to the right. The excess kurtosis of the distributions is greater than 0. It shows that both portfolios may follow the GARCH process.

III. Methodology As mentioned in the literature review, Su et al. (2010) find that Historical Simulation and AR(1, 1)GARCH-M(1, 1) model both perform very well in predicting the change in volatility. But the GARCH model needs less reserve than Historical Simulation. Su and Chiang (2004) test the VaR of two simulated portfolios by ARMA(1, 1)–GARCH-M(1, 1) model and ARMA(1, 1)– EGARCH-M(1, 1) model. They not only prove that the GARCH model fitted the P&L distributions of the two portfolios better but also indicate that the asymmetry GARCH (EGARCH) predicts the change in volatility even better. In order to compare the extant studies, we adopt one of the asymmetry GARCH families, the GJR-GARCH-M model, to test the VaR of the two simulated portfolios. We examine whether or not GJR-GARCH-M is better than former. Sample Group is used to obtain the estimates and to form the equations. We use the equations to forecast the out-sample returns from day 401 to day 616 and calculate the VaR under 95% and 99%

confidence levels. We compare the forward testing VaR with Testing Group to find out the best model to calculate VaR and to evaluate the market risk. The major differences between our article and the extant studies are as follows. First, we adopt three forms of the mean equations, including ARMA(1, 1), AR(1) and MA(1) to explore how the different mean equation affects the return estimates. Second, we open a 5-day moving window in estimating parameters. That is, every 5 days, the parameters are estimated again. Two groups of VaR, which are VaR without updating and VaR with updating, are collected. Theoretically, the more complicate model is associated with the better result in model fitting and updating the estimating parameters should provide some improvement in forecasting. GJR-GARCH-M(1, 1) The GJR-GARCH-M model is developed by Glosten et al. (1993). They examine the intertemporal relation between risk and return. When the standard GARCH-M framework is used, there is a positive but insignificant relation between the conditional mean and the conditional variance of the excess return on shocks. They modify the GARCH-M model by allowing seasonal patterns in volatility, positive and negative unanticipated returns having different impacts on the conditional variance, and nominal interest rates to predict conditional variance. They find a negative relation between conditional expected monthly returns and conditional variance of monthly returns. Moreover, positive unanticipated

1824 returns appear to result in a downward revision of the conditional volatility, whereas negative unanticipated returns result in an upward revision of conditional volatility. In contrast, Nelson (1991) and Engle and Ng (1993) find that large positive as well as negative unanticipated returns led to an upward revision in the conditional volatility although negative shocks of similar magnitude lead to larger revisions. The model proposed by Glosten et al. (1993) is called GJR-GARCH-M. In this article, we adopt GJR-GARCH-M(1, 1) with three forms of the mean equations. They are ARMA(1, 1)-GJR-GARCH-M(1, 1), AR(1)-GJRGARCH-M(1, 1), and MA(1)-GJR-GARCHM(1, 1). We assume that the conditional return may be affected by the last period conditional return and the last period residual. We consider AR process and MA model, both or individually. Those models can be expressed as follows: ARMA(1, 1)-GJR-GARCH-M(1, 1): Rt ¼  þ   ht1 þ Volume  Rt1 þ "t  MA  "t1 8 A þ Bð1Þ  ht1 þ Cð1Þ  ð1 þ Cð2ÞÞ  "2t1 , > > > < " 0 t1 ht ¼ 2 > > > A þ Bð1Þ  ht1 þ Cð1Þ  ð1  Cð2ÞÞ  "t1 , : "t1 5 0 ð1Þ AR(1)-GJR-GARCH-M(1, 1): Rt ¼  þ   ht1 þ Volume  Rt1 þ "t 8 A þ Bð1Þ  ht1 þ Cð1Þ  ð1 þ Cð2ÞÞ  "2t1 , > > > < " 0 t1 ht ¼ 2 > > > A þ Bð1Þ  ht1 þ Cð1Þ  ð1  Cð2ÞÞ  "t1 , : "t1 5 0 ð2Þ MA(1)-GJR-GARCH-M(1, 1): Rt ¼  þ   ht1 þ "t  MA  "t1 8 A þ Bð1Þ  ht1 þ Cð1Þ  ð1 þ Cð2ÞÞ  "2t1 , > > > < " 0 t1 ht ¼ 2 > > > A þ Bð1Þ  ht1 þ Cð1Þ  ð1  Cð2ÞÞ  "t1 , : "t1 5 0 ð3Þ 10

Y. C. Su et al. where Rt is P&L return at time t, ht is the conditional variable at time t, and "t is the residual at time t. If C(2) is negative, the negative residual leads to a greater impact on conditional variance than the positive one does.10 In GJR-GARCH-M, there is an asymmetric effect on the volatility change. Unexpected positive shocks lead to a smaller conditional volatility in the next period than negative shocks do. GARCH-M(1, 1) Engle et al. (1987) firstly include the conditional variance to the conditional mean equation and propose the ARCH-in-Mean model (ARCH-M). Applying this kind of mean equations in GARCH model, we can get GARCH-M. Mean equations have been widely used in time-varying risk premia and the behaviour of stock return variances. For example, Bollerslev (1987) use it to examine foreign exchange rates and stock indices. Besides, EGARCH-M proposed by Nelson (1991), the quadratic GARCH process of Sentana (1995) and Engle (1990), and Threshold GARCH-M (TGARCH-M) proposed by Zakoian (1991) are famous asymmetric GARCH models. Since we adopt the GJR-GARCH-M model in our study, the GARCH-M model is used for comparison. GARCH-M(1, 1) is adopted with three types of mean equations and they can be written as ARMA(1, 1)-GARCH-M(1, 1) Rt ¼  þ   ht1 þ Volume  Rt1 þ "t  MA  "t1 ht ¼ A þ Bð1Þ  ht1 þ Cð1Þ  "2t1

ð4Þ

AR(1)-GARCH-M(1, 1) Rt ¼  þ   ht1 þ Volume  Rt1 þ "t ht ¼ A þ Bð1Þ  ht1 þ Cð1Þ  "2t1

ð5Þ

MA(1)-GARCH-M(1, 1) Rt ¼  þ   ht1 þ "t  MA  "t1 ht ¼ A þ Bð1Þ  ht1 þ Cð1Þ  "2t1

ð6Þ

where Rt is P&L return at time t, ht is the conditional variable at time t, and "t is the residual at time t. Based on the two GARCH models, we can generate two sets of estimates. Comparing with our simulated P&L in Testing Group, we can figure out

There may be different expression of conditional variance  A þ Bð1Þ  ht1 þ Cð1Þ  ð1  Cð2ÞÞ  "2t1 , ht ¼ A þ Bð1Þ  ht1 þ Cð1Þ  ð1 þ Cð2ÞÞ  "2t1 ,

"t1  0 "t1 5 0

Still the news impact curve is steeper for negative shocks than it is for positive shocks when C(2) is positive (Hentschel, 1995). There is no difference to change the sign expression of C(2).

GJR-GARCH model in value-at-risk of financial holdings which model captures the change in volatility better. Then we open the 5-day moving window only in the GJR-GARCH-M model and test whether or not updating in estimates can improve the forward testing ability.

IV. Empirical Results Backward test In Sample Groups, we estimate parameters under GJR-GARCH-M and GARCH-M. The untabulated results show that, in conditional variance equation, there are four parameters in GJR-GARCH-M and three parameters in GARCH-M. B(1) is the coefficient of the conditional variance in the last period. It is obviously significant under GJR-GARCH-M and GARCH-M. Value range of B(1) is from 0.84 to 0.91, indicating that the conditional variance is highly affected by the previous one and is able to adjust from time to time. C(2) is the leverage effect, which provides the impact of unexpected shock on conditional variance. Although the coefficients of C(2) are all negative, none of them are significant, implying that we could conclude there is no strong evidence to support the leverage effect in this set of estimates. The coefficient of the residual squares in the last period, C(1), is almost significant under different models. The coefficients of MA and Volume are significant in ARMA(1, 1) models. Moving average is a short-term serial correlated effect. MA(1) means that the effect will die out in one period. We interpret MA as the effect of economic factors such as the change of foreign exchange rate and the change of government financial policies, which provide the temporary shock. In ARMA(1, 1)-GJR-GARCH-M, MA contributes a significantly negative effect to portfolio A (coefficient is 0.9332) and a positive effect to portfolio B (coefficient is 0.8146).11 Both portfolios A and B are sensitive to the economic policy changes in spite of different directions. On the contrary, they are sensitive in the same positive direction MA under ARMA(1, 1)-GARCH-M. Volume is the coefficient of autoregressive process. The autoregressive process would not die out but tail off as time goes by. AR(1) means the return is affected by the last period return. In ARMA(1, 1)GJR-GARCH-M(1, 1), the autoregressive process

1825

contributes the opposite effect to portfolios A and B. The last period return provides a negative effect in portfolio A and a positive effect in portfolio B. Nonetheless, it contributes positive effect for both portfolios under ARMA(1, 1)-GARCH-M. Although there are many insignificant estimates, we still can use those estimates to do forward test because the key parameter B(1) is significant and large. We also meet the stable condition of GARCH, i.e. B(1) plus C(1) is smaller than one. We will have the forward test results under GJR-GARCH-M and GARCH-M in the next sections. Value-at-Risk in GJR-GARCH-M Tables 4 to 6 present the results under mean equation of ARMA(1, 1), AR (1), and MA(1) at 99% and 95% confidence level respectively. At 99% confidence level, within 216 observations, it only allows two violations. But both portfolios under ARMA(1, 1)GJR-GARCH-M(1, 1) have more violations (five for portfolio A and four for portfolio B). Nevertheless, if we only consider AR(1) or MA(1) in mean equation, the number of violation is decreased to one. Mean VaR12 provides guidance toward capital charge. Under ARMA(1, 1)-GJR-GARCH-M(1, 1), portfolio A needs to charge 1.686146% on the average but only 1.530507% under AR(1)-GJR-GARCH-M(1, 1) and 1.532314% under MA(1)-GJR-GARCH-M(1, 1). Since the P&L distribution of portfolio B is more volatile, portfolio B needs more capital charges than portfolio A. On average, it needs 2.435960% under ARMA(1, 1)-GJR-GARCH-M(1, 1), 2.255777% under AR(1)-GJR-GARCH-M(1, 1) and 2.257213% under MA(1)-GJR-GARCH-M(1, 1). At 95% confidence level, within 216 observations, it allows 10 to 11 violations. ARMA(1, 1)-GJR-GARCH-M(1, 1) still has more violation numbers and large mean VaR. In Table 7, we use the observed violation rate and targeted violation rate proposed by Berkowtz and O’Brien (2002) to make the comparison straightforward. Among all the models, ARMA(1, 1)-GJRGARCH-M(1, 1) has the worst forecasting performance, implying that an overfitting problem exists. Moreover, according to the observed violation rates of portfolios A and B in Table 8, we also find that ARGARCH or MA-GARCH model predicts better than ARMA-GARCH model. It is possible that the model becomes more complete but looses the degree of freedom by adding parameters. The simple model is

11 There is a minus sign before MA in our models. If MA is positive, good news leads to the downward revise of return and bad news leads to the upward revise of return. If MA is negative, good news leads return enlarging and bad news leads return narrowing. 12 The VaR there is not a value-at-risk value. In order to compare with the P&L return, we discuss it in the percentage form. bt1  2:33  b It is calculated as VaRt1 ¼ R t1 at 99% confidence level.

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Y. C. Su et al.

Table 4. VaR comparison under mean equation of ARMA(1, 1) Obs.

Mean VaR

No. of violations

Mean violation

99% confidence level ARMA(1,1)-GJR-GARCH-M(1,1) Portfolio A 216 1.686146% 5 1.060350% Portfolio B 216 2.435960% 4 2.023542% ARMA(1.1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.522111% 3 1.181502% Portfolio B 216 2.300891% 4 2.073424% ARMA(1,1)-GARCH-M(1,1) Portfolio A 216 1.709536% 4 1.200011% Portfolio B 216 2.447956% 5 1.657666% 95% confidence level ARMA(1,1)-GJR-GARCH-M(1,1) Portfolio A 216 1.180043% 12 0.809408% Portfolio B 216 1.709438% 12 1.182317% ARMA(1,1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.058779% 10 0.963102% Portfolio B 216 1.606919% 14 1.332137% ARMA(1,1)-GARCH-M(1,1) 12 0.741928% Portfolio A 216 1.200873% Portfolio B 216 1.718310% 14 0.978065%

Max violation

Aggregate violation

2.464230% 3.667115%

5.301749% 8.094170%

2.464230% 3.667115%

3.544505% 8.293696%

2.464230% 3.667115%

4.800043% 8.288328%

2.464230% 3.667115%

9.712891% 14.187802%

2.464230% 3.667115%

9.631015% 18.649912%

2.464230% 3.667115%

8.903134% 13.692914%

Notes: The table presents the statistics of VaRs of portfolios A and B obtained from the mean equation of ARMA(1,1) at the 95% and 99% confidence levels. We use Sample Group to obtain the estimates and form the equations.

Table 5. VaR comparison under mean equation of AR(1) Obs.

Mean VaR

No. of violations

Mean violation

99% confidence level AR(1)-GJR-GARCH-M(1,1) Portfolio A 216 1.530507% 1 2.464230% Portfolio B 216 2.255777% 1 3.667115% AR(1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.476896% 1 2.464230% Portfolio B 216 2.180534% 1 3.667115% AR(1)-GARCH-M(1,1) Portfolio A 216 1.531998% 1 2.464230% Portfolio B 216 2.261074% 1 3.667115% 95% confidence level AR(1)-GJR-GARCH-M(1,1) Portfolio A 216 1.082025% 4 1.545075% Portfolio B 216 1.602807% 6 2.020295% AR(1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.032792% 5 1.413185% Portfolio B 216 1.514408% 11 1.645233% AR(1)-GARCH-M(1,1) 4 1.545075% Portfolio A 216 1.083027% Portfolio B 216 1.604870% 6 2.020295%

Max violation

Aggregate violation

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

6.180301% 12.121773%

2.464230% 3.667115%

7.065924% 18.097565%

2.464230% 3.667115%

6.180301% 12.121773%

Notes: The table presents the statistics of VaRs of portfolios A and B obtained from the mean equation of AR(1) at the 95% and 99% confidence levels. We use Sample Group to obtain the estimates and form the equations.

more powerful than the complicated one. Overfitting is more evident in backward test. VaR matches daily P&L very well in AR(1)-GJR-GARCH-M(1, 1) and MA(1)-GJR-GARCH-M(1, 1). However, it has

30 to 70 violations under ARMA(1, 1)-GJRGARCH-M(1, 1). Thus, the bad performance of ARMA(1, 1)-GJR-GARCH-M(1, 1) might be due to overfitting.

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GJR-GARCH model in value-at-risk of financial holdings Table 6. VaR comparison under mean equation of MA(1) Obs.

Mean VaR

No. of violations

Mean violation

99% confidence level MA(1)-GJR-GARCH-M(1,1) Portfolio A 216 1.532314% 1 2.464230% Portfolio B 216 2.257213% 1 3.667115% MA(1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.478412% 1 2.464230% Portfolio B 216 2.182488% 1 3.667115% MA(1)-GARCH-M(1,1) Portfolio A 216 1.533595% 1 2.464230% Portfolio B 216 2.261924% 1 3.667115% 95% confidence level MA(1)-GJR-GARCH-M(1,1) Portfolio A 216 1.083306% 5 1.456162% Portfolio B 216 1.604098% 5 2.104026% MA(1)-GJR-GARCH-M(1,1) with a 5-day moving window update Portfolio A 216 1.033807% 6 1.369418% Portfolio B 216 1.516356% 11 1.645233% MA(1)-GARCH-M(1,1) 5 1.456162% Portfolio A 216 1.084171% Portfolio B 216 1.605739% 6 2.020295%

Max violation

Aggregate violation

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

2.464230% 3.667115%

7.280808% 10.520129%

2.464230% 3.667115%

8.216507% 18.097565%

2.464230% 3.667115%

7.280808% 12.121773%

Notes: The table presents the statistics of VaRs of portfolios A and B obtained from the mean equation of MA(1) at the 95% and 99% confidence levels. We use Sample Group to obtain the estimates and form the equations.

Table 7. The observed violation rates and targeted violation rates in VaR comparison under mean equation of ARMA(1,1), AR(1) and MA(1) models

Panel A: 99% confidence level ARMA(1, 1)-GJR-GARCH-M(1, 1) updating ARMA(1.1)-GJR-GARCH-M(1, 1) ARMA(1, 1)-GARCH-M(1, 1) AR(1)-GJR-GARCH-M(1, 1) updating AR(1)-GJR-GARCH-M(1, 1) AR(1)-GARCH-M(1, 1) MA(1)-GJR-GARCH-M(1, 1) updating MA(1)-GJR-GARCH-M(1, 1) MA(1)-GARCH-M(1, 1) Panel B: 95% confidence level ARMA(1, 1)-GJR-GARCH-M(1, 1) updating ARMA(1.1)-GJR-GARCH-M(1, 1) ARMA(1, 1)-GARCH-M(1, 1) AR(1)-GJR-GARCH-M(1, 1) updating AR(1)-GJR-GARCH-M(1, 1) AR(1)-GARCH-M(1, 1) MA(1)-GJR-GARCH-M(1, 1) updating MA(1)-GJR-GARCH-M(1, 1) MA(1)-GARCH-M(1, 1)

Targeted violation rate

Observed violation rate of portfolio A

Observed violation rate of portfolio B

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

0.023 0.014 0.019 0.005 0.005 0.005 0.005 0.005 0.005

0.019 0.019 0.023 0.005 0.005 0.005 0.005 0.005 0.005

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.056 0.046 0.056 0.019 0.023 0.019 0.023 0.028 0.023

0.056 0.065 0.065 0.028 0.051 0.028 0.023 0.051 0.028

Notes: The table presents the observed violation rates and targeted violation rates of VaRs of portfolios A and B obtained from the mean equation of MA(1) at the 95% and 99% confidence levels. We use Sample Group to obtain the estimates and form the equations.

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Y. C. Su et al. Table 8. The observed violation rates and targeted violation rates in VaR comparison under ARMA(1,1)-GARCH, AR(1)-GARCH, and MA(1)-GARCH models

Panel A: 99% confidence level ARMA(1,1)-GARCH AR(1)-GARCH MA(1)-GARCH Panel B: 95% confidence level ARMA(1,1)-GARCH AR(1)-GARCH MA(1)-GARCH

Targeted violation rate

Observed violation rate of portfolio A

Observed violation rate of portfolio B

0.010 0.010 0.010

0.046 0.005 0.005

0.023 0.014 0.014

0.050 0.050 0.050

0.125 0.023 0.023

0.065 0.079 0.079

Notes: The table presents the observed violation rates and targeted violation rates of VaRs of portfolios A and B at the 95% and 99% confidence levels. We use Sample Group to obtain the estimates and form the equations.

The distribution of VaR follows the change of P&L. It is more volatile under ARMA(1, 1)GJR-GARCH-M(1, 1), less volatile under AR(1)GJR-GARCH-M(1, 1) and smooth under MA(1)-GJR-GARCH-M(1, 1). Although portfolio B needs more capital charge, the volatility of VaR distribution is smaller under AR(1) and MA(1) since the estimate parameters of portfolios A and B are different. For portfolio B, B(1) and  become bigger under AR(1) and MA(1) models. Conditional variance is more affected by the conditional variance in the last period. The larger  is associated with stronger effect of conditional variance. The above conditions make conditional returns of portfolio B bigger and more consistent under AR(1)-GJRGARCH-M(1, 1) and MA(1)-GJR-GARCH-M(1, 1) than those under ARMA(1, 1)-GJR–GARCHM(1, 1). If a 5-day moving window is opened for updating estimates, we find that VaR distribution is more sensitive to P&L changes. It needs less capital charges for both portfolios whatever model is applied. Nonetheless, there are also some flaws with updating estimates. At 95% confidence level, portfolio B with a 5-day moving window updating suffers more violations under MA(1)-GJR-GARCH-M(1, 1) and AR(1)-GJR-GARCH-M(1, 1).

Value-at-Risk in GARCH-M For the comprising model GARCH-M, we find that the VaR distributions are quite similar with GJR-GARCH-M. For example, at 95% confidence level, there is one more violation for portfolio B under MA(1)-GARCH-M(1, 1) than that for MA(1)GJR-GARCH-M(1, 1). GARCH-M only underperforms GJR-GARCH-M when using ARMA(1, 1) as the mean equation. It needs to reserve more capital

charges under ARMA(1, 1)-GARCH-M(1, 1) than under ARMA(1, 1)-GJR-GARCH-M(1, 1). Model comparison Overfitting problem exists when we adopt ARMA(1, 1) in mean equations. There will be more violations and larger capital charges whatever under GJR-GARCH-M or GARCH-M. Although the situation will be better by updating parameter estimates for portfolio A, there are still many violations. The number of violations for portfolio A reduces to three at 99% confidence level and 10 at 95% confidence level. The number of violations for portfolio B keeps the same as four at 99% confidence level and increases to 14 at 95% confidence level. Open a 5-day moving window does help VaR forecasting. It reduces the number of violations and capital charges. However, for portfolio B at 95% confidence level, updating does not work very well. Although there are less capital charges, the violations increase simultaneously. There is a trade-off between violations and capital charges.

V. Conclusion We adopt the asymmetric GARCH model, GJR-GARCH-M, to represent the leverage effect. Under the assumption of Su et al. (2010), we form the same simulated portfolios. By using three forms of mean equations – ARMA, AR and MA, we find that the mean equation is associated with VaR performance. We take GARCH-M(1, 1) as a benchmark for comparison and open a 5-day moving window to update parameter estimates. In this article, there are nine different models, which are ARMA(1, 1)-GJRGARCH-M(1, 1), AR(1)-GJR-GARCH-M(1, 1),

GJR-GARCH model in value-at-risk of financial holdings MA(1)-GJR-GARCH-M(1, 1), ARMA(1, 1)-GJRGARCH-M(1, 1) with a 5-day moving window update, AR(1)-GJR-GARCH-M(1, 1) with a 5-day moving window update, MA(1)-GJR-GARCHM(1, 1) with a 5-day moving window update, ARMA(1, 1)-GARCH-M(1, 1), AR(1)-GARCHM(1, 1) and MA(1)-GARCH-M(1, 1). We adopt those models in VaR forecasting and compare the results under different models. The conclusions are as follows. First, a strong GARCH effect exists in portfolios A and B. The coefficients of conditional variance are significant and large under all models. Second, GJR-GARCH-M works very well in VaR forecasting and so does GARCH-M. Third, introducing new information does not always improving the model accuracy. When we use ARMA(1, 1) in mean equation, there is an overfitting problem. On the other hand, updating parameter estimates improves the accuracy of VaR modelling for portfolio A. Although it increases the number of violations and the aggregate violation for portfolio B at 95% confidence level, there is a trade-off between violations and capital charges. Fourth, the evidence of the leverage effect is not pronounced. In backward testing, C(2) is negative but not significant. In forward testing, ARMA(1, 1)-GJR-GARCHM(1, 1) only slightly outperform ARMA(1, 1)GARCH-M(1, 1).

References Alexander, C. and Sheedy, E. (2008) Developing a stress testing framework based on market risk models, Journal of Banking and Finance, 32, 2220–36. Angelidis, T. and Benos, A. (2006) Liquidity adjusted value-at-risk based on the components of the bid-ask spread, Applied Financial Economics, 16, 835–51. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1997) Thinking coherently, Risk, 10, 68–71. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1999) Coherent measures of risk, Mathematical Finance, 9, 203–28. Berkowitz, J. and O’Brien, J. (2002) How accurate are value-at-risk models at commercial banks?, Journal of Finance, 57, 1093–112. Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return, The Review of Economics and Statistics, 69, 542–7. Cuoco, D. and Liu, H. (2006) An analysis of VaR-based capital requirements, Journal of Financial Intermediation, 15, 362–94. Engle, R. F. (1990) Discussion: stock market volatility and the crash of 87, Review of Financial Studies, 3, 103–6.

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Engle, R., Lilien, D. and Robins, R. (1987) Estimating time varying risk premia in the term structure: the ARCH-M model, Econometrica, 55, 391–407. Engle, R. F. and Ng, V. K. (1993) Measuring and testing the impact of news on volatility, Journal of Finance, 48, 1749–78. Giot, P. and Laurent, S. (2004) Modelling daily value-atrisk using realized volatility and ARCH type models, Journal of Empirical Finance, 11, 379–98. Glosten, L., Jagannathan, R. and Runkle, D. (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance, 48, 1179–801. Hendricks, D. (1996) Evaluation of value-at-risk models using historical data, Economic Policy Review, 39, 36–69. Hentschel, L. (1995) All in the family nesting symmetric and asymmetric GARCH models, Journal of Financial Economics, 39, 71–104. Jorion, P. (2006) Value-at-Risk: The New Benchmark for Managing Financial Risk, 3rd edn, McGraw-Hill, New York. Kuester, K., Mittnik, S. and Paolella, M. S. (2006) Value-at-risk prediction: a comparison of alternative strategies, Journal of Financial Econometrics, 4, 53–89. Li, M. Y. L. and Lin, H. W. W. (2004) Estimating value-atrisk via Markov switching ARCH models: an empirical study on stock index returns, Applied Economics Letters, 11, 679–91. Marshall, C. and Siegal, M. (1997) Value-at-risk: implementing a risk measurement standard, Journal of Derivatives, 4, 91–111. Nelson, D. B. (1991) Conditional heteroskedasticity in asset returns, Econometrica, 59, 347–70. Pe´rignon, C., Deng, Z. Y. and Wang, Z. J. (2008) Do banks overstate their value-at-risk?, Journal of Banking and Finance, 32, 783–94. Pe´rignon, C. and Smith, D. R. (2010) The level and quality of value-at-risk disclosure at commercial banks, Journal of Banking and Finance, 34, 362–77. Pritsker, M. (1997) Evaluating value-at-risk methodologies: accuracy versus computational time, Journal of Financial Services Research, 12, 201–42. Sentana, E. (1995) Quadratic ARCH models, Review of Economic Studies, 62, 639–61. Su, Y. C. and Chiang, H. C. (2004) Modeling value-at-risk of financial companies – a comparison of symmetric and asymmetric models, Working Paper, National Taiwan University. Su, Y. C., Huang, H. C. and Wang, S. (2010) Modeling value at risk of financial holding company: time varying versus traditional models, Banks and Bank Systems, 5, 15–25. Wong, W. K. (2008) Back testing trading risk of commercial banks using expected shortfall, Journal of Banking and Finance, 32, 1404–15. Yamai, Y. and Yoshiba, T. (2005) Value-at-risk versus expected shortfall: a practical perspective, Journal of Banking and Finance, 29, 997–1101. Zakoian, J. M. (1991) Threshold heteroskedastic models, Journal of Economic Dynamics and Control, 18, 931–55.

Journal of Risk ResearchAquatic Insects Vol. 14, No. 10, November 2011, 1191–1205

Health and finance: exploring the parallels between health care delivery and professional financial planning Patrick Parnaby* University of Guelph, Guelph, Ontario, Canada (Received 20 January 2011; final version received 10 May 2011) This article compares the role of risk and uncertainty discourse across two rather disparate areas of expertise: personal financial planning and health care delivery. On the basis of 42 semi-structured interviews and eight recorded meetings between planners and their clients, as well as a comprehensive review of extant health literature, it is argued that three points of convergence – the discursive management of uncertainty, the temporalization of risk, and the use of images – offer a modicum of certainty which functions ultimately to legitimize expertise and facilitate courses of remedial action while simultaneously assuaging laypeople’s existential anxieties. The article concludes with a number of reflections on the importance of agency and trust. Keywords: risk; medicine; finance; expertise; professions; uncertainty

Introduction If, as Giddens (1991) suggests, the body is perennially at risk, then the abundance of health-related research pertaining to how medical experts engage in ‘risk talk’ with their patients should not be surprising. There are, however, areas of expertise outside the realm of health care delivery, where ‘focused encounters’ (Candlin and Candlin 2002) between experts and clients occur frequently and risk and uncertainty discourse is equally prominent; for example, the field of professional financial planning. Conceptual advances made by health scholars help us understand the role of risk and uncertainty discourse between financial planners and their clients. Using data gathered from 42 semi-structured interviews with planners and eight recorded meetings between planners and their clients, this article examines three critical areas in which health scholars’ observations regarding the role of risk and uncertainty discourse overlap with, and shed light on, the relational dynamics of professional financial planning. Specifically, there are intriguing similarities with respect to the discursive management of uncertainty, the temporalization of risk, and the use of images. It will be argued that these discursive phenomena construct degrees of certainty which function to bolster professional expertise and facilitate strategic courses of action while mitigating simultaneously existential anxieties.

*Email: [email protected] ISSN 1366-9877 print/ISSN 1466-4461 online Ó 2011 Taylor & Francis http://dx.doi.org/10.1080/13669877.2011.587887 http://www.tandfonline.com

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Connecting health-related research and financial planning Apart from industry magazines and various online publications (see Bachrach 2004; Dorfman 2000; McGuigan and Eisner 2003), literature on financial planners and their clients from a sociological perspective is virtually nonexistent. To be clear: literature on planning strategy, insurance, investments, risk management, investor psychology, and other matters relating to the act of planning is astonishingly large, but theoretically grounded research that examines the social relations between planners and clients and that seeks to understand those relations in some sort of broader context is rare (except see Parnaby 2009). Many financial planners offer ‘full-service’ planning: that is, they offer a wide range of products and/or services to meet their clients’ short- and long-term financial and lifestyle needs (e.g. estate, tax, insurance, and retirement planning). Discussions about investment strategies and related risks usually take place within the context of a holistic financial assessment relative to a client’s position in the lifecycle. Similar dynamics unfold in health care delivery, where physicians seek to identify and assess their clients’ health-related risks. Thus, both professions require the making of diagnoses and prognoses as part of the expert/layperson encounter before moving on to assembly of an effective risk-management strategy. In terms of generic social processes, the two professions appear to have elements in common; however, in the absence of scientific research on the behavioral dynamics between financial planners and clients, decades of health care scholarship provides an intriguing point of analytical departure when it comes to the aforementioned points of comparison.

Medical expertise and uncertainty Health care expertise is not tied exclusively to technical competence. Beyond administering medication, performing surgery, or providing clinical bedside care, there is a highly nuanced and occupationally specific discourse that must be mastered; specifically, physicians must learn to manage medicine’s inherent uncertainty because: medical uncertainty raises emotionally and existentially charged questions about the meaningfulness as well as the efficacy of physicians’ efforts to safeguard their patients’ well-being, relieve their suffering, heal their ills, restore their health, and prolong their lives. (Fox 2000, 409)

The connection between uncertainty management and the bolstering of medical expertise has been well documented (Candlin and Candlin 2002; Cicourel 1999; Fox 1957, 2000; Malcolm 2009; Sarangi and Roberts 1999).1 Malcolm’s (2009) recent study of sports medicine, for example, demonstrates how physicians were able to manage medical uncertainty surrounding what constitutes a concussion by, in some cases, intellectualizing the issue and/or making references to the unfavorable conditions under which they were practicing. While leaving their expertise essentially intact, their discursive strategies allowed them to embrace definitions of concussion that were in line with dominant sports culture and that did not undermine their relations with coaches and players. The importance of managing uncertainty is also revealed in Sarangi and Clarke’s (2002) research on genetics communication, Sarangi et al.’s (2003) work on genetics counseling, Dillard and Carson’s (2005) study of positive screening for cystic fibrosis in newborns and, to a

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lesser extent, Rafalovich’s (2005) work on uncertainty and the diagnosis of attention deficit hyperactivity disorder. Paradoxically, however, uncertainty can also be an expert’s source of power insofar as it sometimes tangles ‘public criticism and the mechanism[s] for professional self-examination and policing’ (Nilson 1979, 577). The power stems from what is believed to be a collective, epistemological uncertainty experienced by all professionals in their relevant field (e.g. ‘We just don’t know’) as opposed to an individual declaration of not knowing (e.g. ‘I have no idea’). If a sense of collective uncertainty is disclosed and managed convincingly, professionals may be able to create a kind of ‘wiggle room’ within which they can explore the boundaries of their knowledge while keeping accountability partially at bay (see Malcolm 2009). Learning how to manage uncertainty to remain an expert in the eyes of others remains fundamental to the practice of modern medicine (Fox 2000). Temporalizing health-related risks According to Luhmann (2008), however probable or improbable an event is thought to be, one has no way of knowing with absolute certainty when that event will transition from a potentiality to a reality. ‘Even if one knows that one suffers a fatal accident driving on the motorway only every twelve million kilometers’ Luhmann argues, ‘death could still be waiting around the next bend’ (49). Likewise, knowing the probability of illness leaves us no further ahead when it comes to knowing exactly when we will become ill. As Rose (2007) remarks: ‘the [genetic test] does not tell the affected individual or their family when they will become ill or how rapidly the condition will progress, let alone when they will die’ (Rose 2007, 52). On the basis of experience-based intuition (see Zinn 2008), physicians will nevertheless temporalize the risk of illness in response to patients’ existential concerns. For example, in Broom and Adam’s (2010) study of indeterminacy in oncology departments, physicians often rearticulated the odds of survival in terms of how long a patient had to live. Though technically uncertain, short- and long-term projections affect differentially patients’ emotional states of mind and sense of agency; fatalism or a renewed sense of hope often hangs in the balance. Equally illustrative is Sarangi’s (2002) work on the language of likelihood among genetics counselors who, in response to patients’ concerns, often express aggregate probabilities of disease in relation to the lifespan, thereby putting a patient’s genetic condition into a kind of temporal context (e.g. ‘the odds of developing heart disease after age 50 are much higher among population X than Y’). Short- or long-term constructions of risk usually parallel an escalation or de-escalation of anxiety, respectively, among patients as they confront the prospects of their new future (Fox 2000; Sarangi et al. 2003; see also Hjelmblink and Holmstrom 2006; O’Doherty, Navarro and Crabb 2009; Pixley 2009). Readily apparent is a tension between a patient and physician’s desire for ontological security on one hand, and the affective dimensions of experience-based intuition while in the presence of uncertainty on the other (see Zinn 2008; Pixley 2002). Physicians are acutely aware of their inability to predict the future; however, they feel pressured to offer intimations of certainty to demonstrate their expertise while simultaneously soothing their patients’ anxieties. As will be demonstrated below, similar dynamics characterize the planner/client relationship.

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Medical images Biomedical informatics scholars have long recognized the importance of using images as a means of communicating health-related risks to patients (see Ancker et al. 2006). Images are thought to make risk-related information easier to understand and are therefore believed to be effective tools for promoting remedial action (Ancker et al. 2006; Severtson and Henriques 2009); however, images are often assumed to be neutral conduits through which information is more or less effectively channeled. Rarely are graphs, charts, and drawings viewed as social phenomena and therefore as reflections of ideology, power relations, and ‘distributed competencies’ (Lynch and Woolgar 1988, 110) within specific organizational and institutional contexts. A doctor’s hand-drawn sketch of a blocked artery, for example, is a material and social construct. Although rendering the artery seemingly visible, the sketch’s meaning and overall sense of relevance are established through social interaction and thus bear the imprint of the socio-structural relations that frame the interaction (see Joyce 2005). Medical sociology has elucidated such processes effectively in myriad environments. Consider, for example, Pfohl’s (1977) classic work on the social construction of child abuse, wherein he demonstrates how organizational, normative, and cultural forces determined partially how images were interpreted, ultimately setting the stage for the discovery of child abuse by radiologists rather than front-line physicians. Nearly 30 years later, Burri’s (2008) research on boundary spanning in radiology labs reached similar conclusions insofar as the introduction of new imaging technology in the 1970s catalyzed important shifts in the distribution of social and symbolic capital among experts, as they struggled to determine who was most qualified to interpret images (see also Rose 2007). Similar to Pfohl’s work, Burri’s highlights effectively how, within particular organizational and institutional frameworks, power relations intersect with, and are reflected in, the production and interpretation of medical imagery. Elsewhere, in their research on the role of images in cancer assessment Prior et al. (2002) argue that deoxyribonucleic acid (DNA) sequences are not reflections of an objective reality. Echoing the work of Lynch (1985) and Lynch and Woolgar (1988), the authors reveal how the risk of disease is rendered visible through a physical and social crafting of DNA images which, in and of themselves, become the subject of analysis and interpretation as they pass through social networks that ultimately ascribe them with meaning. The entire process is oftentimes tacit and tempered by considerable uncertainty. Finally, in his ethnographic examination of neuroscientists, Cohn (2004) demonstrates how, as the actual objects of analysis, brain scan images become embedded in, and reflective of, the social dynamics of biomedicine. While tapping into the ‘ocular-centric metaphor that to see something is to know it’ (Cohn 2004, 57), the images become a kind of canvas upon which neuroscientists construct and render sensible life’s mysteries in ways commensurate with prevailing medical ideologies. Whether it is a diagram sketched hastily on the back of a prescription pad, a double helix, or a high-definition scan of a patient’s brain, images are inevitably crafted, deployed, and rendered meaningful within very specific institutional and organizational contexts: they reflect particular rationalities and ideological frameworks as they function to advance very specific understandings of reality. Perhaps most importantly, and as Cohn (2004) has argued, images appear to render future realities and their risks as knowable phenomena.

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Methods Data for this article stems from a broader research project examining the relationship between discourses of risk and social control in the field of professional financial planning (see Parnaby 2009). Forty-two semi-structured interviews with professional financial planners were conducted, 29 of which took place between 2006 and 2007, whereas the remaining 13 were conducted in 2010. Ranging from 45 to 60 minutes in length, the interviews focused on the relational dynamics between planners and their clients, with particular attention paid to investmentrelated discussions and experiences.2 Of the 42 interviewees, 26 were men and 16 were women. Participants’ experience in the financial services industry ranged from 2 to 32 years. Likewise, ‘book values’ ranged from $500,000 to well over $100 million. Seven of the participants worked for a Canadian bank, whereas the remaining 35 worked for independent money management organizations. Participants were initially compensated $50 for their time; however, when interest began to taper in late summer of 2010, compensation was increased to $75. Approximately one-third of the interviews were conducted by telephone. Although the absence of nonverbal cues during telephone interviews can be problematic (King and Horrocks 2010; Stephens 2007), it was the most practical course of action, given the distance between the researcher and the interviewees. Gaining access to private meetings between experts and laypeople can be difficult. Although health scholars have had some success, others have had to rely on retrospective interviews that are susceptible to memory loss and/or the telescoping of events forward or backward in time (see Broom and Adams 2010; Goodrum 2008). Most importantly, interviews about prior engagements are unable to capture the nuances and spontaneity of real-time social interaction (King and Horrocks 2010). For this study, only the audio from planner/client meetings was recorded in order to minimize obtrusiveness while still capturing the interaction’s essence. Planners contacted clients who were booked previously for an appointment. Interested clients were then sent the necessary documentation for their review and consent. Before their client meetings, planners were given an encrypted digital voice recorder to record the proceedings. The researcher then waited elsewhere for an email or text message from the planner indicating the meeting was over. The researcher then collected the equipment, thanked the participants for their participation, and compensated each individual $75. During the transcription process, all identifying information was removed, including the names of investment companies, places of employment, personal banking information, and all references to particular financial products. The digital audio files were subsequently deleted. The transcripts were then imported into NVivoÒ 8, a qualitative computer data-analysis software application, for manual coding in a fashion largely consistent with the principles of grounded theory (Glaser and Strauss 1967). As hypotheses emerged, interview questions were modified to reflect strategic lines of inquiry. NVivo’s ‘nodes’ function allowed thematic relationships to be readily identified and explored graphically using the software’s modelbuilding function. Uncertainty What makes the comparison between health care delivery and professional financial planning intriguing is uncertainty’s prominent role in the expert/layperson

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dyad: whether at the forefront of discussion or part of a taken-for-granted subtext, uncertainty permeates focused encounters. For planners, uncertainty is usually epistemological (Fox 2000) and external (Kahneman and Tversky 1982), stemming from having limited (if any) knowledge of how future economic relations will unfold (Pixley 2002, 2009). This uncertainty is captured succinctly in the popular professional adage: ‘I don’t have a crystal ball. If I did, I wouldn’t be working.’ As Scott, a planner with more than 10 years of experience, remarked: ‘I mean nobody knows what’s coming down the road for the economy and the markets and so on’. Others make the same point by drawing attention to their lack of influence over market performance. Jamal, for example, commented: ‘One of the major forces I have no control over is the market. I can’t tell the market what to do. Nicholas [his colleague] can’t tell the market what to do. The client can’t tell the market what to do.’ Instrumental to clients’ education, these utterances render the economy’s exact trajectory unknowable and outside any individual planner’s influence. Insofar as planners appear willing to disclose their complete uncertainty, their experiences appear unlike those in professional health care, where a complete and outright admission of uncertainty would likely violate professional norms and invite derision (Dillard and Carson 2005; Fox 2000; Malcolm 2009). Where, then, do these seemingly dissimilar experiences overlap substantively and/or theoretically? First, planners’ forthright disclosure of their uncertainty is at odds with their conduct. Using terms such as ‘might’, ‘could’, and ‘most likely’, planners actually offer hedged predictions about the future. The ‘no crystal ball’ adage, while partially reflective of organizational policies that discourage the making of formal guarantees, actually operates as an overarching disclaimer (Hewitt and Stokes 1975) that allows planners to explore and articulate the kinds of knowledge thought widely to be characteristic of their expertise. Consider, for example, the following 2007 exchange with Laurie, a planner working with an independent money-management organization: Researcher: To what extent do your clients assume you can see the future? Laurie: [laughing] Um, I would say that they do think we can and you know probably you know they think 50% of the time we can see the future . . . and you know I always say to them you know I really love what I do but if I could see the future I’m sorry I wouldn’t be here . . .

Moments later, in response to a question about which clients were more likely to make the assumption, Laurie remarked: Say you’ve been in the industry 20 years and they [clients] figure, ‘Oh well; you must know everything; you must know what the market does and why it does it’ and, yeah, I mean, generally speaking, you can tell when you know the global market’s gonna do better; Canadian markets [are] gonna do better. [italics added]

Laurie begins by rejecting outright the very possibility of predicting the economic future; however, she then modifies her position by offering a hedged assertion about the predictability of market growth. Similarly, in a 2010 exchange between Robert (planner) and Mary (client) about economic stability and planning for retirement, Robert remarks:

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So, the likelihood is that we should have a few good years and it [another market crash] may not happen. But again . . . there is a chance we’re going to die tomorrow, but we don’t go plan our whole lives that way. So, yeah, there’s a chance we’re going to have another bad year like this. Most likely it won’t happen right away and most likely it might not be as bad for everything like it was in 2008. [italics added]

Robert makes numerous hedged claims about the future and the likelihood of market growth in particular. Interestingly, the future’s unpredictability is acknowledged momentarily (i.e. ‘there is a chance we’re going to die tomorrow’), but Robert quickly deems it an illogical realization before reiterating his hedged claims about the future. The discussion concludes moments later with Mary feeling at ease about the safety of her portfolio; however, during his one-on-one interview one hour later, Robert acknowledged the future to be fundamentally uncertain: ‘Isn’t all future uncertain? Do you know if you’re going to be employed five years from now? Do you know if you’re going to be healthy five years from now? In some ways all future is uncertain.’ Finally, Heather has been a planner with a small money-management company for more than a decade. When asked whether her clients expect her to predict the future, she responded: ‘I can’t tell you what will happen in a half-hour [laughing]!’ Moments later, she remarked: ‘This is really a business built on opinions and, quite frankly, educated guesses sometimes.’ Heather was then asked to comment on the feasibility of making long-term projections: Researcher: What about long term? Heather: Long term I am better at [laughs]. Researcher: How so? Heather: Um . . . you can predict a little bit better . . . from an historical perspective what probably will happen. You can’t predict market downturns or upturns; things like that. But you can generally safely predict over a 10- or 15-year period that . . . yeah, it’s [overall market performance] gonna be higher than it is now.

The market’s long-term growth is something most planners take for granted; however, even that assumption is usually hedged. Despite her candor regarding the future’s uncertainty, Heather appeared comfortable, almost certain, when offering her hedged assessment of the long-term future (i.e., ‘generally safely predict’). The ontological status of the future within the context of professional financial planning is tangled, to say the least. On one hand, professional planners proclaim definitively that the future is unknown and unknowable. On the other hand, and as part of how they demonstrate their expertise, planners offer hedged projections, which appear to suggest that parts of the future are at least partially known. Clients are thus confronted by experts who claim the future is unknown while engaging simultaneously in a kind of veiled prognostication: the legitimacy of the latter is often bolstered by planners’ credentials and marked achievements, traces of which are printed on their business cards or hanging on their office walls. The importance of disclaimers and uncertainty discourse in financial planning parallels closely experiences in health care delivery. In Broom and Adams’ (2010) study of oncologists and the articulation of expertise, for example, physicians were forthright about their own sense of external uncertainty. One physician commented: ‘You’ve gotta always say that you don’t know what the future’s going to bring. . . .

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[medical knowledge] is not perfect and you can’t predict what happens to an individual patient’ (Broom and Adams 2010, 1439). At the same time, examples of physicians offering hedged claims about the future are myriad. For example, extracted from Sarangi’s (2002) study on genetic-counseling discourse, the following comments were made by a geneticist while advising a patient: Geneticist: If I think if I didn’t know anything else and just how you’re mother’s your brothers mainly have . . . have had the kidney problem I would say ‘Yes there’s a chance that your mother could also have it, and there’s a chance that you could have it, and a chance that you could hand it on . . . [seven turns of conversation have been removed] . . . by the time someone’s into their 30s, they will show it if they . . . if they have it so by the time someone’s in their 60s, I think it’s . . . it’s . . . it’s . . . as good a well . . . a test, or as good a guarantee as you could have.

Although aware of their external epistemological uncertainties, planners and physicians find themselves pressured by clients/patients to prognosticate (see Christakis 1999 in Fox 2000). For patients, a physician’s hedged assessment of the future can help offset the existential uncertainties surrounding a poor diagnosis. Likewise, for clients, a planner’s hedged assessment can help diminish the anxieties that stem from market turmoil and/or what appears to be a bleak economic future. In both cases, and with the help of an overarching disclaimer, the careful articulation and strategic deployment of well-hedged claims is part of how planners’ and physicians’ expertise is demonstrated and ultimately legitimized in both professions.3 Temporalizing risk Planners’ nuanced and hedged claims about the future have an obvious temporal dimension. Temporal positioning of risk, in particular, plays a role in the discursive interchange between planners and their clients. Despite frequent reminders that there is no crystal ball with which to predict the future, planners offer clients a modicum of certainty by positioning risks along a temporal horizon. Contrary to Luhmann’s (2008) logical argument regarding the disconnect between an event’s probability and the timing of its occurrence, planners position discursively certain (potential) realities closer, while projecting others further away, depending on the context. For example, with more than 10 years of planning experience, Anita works for a Canadian bank. Her clients – David and his 86-year-old mother, Louise – had made an appointment to discuss ways of supplementing Louise’s insufficient monthly income. In the following exchange, Anita projects Louise’s risk of death forward, despite resistance: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Anita: So, Mrs. [Smith] how old are you right now? David: She will be 86 on [birth date]. Anita: 86? Wow. David: Yeah, [birth date] she turns 86. Anita: So, how about you live to 95 or more? Louise: Oh dear! [All three participants then briefly discuss the longevity in Mrs. Smith’s family.] Anita: Well, let’s assume we give you another 10 years or so. Louise: Well, you never know. . . Anita: Right? And that’s the time, eventually, that we will. . . Louise: I just go by year to year.

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(11) Anita: I know we all do – trust me. So our (inaudible) plan will be for another 10 years or so. (12) David: Yep. On line three Anita acknowledges enthusiastically Louise’s age – as if to suggest turning 86 is a noteworthy achievement, given the risks associated with growing old. Anita then discursively positions the risk of Louise’s death 10 years into the future by suggesting they proceed on the assumption that Louise will live until she is at least 95 (line five). Louise’s passive challenge to Anita’s projection (line six) is followed by a brief discussion about longevity in Louise’s family, during which time it is noted that her siblings and mother had lived into their mid and late 80s, whereas her father passed away in his early 70s. Anita then reiterates her initial projection (line seven) in response to which Louise offers more resistance (lines eight and ten). On the basis of a rather dubious claim, Anita then attempts to render Louise’s temporalization of risk entirely common and thus less pertinent by suggesting everyone is in a similar position (i.e. ‘I know we all do – trust me’). David finally interjects to approve Anita’s decision before all three participants, now feeling reasonably comfortable, move on to discuss how best to restructure Louise’s portfolio. The point here is not whether Anita’s planning strategy was appropriate: rather, it is to appreciate how what was essentially an arbitrary temporalization of risk (Anita could have estimated Louise’s death in two or 15 years with as much accuracy) allowed Anita to demonstrate her expertise while meeting simultaneously the bank’s organizational expectations regarding the provision of long-term planning. Likewise, Murray arranged a meeting at his clients’ home for a review of their finances and a discussion about long-term care. The broader context from which the following excerpt was taken involved a discussion about different types of longterm care plans. Here again, the temporalization of risk is seen in subtle ways: Murray: Over the course of 20 years for this instrument . . . um . . . I didn’t do it for this particular one, but if you do the math, you think about it, um . . . I did a scenario where I took this much money that you’re [already] paying every month – $316 . . . and I invested it at 6% over 20 years, and it ended up at something like $95 thousand dollars. And if it was 8%, it was like $127 thousand dollars, or some number in that ballpark. And you could potentially, at age 80, be getting, uh, rounded off, $40 thousand a year, say. Sean: Mm hmm. Murray: Well, that’s two to three years of that savings pool, had you invested the money. That money will last, like, three years. Whereas this [referring to a specific insurance policy] will go [pay out] as long as you live.

For illustrative purposes, Murray begins by constructing an investment strategy which, under ‘normal market conditions’, would ‘likely’ lead to the clients’ insolvency by 83 years of age (again, note the hedged terminology). Murray then presents his alternate financial strategy which, in terms of maintaining a steady source of income, appears to offer a risk-free financial future. Once again, risks are strategically constructed along a temporal horizon. The initial risk of insolvency was constructed in the short term insofar as it paralleled the clients’ turning 83 years of age with years of healthy living ahead of them (an entirely arbitrary claim in and of

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itself). Murray then presented the second financial strategy in a way that rendered the temporal dimension almost irrelevant by removing references to risk entirely. The somewhat bitter truth, however, is that planners have no way of knowing with certainty whether the economy will be stable in the near or long-term future, whether a person will live or die in the next 10 years, or whether an insurance company will be able to make its payments 15 years hence – a lesson many planners learned following the 2008 market collapse.4 Thus, the temporal positioning of risk is, and always will be, nothing more than a leap of faith conducive to particular states of mind, courses of action, and ongoing capital exchange. Planning images The use of planning imagery may not generate the wonder and excitement observed by Joyce (2005) and Cohn (2004) in advanced medical environments. Nevertheless, there is an intriguing sociological parallel worth noting. In both professions, images become focal points around which experts construct an otherwise uncertain and unknowable future in terms of specific probable outcomes. At the same time, however, images operate as a critical means by which risks are constructed in ways that legitimate and reinforce the existing corpus of knowledge, reaffirm institutional and organizational logics, and, by implication, lead to particular courses of action. Diagrams, charts, and graphs are not objective reflections of reality. Although certainly things in a material sense, images are inherently social phenomena created and rendered meaningful by people. At each stage in an image’s development, certain interpretations are embraced, whereas others are cast aside (see Latour and Woolgar 1986). In planning, for example, investment strategies are often explained diagrammatically: arrows usually indicate a transfer of funds, boxes represent areas of investment and/or future goals, and graphs illustrate market volatility while emphasizing growth over time. For example, in response to his client’s confusion, Marco offered to draw a diagram: Michael: This is kind of like . . . this is kinda’—is this the Smith maneuver? With the RSVPs? Or the RRSPs? Marco: No, not particularly. First of all, do you under—do you want me to draw it out for you really quickly? . . . OK? Traditionally, it looks like this . . . [Marco begins sketching] Michael: So it kinda’ looks like this because the money goes in the subaccount [the client then amends the sketch]? Marco: Yep. Michael: OK. Marco: I was going to do that part of the diagram next [laughs].

Practically speaking, diagrams are an effective means by which clients can be educated. That said, they often imply an unfettered, predictable, and rational movement of capital, a depiction that has little resemblance to the messiness and unpredictability of socioeconomic life, but, nevertheless, makes pragmatic decision making possible. Diagrams also function as symbolic extensions of a planner’s expertise. Notice, for example, how at the end of the excerpt Marco reaffirms his expertise by assuring Michael that he, too, would have adjusted the diagram.

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Unlike hand-drawn diagrams, industry-produced images suggest objectivity, order, and scientific rigor to a greater extent. The MorningstarÒ Andex chart, for example, is one of the most widely recognized planning images.5 The chart tracks the investment and growth of one dollar since the 1950s. Its horizontal axis highlights the social, political, and economic crises that have shaped the course of western history in meaningful ways. Tanya, a salary-based planner, uses the Andex chart to help shape her client’s expectations: ‘When we’re looking at the long-term horizon I might take out the Andex chart and go over five and 10 years’. Similarly, while discussing the challenges associated with managing clients during market downturns, Laura, a planner with a large money-management firm, remarked: ‘You know what? Now is the time [to buy],’ and you have to go through and you have to cushion it and make them feel good, and I pull out the Andex chart that I carry around.

For many planners, the chart is a valuable tool. Of particular importance, however, is what the Andex chart does not reveal. Meaningful references to the billions of dollars in personal wealth lost and/or the lives and communities utterly destroyed by market ‘corrections’ around the world are entirely absent: global markets might have a tendency to bounce back, but individuals are tragically less resilient (see Harvey 2010). The chart is therefore a blatant reflection of the ideological and organizational imperatives that led to its creation and distribution in the first place: it functions to reinforce the validity of free market capitalism and long-term investment logic. The Andex chart is only one of many tools planners use. Laura, for example, uses magazine covers from previous decades to teach clients about the market’s tendency to correct itself: I have a bunch of articles, or not . . . um the covers of [names magazine] magazine over the years, right? To show them, you know, this isn’t the only time we’ve suffered a downturn like this, here are some examples. Look these are the covers of [names magazine] magazine over the decades and, um . . . talking about all the things . . . so educating the clients – you have to educate them a lot more. . .

Like the Andex chart, Laura’s magazine covers serve to highlight the market’s continued existence as the primary justification for present-day, long-term investment. With a similar effect, Robert attempts to allay Mary’s uncertainty about a specific fund in her portfolio by sharing images on his computer screen: Robert: The idea behind it is to have growth without too much risk. And you will see, you know, if I show you other things, they will be like maybe 35% here and 28% here [pointing to a graphic on his computer screen], right? So those graphs, those, um. . . . bars . . . here . . . the fact they don’t stick out in the good year or the bad year tells you that they are more conservative. Mary: Yeah, yeah. Robert: That’s why it looks closer to a straight line than most balanced funds even. Mary: And we’re looking at a five-year plan? Robert: Yeah.

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Robert uses the fund company’s graphic as evidence of the fund’s future stability and thus its suitability for a conservative portfolio. Derived from a remote database and appearing on Robert’s screen with an aura of objectivity, the image functions to mitigate Mary’s uncertainty. Planning images are inherently reductionist, unable to account for the notorious unpredictability of the market in late modernity (see Borges et al. 2001). The use of images therefore provides a seemingly concise etching together of economic realities conducive to pragmatic decision-making and the ongoing exchange of investment capital: it all bears a striking resemblance to Joyce’s (2005) research on magnetic resonance imaging where: anatomical images ‘etch together’ local decisions and priorities, technology, and aspects of the physical body to produce what is perceived as cutting-edge, authoritative knowledge. (Joyce 2005, 438)

In the same way that genetics images or brain scans facilitate the practical assembly and visibility of risk in medical settings (see Prior et al. 2002), planning images also cater to the ocular-centric metaphor identified by Cohn (2004), whereby that which is made visible is thought to be known. However, planning images, like their medical counterparts, are not mirrors of reality, but rather social constructs both in terms of their material existence and ascribed meaning: they are products of organizational and institutional forces, distributed competencies, and strategically informed editorial and aesthetic decision-making. Discussion and conclusion There are, I would argue, relatively few areas of professional practice where focused encounters are as profoundly tied to a person’s desire for a known future as financial planning and health care delivery. ‘Will I be okay?’ is often the question at the forefront of clients’ and patients’ minds as they explore the boundaries of the expert’s knowledge. However, with the future technically unknown and with there being nothing self-sustaining about expertise in and of itself (Bauman 1992), planners and physicians deploy discourses of risk and uncertainty in the form of hedged projections, temporalized risks, and strategically deployed images to protect and/or bolster their expertise, provide a sense of ontological and/or existential security, and facilitate courses of remedial action that coincide with expectations and entrenched organizational logics. Of course, blatant cultural and organizational differences between professional planning and health care delivery do exist: the two areas of expertise have, for example, entirely different genealogies, cling to distinct ‘market shelters’ (Freidson 2001), and demand different levels of professional credibility. There is also a profit motive that is more readily apparent in planning. Nevertheless, many of the dynamics identified by Sarangi et al. (2003), Fox (2000), O’Doherty, Navarro, and Crabb (2009), and Malcolm (2009), for example, appear to cross the occupational divide into financial planning. There are, however, two interrelated issues that must be addressed: human agency and the importance of trust. Although equally relevant to health care delivery, the focus will be on financial planning in particular. This analysis has presupposed a degree of autonomy among planners (and their clients, for that matter): it has been argued that planners manage discourse ‘strategically’; thereby implying motive and a level of conscious deliberation. That said, oftentimes their choices are made in response to conditions they had little or no role

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in creating; therefore, their agency simultaneously defines, and is defined by, overlapping structural conditions. For the most part, they remain unabashed emissaries of free market capitalism – their utterances reflect routinely organizational imperatives and/or dominant market ideologies. However, that need not imply their motives are necessarily less genuine or less benevolent than anyone else’s. What, then, is the role of trust? The data gathered for this study yielded hundreds of remarks by planners about the importance of trust: it is, as they said time and again, ‘everything’. Loosely defined as a belief that ‘others will perform in a way that is beneficial, or at least not detrimental, to us’ (Robb and Greenhalgh 2006, 435), trust plays a critical role in the expert/layperson dyad (Evetts 2006). In its voluntary form, trust is extended without meaningful calculation. Alternatively, trust can stem from forced dependency whereby a layperson is forced to trust because he or she has no alternative (see Robb and Greenhalgh 2006). In the financial planning industry, planners nurture a sense of voluntary trust by, among other things, exercising patience, listening carefully, meeting for dinner, asking about family, and sharing personal stories. As trust is established over time, the line between friend and professional advisor becomes less obvious. Although this study was not designed methodologically to measure the presence and/or outcomes of trustworthy planner/ client relations, one could logically assume that the effects of the aforementioned discourse would vary accordingly. There is, I would argue, always a need for comparative sociology. This article has taken advantage of what health scholars have achieved over the course of nearly four decades in order to shed light on an under-researched area of professional expertise. Admittedly, the points of analytical convergence are not seamless, but are perhaps indicative of the kinds of generic social processes that probably characterize expert/layperson relations in fields of expertise where risk and uncertainty feature prominently. At the same time, the secondary health literature and the primary data collected among planners and their clients appear to speak to more fundamental features of the human condition: a desire to feel needed, secure, and at least partially in control of what lies ahead. Notes 1. Although there have been debates about the extent to which uncertainty permeates medical culture [see the classic debate between Fox (1957) and Atkinson (1984), for example], there is little doubt among scholars that uncertainty management is critical to almost all forms of professional expertise (see Zinn 2008). 2. The majority of those who participated in this study were licensed to sell mutual funds, not individual stocks/securities. 3. Although Broom and Adams (2010) are reluctant to connect uncertainty management to a general support of professional expertise (at least in the medical context), their argument is countered elsewhere by more theoretically and methodologically rigorous research (see Sarangi et al. 2003) demonstrating how such discourses factor into the delineation and preservation of expertise. 4. Sometimes planners find their leaps of faith punished by what Taleb (2001) calls ‘the black swan’ – an economically disastrous event that, once thought to be highly improbable, becomes a reality when least expected. Many of the planners interviewed for this study, for example, admitted they were caught completely off guard in 2008 when the global economy tumbled into a recession which, according to the International Monetary Fund, resulted in an estimated loss of $50 trillion worldwide (Harvey 2010). 5. For example images see: http://corporate.morningstar.com (accessed 28 December 2010).

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Journal of Risk Research, 2014 Vol. 17, No. 2, 263–279, http://dx.doi.org/10.1080/13669877.2013.808685

How did enterprise risk management first appear in the Korean public sector? Eun-sung Kim* Department of Sociology, Kyung Hee University, Seoul, Korea (Received 17 February 2012; final version received 8 May 2013) This research addresses the rise and fall of the Crisis Management Guideline of Public Organizations (CMGPO) from a historical perspective. In the Korean public sector, as a form of enterprise risk management (ERM), CMGPO is not designed to be merely a tool of financial risk management but also to be a policy tool for crisis management. CMGPO emerged within the conflict between integrated crisis management and dispersed crisis management. The purpose of CMGPO is to bureaucratically integrate the crisis management of public organizations with the governmental crisis management system. ERM as a form of self-regulation is entangled with the pre-existing command and control of the Korean government over integrated crisis management. As a result, CMGPO is characterized as ‘enforced self-regulation’ rather than self-regulation; this is a fundamental idea in ERM. Keywords: crisis management guideline for public organizations; enterprise risk management; integrated crisis management; internal control

Introduction Enterprise risk management (ERM) has grown since the financial scandals of the 1990s that involved Barings Bank, Polly Peck, Maxwell, and Guinness (Dickstein and Flast 2009; Olson and Wu 2010). By building a self-regulated system of risk through an internal control system, ERM integrates organizational management with risk management (Power 1999, 2003, 2004, 2005, 2007). ERM faced two big transformations in the late 1990s and early 2000s. First, in addition to the previous preoccupation of ERM with financial risk management, ERM adopted an integrated or comprehensive approach to the management of all risks-facing organizations. Second, in the wake of new public management, ERM started to shift from the private sector to the public sector (Hood and Rothstein 2000), as in the UK, for instance.1 Recent ERM scholarship has stressed the contextuality and contingency of ERM (Power 2007; Mikes 2009; Arena, Arnaboldi, and Azzone 2010, 2011). This scholarship tends to avoid the deductive approach that applies rigid ERM rules to business management. Instead, while framing ERM as an ‘umbrella’ concept (Power 2007; Mikes 2009), this scholarship attempts to seek an inductive or descriptive approach that interprets ERM as a wide variety of practices in specific *Email: [email protected] Ó 2013 Taylor & Francis

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organizational and cultural contexts. The meaning of ERM is not static but fluid and malleable depending on specific organizational contexts, where ERM can become entangled with, or decoupled from, the organizations’ pre-existing governance styles and organizational culture (Power 2007; Arena, Arnaboldi, and Azzone 2010; Mikes 2011; Huber and Rothstein, 2013). In line with this scholarship, this paper explores how ERM entered the Korean public sector, with a particular focus on Korean public enterprises. This paper examines ERM in a non-Western context from two perspectives. First, the Korean government planned to build ERM into the Korean public sector not as a mere tool of financial risk management, but instead as a policy tool for crisis management. Second, in the course of this process, ERM as a form of self-regulation became entangled with the pre-existing bureaucratic command and control of the Korean government in relation to crisis management. The first entry of ERM into the Korean public sector was initiated by the Crisis Management Guideline for Public Organizations (CMGPO). This guideline was created by the Roh administration’s National Security Council (NSC) in 2007 in order to establish integrated crisis management. Prior to the emergence of CMGPO, in July 2004, the NSC had established the Framework Guideline for National Crisis Management in order to establish a pan-ministerial standardized model of national crisis management. NSC also wanted to standardize the crisis management of public organizations (NSC Executive Office 2004). In the beginning, 17 public enterprises2 were subject to CMGPO; this guideline was intended to be extended to other public organizations. However, public enterprises expressed concerns about the integration and standardization of crisis management under CMGPO, because they had their own crisis management systems designed according to the existing governmental laws and guidelines. They were worried about overlap or conflict between CMGPO and pre-existing governmental laws and the possibility that CMGPO could undermine the managerial autonomy of public enterprises. However, they had no choice but to modify their crisis management system according to the standardized manual of CMGPO, because the President’s Office was the driving force behind the new guideline. Therefore, ERM appeared within the conflict between integrated crisis management and dispersed crisis management in the Korean public sector. This research examines the implications of ERM through a historical analysis of CMGPO. It first addresses the institutionalization of integrated crisis management in Korea and the implications of this process from the ERM perspective. Second, the paper analyzes the relationship between ERM and CMGPO through content analysis. Third, this paper investigates the historical context surrounding the establishment of CMGPO. This research is built on literature review and interviews. I conducted the content analysis of CMGPO and relevant contemporary documents of the associated task force. Oral history interviews were conducted with two people involved in the establishment of the guidelines. The first interviewee was Cheol Hyun Ahn, former Director-General of the Crisis Management Center under NSC. He was in charge of writing the guidelines. The second interviewee was Kyoung Hwan Kang from the Korea Minting, Security Printing & ID Cards Operating Corporation, another member of the taskforce. In order to collect information about the response of public enterprises, I also conducted in-depth face-to-face interviews with crisis management personnel for seven public enterprises from March to November 2011.

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Moreover, focus group interviews were also conducted through two workshops on 30 September and 3 November 2011. More than 13 risk managers from public enterprises attended both workshops, where participants addressed the impact of CMGPO on their risk management as well as their current state in public enterprises. ERM and crisis management Most Korean academic discussion of national crisis management systems tends to favor an integrated approach over a dispersed approach (Lee 1999, 2005; Lee et al. 2004; Kim 2005; Lee, Oh, and Jeong 2009). According to Lee (2005), an integrated crisis management system refers to the integration and standardization of crisis typology, the crisis management process, crisis management organizations, and the resources necessary for crisis management. First, the integration of crisis typology means the integrated management of natural disasters, man-made disasters, national infrastructure crises, and traditional security crises. Second, the integration of crisis management processes denotes the integration and connection of crisis management activities centered on prevention, preparedness, response, and recovery (Petak 1985). Third, the integration of crisis management organizations refers to the connection of the NSC Crisis Management Center, the Emergency Planning Committee of the Prime Minister’s Office, the Ministry of Public Administration, and the National Emergency Management Agency, together with local governments, companies, and non-governmental organizations. Fourth, the integration of resources refers to integrated management systems of human and material resources for disaster management. Integrated crisis management and dispersed crisis management have different strengths and weaknesses (Lee et al. 2004, 357; Korea Disaster Prevention Association 2008, 40; We et al. 2009, 33). According to Lee et al. (2004), dispersed crisis management can clarify the accountability of ministries for specific disasters. It also can utilize the expertise of specific public organizations for particular disasters. However, the dispersed crisis management system has limitations in responding to large and complex disasters. This system also faces various challenges, including the complexity of disaster fund preparation and distribution, the dispersion of authority and responsibility, and the duplication of tasks among relevant agencies. This system can also result in a lack of comprehensive mediation in the face of a national crisis. By contrast, the integrated crisis management system enables the total mobilization of resources in the face of large and complex disasters. However, the tasks and accountability of crisis management can be heavily concentrated in a specific organization under this system. Prior to the 2000s, crisis management in the Korean government was dispersed into several ministries that were responsible for managing specific types of disasters or crises – natural disasters, human-made disasters, and social disasters, respectively. However, as integrated crisis management became internationally prominent in the 2000s,3 both the Roh (2003–2007) and Lee administrations (2008–2012) took this new approach as a core piece of their crisis management projects. However, they pursued different approaches to integrated crisis management in terms of the extent of integration. The Roh administration established a Crisis Management Center under the Executive Office of the NSC in the President’s Office. In March 2004, the administration

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integrated existing laws, such as the Disaster Management Act and the Natural Disaster Response Act, into the Framework Act on Disaster and Safety Management. This act employed a more extensive notion of disasters, including natural and man-made disasters, extending even to the paralysis of national core infrastructures such as electricity and communication (Lee et al. 2009). Along with this act, on 1 June 2004, the National Emergency Management Agency was created to strengthen the function of comprehensive mediation and policy deliberation and to unify the task system related to disasters. Although the Framework Act on Disaster and Safety Management integrated natural disasters and man-made disasters, it did not address national security crises, such as war and terrorism, or financial crises. As a result, in July 2004, the Roh administration established the Framework Guideline for National Crisis Management (NSC Executive Office 2004). The guideline produced many standardized manuals in order to effectively respond to a wide array of disasters (Kim 2005). The guideline is based on a notion of ‘comprehensive security’ (Lee 2008) that integrates various disasters, national security, and national core infrastructure crises. By contrast, the Lee administration has taken a different approach toward integrated crisis management. The establishment of an integrated system concerning disaster management was also vital to this administration, and therefore it was adopted as a core national project. In December 2008, the Ministry of Public Administration and Security formulated a Comprehensive Plan for Integrated Disaster and Safety Management. The Second National Plan for Safety Management (2010–2014) carefully studied this plan as a core project for disaster and safety management. However, the Lee administration was reluctant to integrate disaster management and national security. In this vein, the Lee administration dismantled the Executive Office of the NSC, which had played a major role in integrating national security, disasters, and the crisis of national infrastructure under the Roh administration. ERM is related to both integrated crisis management and dispersed crisis management. So, it is significant to explore how ERM appeared in the Korean public sector in terms of this conflict. On the one hand, ERM is very similar to integrated crisis management in the Korean government. ERM integrates every type of risk that an organization faces. ERM also involves the standardization of risk management by the Committee of Sponsoring Organizations of the Treadway Commission (COSO), International Organization for Standardizations (ISO), and the Institute of Internal Auditors (IIA). This standardization is compatible with an integrated crisis management approach that attempts to standardize various crisis management manuals. On the other hand, ERM is different from integrated crisis management in the Korean government. In fact, integrated crisis management depends on the idea of bureaucratic command and control. In this sense, the principal-agency theory of bureaucracy can be applied to integrated crisis management: the NSC of the President’s Office is the principal, while public organizations are the agents. There is a hierarchical connection between the principal (government) and agents (public organizations). However, initiated by corporate governance, ERM emphasizes the self-regulation of organizations in terms of risk management. Therefore, in theory, the power relationship among organizations is not hierarchical but decentralized. In other words, ERM sees an autonomous organization as a basic organizational unit of risk management that is able to manage risk by itself. This is quite different from the bureaucratic idea of an organization. In this sense, the self-regulation of ERM is

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theoretically different from the command and control of integrated crisis management in Korea. The CMGPO and ERM This section analyzes the relationship between ERM and CMGPO through content analysis. From the ERM point of view, CMGPO has three implications: first, CMGPO takes an integrated approach to the crisis management of public organizations. Second, CMGPO suggests an internal control system. Third, CMGPO introduces the notion of a chief risk officer (CRO). Integrated approach Even though there are various models of ERM, one of the most dominant models of ERM is the integrated approach. CMGPO takes an integrated approach to crisis management from two perspectives. First, the guideline standardizes the crisis typology of public organizations into four types: management crisis, disaster, communication crisis, and conflict crisis.4 Key risk indicators of public enterprises are developed under this category. Second, CMGPO classifies the crisis management process into common activities, such as prevention, preparedness, response, and recovery, according to the standardized process of disaster management (Petak 1985, 3; Lee 2005, 3). CMGPO mixes the ERM process with the traditional process of disaster management. Basically, the crisis management process follows the disaster management process, while, as in the ERM process, common activities include risk identification and risk assessment. According to CMGPO, ‘common activities’ refer to those that precede the activities of prevention, preparation, response, and recovery, or common functions that should continue throughout the entire process, including the discovery of risk types and threat factors, monitoring, identification, diffusion of symptoms, threat characterization and warning, and the establishment and maintenance of cooperative relationships inside and outside (NSC Executive Office 2007). The reason why CMGPO is connected to the disaster management process, even though the ERM process differs from the disaster management process, is that CMGPO is designed to be consistent with the Framework Guideline for National Crisis Management, which was established by the NSC in July 2004. Internal control system The establishment of an internal control system is vital to ERM. Internal control was regarded as a benchmark of good corporate governance in the mid-1990s (Power 1999). Internal control refers to the establishment of internal departments in charge of evaluating the risk management performance of executive departments. ERM is ‘a managerial turn’ in risk management that combines ‘command and control’ regulation with organizational management. ERM internalizes state regulation by establishing an internal control system (Power 2007). First, executive organizations conduct first-order risk management of the outside risks facing the organizations. Second, either general or auditing organizations conduct the secondary risk management of operational risk concerning risk management activities of the organizations. Internal control refers to ‘secondary risk management’ (Power 2004).

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According to CMGPO, both general departments and audit departments are capable of inspecting and evaluating the risk management performance of their organizations. These departments are able to inspect and evaluate the operation of the crisis management system and activities, either regularly or only occasionally, to establish an improvement plan according to the result of the inspection, and to report the result to management headquarters (NSC Executive Office 2007). Accordingly, CMGPO is designed to form a double internal control system through both general departments and audit departments. The first internal control is exercised by general departments on the activities of executive departments, while the second internal control is performed by audit departments on the activities of both general departments and executive departments. This double internal control is a kind of ‘control of control’ (Power 2007).

Chief risk officer The CRO is a new occupation that appeared along with the development of ERM in the 1990s (Power 2005). CROs began to play a prominent role in the financial risk management of corporate governance because of the rise of strategic management. In the late 1990s, the role of CROs was not limited to financial risk management but also was extended to the management of all risks-facing corporations, including those related to energy, utility, and corporate service. CMGPO designs core departments related to crisis management, including general departments,5 executive departments,6 emergency departments, and inspection/ evaluation departments. The CRO is the leader of the general departments, which comprehensively plan, mediate, and control the entire crisis management of public organizations and advise or assist management headquarters or top decision-making departments. According to CMGPO, the CRO is selected from internal members who have expertise and special knowledge about crisis management. The CRO can be independently appointed in the case of big organizations or be a position held concurrently by another board member in strategic planning departments or emergency management departments in the case of small organizations (NSC Executive Office 2007). On the other hand, the CRO is also connected to the Framework Guideline for National Crisis Management (Presidential Executive Order 124) that already institutionalized a ministerial CRO who presides over crisis management at the ministerial level. In this sense, the institutionalization of the CRO in public organizations aims to coordinate the crisis management of public organizations in tandem with those of corresponding high-level ministries. It appears that the CRO plays a crucial role in the command and control of ministries over public organizations.

History of the CMGPO This section addresses the emergence and decline of the CMGPO from a historical perspective. In analyzing this topic, it addresses the conflict between integrated and dispersed crisis management that plays out in both government and public enterprises. Given that ERM involves both integrated and dispersed crisis management, as described earlier, this paper examines how ERM emerged in the Korean public sector in terms of this conflict.

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Preparatory stage: invitation of auditors and establishment of a task force The plan for CMGPO resulted from two critical events: these were the electricity outage on Jeju Island in 2004 and the paralysis of highways due to snowfall in 2005. Because of these events, the President’s Office recognized the necessity for crisis management at the level of public organizations following the Roh administration’s (2003–2007) establishment of governmental crisis management through the Framework Guideline for National Crisis Management in 2004. This system hailed as an integrated crisis management system. The Roh administration planned to extend this system to the level of public organizations through CMGPO. Dr Cheol Hyun Ahn is the former Director-General of the Crisis Management Center, under the NSC Executive Office, and presided over the establishment of both the Framework Guideline for National Crisis Management and CMGPO. According to Dr Ahn,7 while the establishment of a governmental crisis management system had been well executed until 2006, the crisis management systems of public enterprises and quasi-governmental organizations, which significantly affected governmental functions and citizens’ lives, tended to be unsystematic, lacked adequate attention, and were characterized by huge variations from organization to organization. Ahn insists that this crisis within these organizations could result in a national crisis. Public enterprises only paid attention to management evaluation but lacked efforts for crisis or disaster management. He asserted that the NSC Crisis Management Center launched the institutionalization of CMGPO in order to both systemize the crisis management of public organizations and make their standardized manuals parallel to those of the governmental crisis management system.8 Furthermore, according to Dr Ahn, the government also had a plan to insert crisis management into the annual management performance evaluation of public organizations, conducted by the Ministry of Planning and Budgeting, in order to efficiently and persistently drive the crisis management of public organizations.8 At first, the NSC Crisis Management Center intended to use auditors of public organizations to build crisis management systems in public organizations. On 24 May 2006, the Center invited auditors from government investment agencies and explained the operating system of governmental crisis management. While the roles of auditors are integral to ERM (Power 1999), and Ahn was aware of ERM at this time, he did not intend to consider ERM for crisis management when he initially invited auditors to the President’s Office. Ahn7 argued that it would be very important to mobilize high-ranking auditors in the management of public organizations and empower them to use their authority to elevate the crisis management of public organizations to the level of a governmental crisis management system. The Center wanted to let auditors know the significance of crisis management and encourage them to take responsibility for crisis management. The Center thus achieved ERM spontaneously as it tried to extend a governmental crisis management system into public enterprises. From June to July 2006, the Center invited auditors of government investment agencies to the NSC Executive Office and asked them to strengthen the function of inspection and evaluation concerning crisis management in their internal auditing. In July 2006, the NSC Executive Office reported to President Moo Hyun Roh on a plan concerning the establishment of CMGPO, and then launched a task force called ‘CMGPO Task Force’ to enact the plan. The CMGPO Task Force did not initially intend to focus on all four areas of crises – management, disasters,

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communication, and conflict – facing public organizations. According to Kyoung Hwan Kang, from the Korea Minting, Security Printing & ID Cards Operating Corporation, who was a member of the CMGPO Task Force, the NSC Executive Office initially planned to focus only on disasters but later expanded the scope of organizational crises to the four standardized types in order to make CMGPO suitable to the context of public organizations. He said: Crisis conceptualized at that time mainly focused on disasters because the NSC intended to use a national crisis management system. So, I argued that CMGPO should be userscentred and suitable to pubic organizations. CMGPO would be useless if it were made simplistically in the way the government wanted. So, in cooperation with professional organizations associated with crisis management, Director-General Ahn and I decided to make a preliminary study on the crisis management of public organizations.9

The CMGPO Task Force consisted of members from Samil PricewaterhouseCoopers (whose specialty was ERM), Samsung Loss Control Center (whose specialty was disasters), Samsung Economic Research Institute (financial management), SookMyung Women’s University (communication), TRC Korea (business continuity plans), and the Korea Minting, Security Printing & ID Cards Operating Corporation (a representative public enterprise) (CMGPO Task Force 2006a). On 4 August 2006, the NSC Executive Office launched the CMGPO Task Force to develop a standardized model of crisis management at public organizations in cooperation with the main participatory agencies (CMGPO Task Force 2006a, 2006b). Inquiry into the status of crisis management of public organizations The CMGPO Task Force first planned to conduct a preliminary study of the crisis management of public organizations. In order to accomplish this investigation, the task force classified public organizations into the categories of manufacturing, social overhead capital, and promotion and service, and visited eight organizations from these categories on 24 August and 25 August 2006.10 The aim of the interview was to understand the function of government investment agencies, diagnose the status of their crisis management, and then use the results as basic information for developing the standardized model of crisis management (CMGPO Task Force 2006b). According to their interviews (CMGPO Task Force 2006b), the task force asked about policies and manuals, education, organizational structure, risk types, and public relationships associated with crisis management. In the classical wisdom of poststructuralist policy analysis, the identification and definition of policy problems is not separate from but actually closely connected to policy recommendations for resolving these problems (Bacchi 1999). Therefore, the analytic framework of the CMGPO Task Force concerning the crisis management of public organizations was not objective but reflected the will of the government toward integrated crisis management. In this vein, the task force investigated the degree of integrity concerning the crisis management of public organizations, as well as the existence of general departments that comprehensively mediate and control the crises of organizations. The task force also examined whether or not public organizations had all-inclusive crisis management policies or guidelines from the whole organizational perspective. According to the results of this inquiry (CMGPO Task Force 2006b), public enterprises having an integrated crisis management system included the Korea

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Electricity Power Corporation (KEPCO), the Korea Land Corporation, and the Korea Minting, Security Printing & ID Cards Operating Corporation. Public enterprises having a dispersed crisis management system were the Korean Expressway Corporation, KORAIL, the Korea Gas Corporation (KOGAS), and K-water. Moreover, there was no general department that comprehensively mediated all crises in these public organizations, although there were executive departments in charge of managing specific types of crisis such as rainfall and snowfall. The task force recommended that public organizations establish such a general department. Moreover, the task force presented the will of the government concerning the standardization of the crisis management organizational structure. It argued that although all organizations commonly had disaster-response teams as part of a disaster management committee, along with a disaster management headquarters controlling the operation of emergency management organizations, the role of the disaster management committee and disaster management headquarters, as well as the composition of disaster-response teams, varied among public organizations. Therefore, the task force recommended standardizing crisis management organizations both by unifying the disaster management committee with the disaster management headquarters and by placing response teams under this unified organization (CMGPO Task Force 2006b). When it comes to the relationship between public organizations and the public, the task force argued that departments in charge of public communications varied; for instance, there were differences between customer departments, planning departments, and public relations offices. The task force additionally suggested that it would be necessary to comprehensively manage the function of public communications when a crisis occurs, although it is desirable to designate a specific department according to the type of public relationship (CMGPO Task Force 2006b). The task force also examined crisis management manuals in terms of how public organizations had handled four types of crises. It concluded that the types of crises varied with the characteristics of public organizations but lacked standardized manuals and standardized systems. Most organizations classified types of crises peculiar to the nature of their own businesses. Some organizations focused more on financial risk rather than on disasters. Most public organizations lacked manuals involving conflict management, although they had disaster management manuals. For instance, the Korea Expressway Corporation established a disaster management system because of heavy snowfall in 2005, but had no manuals about financial risk management and public communications. Furthermore, public relations offices were in charge of establishing the crisis management communication system. However, they lacked practical standardized communication manuals designed for each type of crisis (CMGPO Task Force 2006b). Finally, with regard to internal control, the task force emphasized the role of audit departments in terms of the inspection and evaluation system of crisis management. The task force argued that disaster management departments periodically inspected crisis management systems, while audit departments did not have audit functions for crisis management (CMGPO Task Force 2006b). The anxiety of public enterprises At first, the government planned to produce a collection of manuals rather than a set of guidelines. However, in that moment, the director of the NSC Executive Office – Hee In Ryu – thought that the manuals would be useless without the

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power of enforcement. Therefore, in early 2007, the form of crisis management documents shifted from a huge volume of manuals filled with many cases to a condensed volume of guidelines.9 On the other hand, the KOGAS, Korea Airports Corporation, KEPCO, and KORAIL, which had their own crisis management systems, expressed concern about the government’s attempt to build a new guideline for crisis management. For example, KEPCO already had produced a practical manual for crisis response in the field of electricity in 2004.11 The Korean Expressway Corporation also built a disaster prevention system in 2004 because of snowfall that caused the paralysis of the expressways. In 2005, KOGAS had a safety management system (SMS) named Environment Health Safety Quality. The Korea Airports Corporation had managed the safety of airport facilities according to the Framework Act on Disaster and Safety Management.12 In this regard, public enterprises, which had already complied with the guidelines of governmental ministries or international organizations, were concerned about the overlap of pre-existing guidelines or systems with CMGPO, because this overlap could result in inefficient crisis management. However, in spite of the conflict between the pre-existing crisis management systems and CMGPO, public enterprises had no choice but to follow CMGPO because CMGPO was driven by the NSC under the President’s Office and could be applied to the management evaluation of public enterprises. This is what I call the conflict between the integrated and dispersed crisis management systems. CMGOP as a form of ERM appeared in conflicts between two different styles of crisis management systems. The anxiety of public enterprises was also found in the recommendations of the CMGPO Task Force (2006b) to the NSC. The task force recommended that the government should suggest the standardized model of crisis management as a comprehensive framework in which public enterprises would be allowed to be more flexible in building their own crisis management systems according to their organizational nature. The task force also pointed out that it was necessary to address the opinion that the standardized model driven by the government could be a new coercive burden on public enterprises and to collect the opinions of public enterprises about the efficiency and timeliness of the standardized model. The task force agreed with the possibility of overlap between existing laws (and policies) and CMGPO. It recommended that integrated management will be required to prevent overlap or conflict with the SMS operated by other governmental organizations (CMGPO Task Force 2006b). However, CMGPO faced challenges because laws concerning disasters and national crisis were not integrated at that time. CMGPO could not take precedence over these laws.13 In my interview, one official from the Korea Airports Corporation insisted: In fact, CMGPO was effective at first because it was made by the NSC at the President’s Office, although CMGPO was not a law. However, the biggest weakness of CMGPO is that they cannot prevail over laws. A guideline is just a guideline. A law takes precedence over CMGPO. Therefore, whether or not pre-existing dispersed laws were able to conflict with CMGPO was of the utmost concern.14

In this regard, NSC allowed public organizations to follow existing manuals. In the case of crises without existing manuals, CMGPO could be used for establishing a new crisis management system. For instance, KEPCO is subject to existing laws,

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under the national disaster and safety management system, in the cases of such disasters as electricity accidents and rainfall. By contrast, financial risk management and conflict management without manuals operate in accordance with CMGPO. However, the integration and standardization of crisis typology into four types – management, disasters, public relations, and conflict – were severely critiqued by public enterprises because of the wide variety of specialties and priorities in the tasks of public enterprises. Public enterprises held dissenting opinions in terms of whether or not they had to deal with unnecessary crises because they differed greatly in their prioritizing of crises. For instance, the Korean Airports Corporation was skeptical of dealing with management risk and the risk of public relations, although they frequently dealt with disasters and safety problems.14 In particular, crisis management could be different for disasters in terms of risk identification, evaluation, and perception. The crisis management process can differ according to the risk appetites of public enterprises in that prevention is vital to financial risk management, while disaster response tends to be dominant in disaster management. However, with regard to the previously mentioned critique made by public enterprises, Dr Cheol Hyun Ahn, a former Director-General of the NSC Crisis Management Center, had the following opinions.8 First, a public enterprise is responsible for pursuing the public interest, including crisis management (even as business profitability is still important), particularly because the crises of public enterprises can result in a national crisis. Second, standardization and integrated operation at the governmental level is needed to produce organic relationships among relevant ministries, although these ministries differ in their risk appetite and tasks. Therefore, it is necessary to standardize the crisis management systems of public enterprises as well as to integrate these systems with the governmental crisis management system. Third, the four types of crises defined by CMGPO are commonly found in public enterprises, although public enterprises differ in the degree, extent, and priority of these crises. CMGPO does not establish a uniform priority for crisis management. Fourth, Ahn stressed that what CMGPO tried to do was not enforce the specific content of crisis management but instead suggest the common elements of crisis management with which any public enterprise is able to manage its own crises. Revision of the guideline on the management and innovation of public enterprises and quasi-governmental agencies The ‘guidelines’ in CMGPO are just recommendations without any legal sanction for public enterprises when they violate the CMGPO guidelines. For this reason, NSC searched for a way to improve the effectiveness and utility of CMGPO. At that time, ‘the Guideline on the Management and Innovation of Public Enterprises and Quasi-governmental Agencies’, of the Ministry of Planning and Budgeting, was under revision. The guideline was vital to the annual management performance evaluation of public enterprises. Therefore, to insert the realm of crisis management into this guideline, Dr Cheol Hyun Ahn, Director-General of the NSC Crisis Management Center, visited the Ministry of Planning and Budgeting to persuade the personnel working on budgeting and the management of public enterprises. Without such an enforcing mandate, he thought, public enterprises would not be willing to comply with CMGPO. As a result, in May 2007, the Guideline on the Management and Innovation of Public Enterprises and Quasi-governmental Agencies was revised to include the use of CMGPO in Article 38 as follows:

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Public enterprises and quasi-governmental agencies should plan and enforce crisis management policies in order to systemically manage threat factors and to minimize damages in the wake of crises according to the CMGPO built by the NSC Executive Officer.

The dismantling of the NSC Executive Office and the rise of autonomous crisis management of public enterprises After CMGPO released in 2007, several public enterprises worked to produce new crisis management manuals, business continuity plans, and a situation management system or a crisis management center according to CMGPO. However, as the Lee administration dismantled the NSC Executive Office and the Crisis Management Center, CMGPO was in a state of drift without any driving power. Instead, the Lee administration downsized the NSC Executive Office and created the Crisis Information Team in its place. On 22 July 2008, this team was expanded into the National Crisis Situation Center in the wake of the gunshot murder of a Korean tourist at Kumgang Mountain in North Korea. On 1 August 2009, the National Crisis Situation Center decided to leave CMGPO to the full autonomy of public enterprises by nullifying the content of CMGPO, as the following passage from Article 38 of the Guideline on the Management and Innovation of Public Enterprises and Quasi-governmental Agencies reveals: Public enterprises and quasi-governmental agencies should make crisis management plans voluntarily in order to minimize damage in the face of threats such as the aggravation of the management environment and disasters.

So, public enterprises did not need to comply with CMGPO. This decision appears to accept the critique made by public enterprises that it would be difficult to uniformly apply CMGPO to the crisis management system of public enterprises, because of the unique characteristics of each public enterprise. Ik-Dong Lee, the Director of the National Crisis Situation Center, claimed that the crisis management manuals of public organizations were adequate, and the Center should revise the guidelines so that public organizations could conduct crisis management more autonomously (Disaster Focus, 6 September 2009). The Ministry of Strategy and Finance also did not insert crisis management into the annual performance evaluation of public organizations, instead recommended the activation of crisis management in the Budgeting Guideline of Public Enterprises and Quasi-governmental Agencies. Without mentioning the evaluation of crisis management performance, this guideline urges public enterprises to strengthen crisis management in the face of various threat factors and a rapidly changing management environment. The annual evaluation of management performance mainly focuses on visible business performance such as financial gains, while downplaying crisis management performance. Auditing performance in the public sector is mainly limited to financial auditing and duty surveillance. After all, instead of a standardized, unified, and integrated crisis management, most public enterprises have performed a wide variety of autonomous, dispersed crisis management.

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Discussion and conclusion CMGPO provides a standardized framework for the integrated crisis management of public organizations, based on the concept of ERM, in terms of standardized types of crisis, internal control, and the CRO. The initial entry of ERM into the Korean public sector has the following three implications. First, CMGPO aims to connect crisis management systems of the government and public organizations by building an integrative crisis management system. Within this guideline, the Roh administration worked to innovate the dispersed crisis management of public organizations. In particular, CMGPO conceptualized the crisis management process as similar to a traditional process of disaster management (Petak 1985). However, the guideline invited a warning from Samil Pricewaterhouse Coopers that this process would result in the confusion of public organizations about ERM methodologies because the ERM process of COSO and ISO differs from traditional disaster management processes (preventionpreparedness-response-recovery). So, the NSC planned to first build a standardized process of crisis management similar to the disaster management process, and then introduce the ERM process through the standardized model later.15 CMGPO aimed to operate the crisis management systems of public organizations in tandem with the national crisis management system built on a disaster management processes. In the same vein, the CRO of public organizations also intended to play a significant role in connecting the crisis management of public organizations to the national crisis management system. CROs of public organizations are comparable with ministerial CROs, as established in 2004 by the Framework Guideline for the National Crisis Management. The command and control of the government over public organizations takes place through the interaction of two CROs. CROs of public organizations are responsible for controlling and managing the crisis management of public organizations, as well as for communicating with ministerial CROs. In this light, I can see how ERM is tied to the integrated crisis management of the Korean government. The internal control system of ERM and the standardization of crisis management from the ERM perspective are compatible with the integrative crisis management of the Korean government, which is built on a bureaucratic command and control system. Actually, in theory, the idea of internal control in ERM is not necessarily identical to the idea of bureaucracy. Rather, it is more connected to the new public management theory. However, the Korean government was willing to perform ERM in the public sector in a bureaucratic manner. The Korean government intended to build a uniform system of crisis management that covered public organizations as well as the government. The ERM ideas of both internal control and a CRO are useful for building an integrated system of crisis management within public organizations in close association with the national crisis management system. Second, CMGPO is built on the idea of ‘enforced self-regulation’ (Braithwaite 1982). Enforced self-regulation refers to the connection of state regulation to the self-regulation of companies, in that the government forces public organizations to follow ‘self-regulation’ (Gunningham and Rees 1997; Sinclair 1997). Under enforced self-regulation, internal control is connected to the command and control of outside regulatory agencies. Enforced self-regulation is also called co-regulation, meta-regulation, and hybrid regulation. Enforced self-regulation is different from the traditional regulation of command and control that come from governmental

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regulatory agencies. While traditional regulation in a bureaucratic manner means external control, ERM connects internal control to risk management. It blurs the boundary between the externalization of internal control and internalization of external control (Power 2007, 62). The ‘guidelines’ offered by CMGPO are not mandatory but just recommendations, constituting a voluntary code of conduct. Therefore, the guideline actually lacks the power of implementation. For this reason, the NSC worked to enforce crisis management of public organizations by inserting language concerning crisis management in Article 38 of ‘the Guidelines on the Management and Innovation of Public Enterprises and Quasi-governmental Agencies’. This guideline is related to the annual management performance evaluation of public organizations. Therefore, CMGPO is not built on the concept of self-regulation of traditional ERM in private companies, but on the notion of enforced self-regulation in which the government affects the crisis management of public organizations. Here, bureaucratic command and control become tied to self-regulation within the concept of enforced self-regulation. Finally, CMGPO enables public organizations to run a double internal control system in which the general department first controls executive departments, while the audit department controls both the general and executive departments. That is to say, the double internal control system means ‘control of control’ (Power 2007), in that the audit department conducts ‘secondary risk management’. However, there is no clear rule to mandate that both general departments and the auditing department should conduct internal control together. Whether public enterprises will perform as a single internal control system or a double internal control system is decided by each public enterprise. Moreover, the internal control system of CMGPO resembles post-damage control that depends on periodic audits. It is not aimed at running real-time monitoring that can prevent a crisis before the crisis occurs based on scenario-based foresight of the symptoms of crisis. It has been five years since CMGPO first introduced ERM to the crisis management systems of public enterprises in 2007. In spite of the de facto annulment of CMGPO, Korean public enterprises did not abandon the project, but have established the ERM system voluntarily, in consultation with professional ERM companies and according to either the CMGPO or the international standards of ISO and COSO. In the future, it would be worthwhile to explore how ERM is currently practiced in the Korean public sector. Acknowledgements This research is funded by the Korea Institution of Public Administration. It is partially based on my 2011 report on the Korea Institute of Public Administration entitled ‘Exploring Enterprise Risk Management in the Crisis Management of the Korean Public Sector’. This publication is permitted according to the Principle of Research Documents Management. Special thanks to Cheol-Hyun Ahn, Ji-bum Chung, and Da-Hye Lee for their help.

Notes 1. The UK Prime Minister’s Office launched a two-year cross-Whitehall Risk Program on 2 November 2002. The evaluation of this program in terms of the risk management performance of ministries occurred once a year, in December 2003 and July 2004 (UK HM Treasury 2004).

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2. The 17 public enterprises include the Korea Electricity and Power Corporation, the Korea Minting & Printing Corporation, the Korea Expressway Corporation, the Korea Land Corporation, the Korea Gas Corporation, the Korea Tourism Organization, KORAIL, the Korea Airports Corporation, the Korea National Oil Corporation, the Korea Coal Corporation, K-Water, Incheon International Airport Corporation, the Korea Resources Corporation, the Korea National Housing Corporation, the Korea TradeInvestment Promotion Agency, the Korea Agro-Fisheries Trade Corporation, and the Korea Rural Community Corporation. 3. For instance, integrated crisis management in the USA was on the rise with the establishment of FEMA in 1993 and the Department of Homeland Security after 11 September 2001. 4. According to these guidelines, management risk includes the aggravation of the management environment, financial risks such as money exchange risk, the leak of major technologies and significant information, and systemic paralysis. The category of disasters includes casualties and accidents caused by natural and human-made disasters, as well as technological and operational malfunctions. Communication risk includes the loss of reputation and trust and the rise of blame due to negative news. Finally, conflict risk is the long-term conflict of interest groups as well as the problem of public service in the aftermath of social conflict. 5. The organizational structure has some degree of freedom. The general department can be built independently or can be composed of existing departments such as strategy planning departments, safety management departments, and emergency management departments. 6. An executive department is in charge of performing the actual practices of crisis management, including prevention, preparation, response, and recovery after the crisis. 7. Interview with Dr Cheol Hyun Ahn, former Director-General of the NSC Crisis Management Center (12 October 2011). 8. Email communication with Dr Cheol Hyun, Ahn, former Director-General of the NSC Crisis Management Center (6 November 2011). 9. Interview with Kang, Kyung Hwan from the Korea Minting & Printing Corporation (20 October 2011). 10. The manufacture-related organization is the Korea Minting & Printing Corporation. SOC-related organizations include the Korea Electricity Power Corporation, the Korea Expressway Corporation, Korea Land Corporation, and the Korea Gas Corporation. The service-related organization is the Korea Tourism Organization (CMGPO Task Force 2006b). 11. Interview with an official from the Korea Electricity Power Corporation (12 October 2011). 12. Navigation safety facilities were inspected by the Ministry of Construction and Transportation. Aviation safety had been managed since February 2005 through the SMS. Airport security had been inspected by the International Civil Aviation Organization (Interview with an official from the Korea Airports Corporation, 25 October 2011). 13. For instance, the Korea Airports Corporation deals with laws concerning the management of national critical infrastructure and disaster and safety management. 14. Interview with an official from the Korea Airports Corporation (25 October 2011). 15. Kang-Kyung Hwan, from the Korea Minting, Security Printing & ID Cards Operating Corporation, indicated this at the first workshop concerning the crisis management of public organizations (30 September 2011).

References Arena, M., M. Arnaboldi, and G. Azzone. 2010. “The Organizational Dynamics of Enterprise Risk Management.” Accounting, Organization and Society 35 (7): 659–675. Arena, M., M. Arnaboldi, and G. Azzone. 2011. “Is Enterprise Risk Management Real?” Journal of Risk Research 14 (7): 779–797. Bacchi, C. L. 1999. Women, Policy, and Politics: The Construction of Policy Problems. Thousand Oaks, CA: SAGE.

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Applied Financial Economics, 2012, 22, 59–70

Implied risk aversion and volatility risk premiums Sun-Joong Yoona,* and Suk Joon Byunb a

Department of Finance, Hallym University, Hallymdaehak-gil 39, Chuncheon 200-702, Korea b Graduate School of Finance, KAIST Business School, Seoul, Korea

Since investor risk aversion determines the premium required for bearing risk, a comparison thereof provides evidence of the different structure of risk premium across markets. This article estimates and compares the degree of risk aversion of three actively traded options markets: the S&P 500, Nikkei 225 and KOSPI 200 options markets. The estimated risk aversions is found to follow S&P 500, Nikkei 225 and KOSPI 200 options in descending order, implying that S&P 500 investors require more compensation than other investors for bearing the same risk. To prove this empirically, we examine the effect of risk aversion on volatility risk premium, using delta-hedged gains. Since more risk-averse investors are willing to pay higher premiums for bearing volatility risk, greater risk averseness can result in a severe negative volatility risk premium, which is usually understood as hedging demands against the underlying asset’s downward movement. Our findings support the argument that S&P 500 investors with higher risk aversion pay more premiums for hedging volatility risk. Keywords: risk aversion; volatility risk premium; S&P 500 index options; Nikkei 225 index options; KOSPI 200 index options JEL Classification: G13; G15

I. Introduction Stimulated by the seminal work of Black and Scholes (BS, 1973), derivatives markets have been extensively developed in terms of not only size, but also variety of products. This rapid expansion has led to a tremendous amount of theoretical and empirical literature on the subject. Among them, one popular issue has been to estimate the implied risk aversion of investors from the prices of derivatives securities. The wellknown failure to extrapolate a reasonable risk aversion from underlying stock prices, denoted the equity

premium puzzle, has led to successful analyses estimating risk aversion from derivatives securities, for example, Jackwerth and Rubinstein (1996), AitSahalia and Lo (2000), Jackwerth (2000), Rosenberg and Engle (2002), Bliss and Panigirtzoglou (2004), and Bakshi and Madan (2006). Theoretically investor risk aversion influences the premium required for bearing risk and a comparison can therefore be useful in understanding the different risk premium structures across derivatives markets, with high risk aversion indicating high risk premium and low risk aversion indicating low risk premium for the

*Corresponding author. E-mail: [email protected] Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online ß 2012 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/09603107.2011.597723

59

60 same risk. Despite a surge in studies empirically analysing risk premium, very few have focused on such a comparison.1 This article aims to estimate and compare the risk aversion of three actively traded derivatives markets: the S&P 500, Nikkei 225 and KOSPI 200 index options markets. Risk aversion is commonly estimated using the difference between physical and riskneutral densities, but the specific methodologies in the aforementioned studies differ considerably from one another. They differently assume the preference structure of investors and/or differently estimate the distribution of risk-neutral densities. In contrast, this study adopts the identical method proposed by Bakshi and Madan (2006) in the three options markets, thereby allowing a direct comparison. Bakshi and Madan (2006) define volatility spreads as risk-neutral volatility minus physical volatility divided by physical volatility and find them to be determined by higher moments of physical distribution and the risk aversion of investors. Unless physical density follows a normal distribution and investors are risk-neutral, the volatility spreads will not be zero. Using this theoretical restriction, we can calculate risk aversion. Empirically, volatility spreads are positive and the recent literature including Bollerslev et al. (2009), Carr and Wu (2009) and Todorov (2009) denotes these positive volatility spreads as the variance risk premiums.2 Furthermore, using delta-hedged gains, we investigate the influences of differences in investor risk aversion to volatility risk premiums. The frequently observed negative volatility risk premiums correspond to investor hedging demands. As a market moves down, volatility usually increases. Because the values of options are proportional to volatilities, holding both an underlying asset and options creates a hedging effect against significant market declines. Hence, options buyers are willing to pay a premium to protect against market depreciation (French et al., 1987; Glosten et al., 1993). Since the volatility risk premiums here can be approximated using deltahedged gains, we examine the pattern of such gains to measure the volatility risk premiums in the three previously mentioned options markets. Our findings indicate that the higher the risk aversion, the higher the magnitude of negative risk premium (the effect of hedging demand). 1

S.-J. Yoon and S. J. Byun The main results can be summarized as follows. The implied risk aversion of the S&P 500 options market is highest and that of the KOSPI 200 options market is lowest. This implies that investors in the S&P 500 options market trade more conservatively, while those in KOSPI 200 options market trade more aggressively. This result is consistent with the pattern of volatility risk premiums observed in the three options markets. When estimating them using the delta-hedged gains, the evidence of negative volatility risk premium is prominent in the S&P 500 options market, which has the highest risk aversion investors, and inconspicuous in the KOSPI 200 options market, which has the lowest risk aversion investors. Thus we can conclude that the more risk-averse investors are those who are willing to pay premiums for taking volatility risk. Additionally, to explain the difference in risk aversion and volatility risk premium across the three options markets, we introduce a rationale associated with individual investors who are known to trade speculatively. The remainder of the article is organized as follows: Section II introduces the theoretical background to estimate risk aversion and volatility risk premiums. Section III describes the data used in our empirical study and documents the descriptive statistics of our samples. Section IV presents and compares the volatility spreads and implied risk aversion of the three options markets. Finally, Section V presents our conclusions.

II. Theoretical Background This section briefly reviews the method of Bakshi and Madan (2006) for estimating risk aversion and the relation between volatility risk premiums and deltahedged gains. Volatility spreads and risk aversion Bakshi and Madan (2006) define volatility spreads as V:S: 

q2  p2 p2

ð1Þ

where p2 and q2 represent physical volatility and riskneutral volatility, respectively. The authors also build

Jackwerth (2004) uses the option prices traded in the S&P 500, DAX, FTSE 100 and Nikkei 225 index options markets to show that the U-shape of implied risk aversion is not only a local aspect but also a worldwide phenomenon. However, that study uses only one cross-sectional dataset observed at a particular date. 2 Carr and Wu (2009) define the variance risk premium as realized variance minus risk-neutral variance, while others define it oppositely, i.e. risk-neutral variance minus realized variance. To avoid confusion, in this article, the variance premium indicates the latter case.

Implied risk aversion and volatility risk premiums

61

on the analytic expression of volatility spreads in Theorem 1 of their original paper.

et al. (2003) and Jiang and Tian (2005). According to these studies, the risk neutral variance can be represented as

Theorem 1 of Bakshi and Madan (2006): A class of pricing kernels satisfying the Taylor series expansion around zero can be expressed as 1 m½R  1  A1 R þ A2 R2 þ O½R3  2

ð2Þ

where m½0 ¼ 1, A1  @m=@RjR¼0 , and A2  @2 m=@R2 jR¼0 . Under the power utility class with the pricing kernel m½R ¼ erR for the coefficient of relative risk aversion , up to the second-order of , the -period volatility spreads are theoretically determined as q2 ðt, Þ  p2 ðt, Þ p2 ðt, Þ

 1=2    p2 ðt, Þ  p ðt, Þ þ

 

2  p2 ðt, Þ  p ðt, Þ  3 ð3Þ 2

where q ðt, Þ is the volatility of risk-neutral density, and p ðt, Þ, p ðt, Þ and p ðt, Þ are the volatility, skewness and kurtosis of the physical density, respectively. According to the authors’ conclusion, the divergence between physical and risk-neutral volatilities is determined by the second, third and fourth moments of the physical distribution and the risk aversion of investors. More specifically, if investors are riskaverse and if the physical density is more left-skewed and leptokurtic than the normal distribution (p ¼ 0, p ¼ 3), the volatility spreads will be positive. To investigate volatility spreads empirically, it is necessary to compute several physical moments and the risk-neutral volatility. The moments of physical density, that is, variance, skewness and kurtosis (p2 , p , p ), are defined as follows: Z p2  p2 ðt,  Þ ¼ ðR  P Þ2 pðRÞdR 3 R R  p pðRÞdR p  p ðt,  Þ ¼ p3 R 4 R  p pðRÞdR p  p ðt,  Þ ¼ p4

ð4Þ

where R is the return and p ðq Þ is the mean of the physical (risk-neutral) density. Similarly, the riskneutral variance (q2 ) is defined as follows: Z  2 2 2 q  q ðt,  Þ ¼ R  q qðRÞdR ð5Þ Empirically, the risk-neutral variance can be calculated using the model-free approach of Bakshi

^ q2 ðt,  Þ ¼ er Vt,tþ  ^ 2q

ð6Þ

where r is the risk-free rate between t and t þ ; and t,tþ and Vt,tþ represent the following equations:  Z St  Z1 1 1 r r PðK ÞdK þ CðK ÞdK ^ q ¼ e  1  e 2 2 0 K St K Z1 2ð1  lnðK=St ÞÞ CðK ÞdK Vt,tþ ¼ K2 St Z St 2ð1 þ lnðSt =K ÞÞ þ PðKÞdK ð7Þ K2 0 where C(K ) and P(K ) are the prices of European calls and puts at time t with a strike price of K and maturity of . The physical variance also can be calculated in a model-free manner according to the works of Merton (1980) and Anderson et al. (2003), as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 20  u250 X 2 ð8Þ p ðtÞ  t Rðt þ l Þ  R 20 l¼1 where R is the average return during the calculated period. Volatility risk premiums and delta-hedged gains Bakshi and Kapadia (2003) derive the relation between delta-hedged gains and volatility risk premiums and we summarize their results below. The stock price and its volatility are denoted by St and t , respectively, and their processes are as follows: dSt ¼ t ðS, Þdt þ t dZ1t St dt ¼ t ðÞdt þ t ðÞdZ2t

ð9Þ

where the correlation between two Brownian motions is . Let Cðt, ; K Þ represent the price of a European call maturing in periods from time t with exercise price K; and Dðt, ; K Þ indicate the corresponding option delta. The delta-hedged gains, t,tþ , on the hedged portfolio are given by Z tþ t,tþ ¼ Ct,tþ  Ct  Du dSu t Z tþ  ru ðCu  Du Su Þ du ð10Þ t

where St is the underlying stock price at time t and rt is the risk-free rate at time t.

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S.-J. Yoon and S. J. Byun

Table 1. Specification of samples

Underlying asset Source Sampling time Type of price Sample period Risk-free rate Moneyness (S/K )

S&P 500

Nikkei 225

KOSPI 200

S&P 500 index OptionMetric Closing price Midpoint of best bid and best offer From January 1999 to June 2007 3-month Treasury Bill rate 0.9–1.1

Nikkei 225 index Nikkei Database Closing price Contract price

KOSPI 200 index Korea Exchange Price at 2:50 pm. Contract price

3-month Treasury Bill rate 0.9–1.1

3-month CD rate 0.9–1.1

Note: This table presents the source of the sample, the underlying index, the sampling time, the price type, the sampling period and the range of moneyness with respect to data from each index options market.

In an economy with a constant volatility, the gains on a continuously hedged portfolio will be zero. Even if discretely hedged, the mean of the gains converges asymptotically to zero (Bertsimas et al., 2000). The discretely N times hedged gains are given by

t,tþ ¼ Ct,tþ  Ct  

N 1 X n¼0

N1 X n¼0

Dtn ðStnþ1  Stn Þ

rn ðCt  Dtn Stn Þ

 N

ð11Þ

where t0 ¼ t, tN ¼ t þ . Bertsimas et al. (2000) derive the asymptotic distribution of the discrete delta-hedged gains and show that it is around zero, regardless of the rebalancing frequency. Similarly, Bakshi and Kapadia (2003) show the properties of delta-hedged gains when volatility is stochastic



Et t,tþ ¼

Z

t

tþ

  @Cu Et u du @u

ð12Þ

where  is the volatility risk premium, covðdmt =mt , dt Þ, and mt is the pricing kernel. The delta-hedged gains are determined by the volatility t risk premium, , and the option vega, @C @t . Based on the Ito–Taylor expansion, the delta-hedged gains Et ½t,tþ  are related to the current underlying asset, volatility, maturity and moneyness. For a broad class of option pricing models, option prices are homogeneous of degree one in stock prices and the exercise price (Merton, 1976). In addition, given a fixed moneyness and maturity, the option’s vega is linearly correlated with stock prices. As a result, delta-hedged gains can be expressed as

Et t,tþ ¼ St  gt ðt , , y; t Þ

ð13Þ

where gðÞ is the model-specific function of volatility, time-to-maturity, and moneyness, given the risk premium (Bakshi and Kapadia, 2003; Loudon and Rai, 2007). Finally, the volatility risk premium  is related to the variance risk premium (volatility spread). Chernov (2007) shows that the nonzero volatility risk premium ( 6¼ 0) affects (risk-neutral) implied volatility (variance) and thus changes the volatility spreads (variance risk premium), which is defined as the difference between risk-neutral and physical volatilities (variances). Therefore, the delta-hedged gains, volatility risk premiums, and variance risk premiums are closely related each other.

III. Data Data sampling This article uses the S&P 500, Nikkei 225 and the KOSPI 200 index options traded on the Chicago Mercantile Exchange, Osaka Security Exchange, and Korea Exchange, respectively, together with corresponding underlying indices. The sample period is from January 1999 to June 2007. Table 1 presents the description of each dataset. For the S&P 500 and Nikkei 225 index options, we apply the closing prices, whereas for the KOSPI 200 index options we apply the transaction data captured at 2:50 pm to avoid the synchronizing problem that occurs when we select the index and corresponding options. In addition, the midpoint of the best-bid and best-ask prices are used for the S&P 500 options and the contract prices are used for the Nikkei 225 and KOSPI 200 options. The risk-free rates are approximated by the 3-month London Interbank Offered Rate (LIBOR) for the S&P 500 options and Nikkei 225 options and by the 3-month Certificate of Deposit (CD) rates for the KOSPI 200 options.

63

Implied risk aversion and volatility risk premiums Table 2. Cross-sectional properties of option samples S&P 500

Nikkei 225

KOSPI 200

Number of crosssection

Average number of OTM (Range)

Number of crosssection

Average number of OTM (Range)

Number of crosssection

Average number of OTM (Range)

1 month (20 trading days)

104

102

103

C P Total C P Total

102

2 month (40 trading Days)

C P Total C P Total

C P Total C P Total

19.01 18.84 37.85 15.01 13.35 28.36

(3–56) (3–54) (9–104) (2–56) (3–52) (7–102)

102

4.89 4.23 9.11 5.32 4.10 9.42

(2–9) (2–6) (5–14) (2–9) (2–6) (6–15)

99

6.34 6.86 13.21 6.04 6.08 12.14

(2–12) (2–15) (7–22) (1–20) (1–13) (2–32)

Notes: This table presents the cross-sectional properties of option samples that satisfy the no-arbitrage condition with maturities of 20 and 40 trading days. All samples reported are for Out-Of-Money (OTM) options. The numbers of the crosssectional data range from 99 to 104 and the numbers in parentheses denote the range of options samples.

Table 3. Summary statistics of underlying assets Annualized average return

SD

Skewness

Kurtosis

Autocorrelation

Min

Max

JB-statistic

0.07 0.15 0.04

5.40 4.13 4.80

0.01 0.00 0.01

1501.13% 1501.13% 870.34%

1393.31% 1393.31% 870.34%

515.51 57.16 152.34

1805.44% 1805.44% 88.05%

254.80 82.62 93.96

2104.31% 2104.31% 1259.02%

852.25 108.88 117.06

S&P 500

Full sample Sub 1 Sub 2

2.36% 8.33% 11.83%

17.55% 22.03% 12.27%

Nikkei 225

Full sample Sub 1 Sub 2

3.25% 12.27% 16.94%

21.64% 24.74% 18.49%

KOSPI 200

Full sample Sub 1 Sub 2

14.62% 5.28% 22.83%

31.48% 39.96% 21.42%

0.15 0.03 0.41

0.32 0.25 0.30

4.69 4.42 4.16

0.01 0.01 0.00

6.06 4.56 4.47

0.03 0.03 0.01

1808.50% 1808.50% 1306.46%

3184.74% 3184.74% 1517.04%

Notes: This table presents the summary statistics for daily returns of the S&P 500, Nikkei 225 and KOSPI 200 indexes from 1 January 1999 to 31 June 2007. The dataset is also classified into two subsamples. Subsample 1 includes observations from 1 January 1999 to 31 December 2002, while subsample 2 includes observations from 1 January 2003 to 31 June 2007. For each sample period, this table reports the annualized returns, SD, skewness, kurtosis, autocorrelation, minimum, maximum and Jarque–Bera (JB) statistics for each index. The annualized return is calculated as logðPt =Pt1 Þ  250, where Pt is the index price.

The sample is screened according to the following criteria. First, we delete options that violate arbitrage bounds and options whose implied volatility is less than 5% or more than 100% to reduce the impact of mispriced data. Second, we adopt only 1- and 2-month maturity options to construct the volatility spread. Since options with too short or too long maturity are rarely traded, 1 and 2 months are appropriate maturities for conducting empirical studies. Table 2 shows the cross-sectional properties of the options selected above. For options with maturities of 1 and 2 months, the number of cross sections ranges from 99 to 104. For each cross section, the number of out-of-the-money options ranges from 7 to 104, from

5 to 15, and from 2 to 32 in the S&P 500, Nikkei 225 and KOSPI 200 options, respectively. Descriptive statistics Table 3 shows the descriptive statistics of daily returns for the S&P 500, Nikkei 225 and KOSPI 200 indices. The total sample periods are divided into two subsamples: January 1999 to December 2002 and January 2003 to June 2007. For all indices, the average returns increase when volatility decreases over time. In particular, the shape of changes over time is very similar for the S&P 500 and Nikkei 225 indexes. The skewness of the S&P 500 and Nikkei 225 indexes is approximately zero, albeit with a difference

64

S.-J. Yoon and S. J. Byun

Table 4. Volatility spreads (1 month) Volatility spreads q =p  1 (%)

S&P 500

Nikkei 225

KOSPI 200

Sample period

Risk-neutral volatility q (%)

Physical volatility p (%)

Mean

t-stat.

Indicator ZT (%)

Total sample Subsample 1 Subsample 2 1999 2000 2001 2002 2003 2004 2005 2006 January 2007–June 2007 Total sample Subsample 1 Subsample 2 1999 2000 2001 2002 2003 2004 2005 2006 January 2007–June 2007 Total sample Subsample 1 Subsample 2 1999 2000 2001 2002 2003 2004 2005 2006 January 2007–June 2007

20.53 20.23 20.79 21.1 18.05 19.54 22.21 22.19 18.04 21.29 24.06 17.14 23.43 26.8 20.43 23.23 22.3 31.33 30.33 25.74 21.35 15.08 20.7 18.12 30.46 38.47 23.34 43.14 40.5 33.83 36.43 30.36 25.15 19.72 20.53 18.51

15.83 20.93 11.46 17.52 22.11 19.11 24.99 15.69 10.73 9.64 9.56 11.75 20.42 23.79 17.41 20.3 21.41 29.49 23.98 22.77 17.26 12.98 18.72 13.24 28.75 38.9 19.73 42.07 46.34 35.15 32.03 25.27 22.98 16.25 17.1 14.35

54.05 3.20 97.64 23.19 10.81 6.06 5.64 49.23 72.92 128.22 162.95 63.46 21.19 18.10 23.94 18.39 10.33 12.38 31.32 15.24 27.86 25.04 16.37 46.42 17.63 6.50 27.53 6.16 6.36 8.84 17.36 25.79 22.40 26.96 29.81 37.80

0.75 0.10 1.41 0.85 0.3 0.22 0.24 1.48 2.03 1.98 3.04 0.72 0.77 0.70 0.84 0.76 0.40 0.49 1.21 1.00 1.15 0.71 0.61 1.15 0.49 0.19 0.78 0.19 0.20 0.20 0.69 0.89 0.52 0.90 0.77 0.87

75.96 50 98.21 75 25 58.33 41.67 100 100 100 100 87.5 74.51 72.92 75.93 75 58.33 66.67 91.67 66.67 83.33 75 75 83.33 67.65 56.25 77.78 50 33.33 66.67 75 66.67 75 83.33 83.33 83.33

Notes: This table reports (i) the average risk-neutral SDs (rn ), (ii) the average physical SDs (p ), and (iii) the percentage volatility spreads as (rn =p  1). The t-statistics are calculated as the average estimate divided by its SE. Here, ZT is an indicator that assigns a value of one if rn 4 p , and zero otherwise. The risk-neutral volatility and physical volatility are computed in Equations 6 and 9, respectively. To calculate the risk-neutral volatility, options with a maturity of 20 trading days are selected. The volatility spreads on samples with a maturity of 40 trading days are omitted due to their similarity to the volatility spreads of the data with a maturity of 20 trading days. All numbers are annualized, and the sample period is from January 1999 to June 2007.

in their signs, whiles the KOSPI 200 index has a relatively severe negative skewness. In addition, the distributions of all the indexes are commonly leptokurtic. To compare the volatility spreads, Table 4 documents the four sets of results for the three options markets: the average risk-neutral volatility, q , the average physical volatility, p , the volatility spread,

q =p  1, and the frequency of positive volatility spreads, ZT . All the index options have positive volatility spreads, although their degrees are slightly different. According to Bakshi and Madan (2006), the positive volatility spreads correspond to negative skewness, excess kurtosis of physical density, and/or the nonrisk neutrality of investors. All values are calculated using options with maturities of 1 month.

Implied risk aversion and volatility risk premiums

65

The results of using options with maturities of 2 months are similar to those for options with maturities of 1 month, and thus, for the sake of space, this article omits them. For the total sample periods, the volatility spreads of the S&P 500 options, Nikkei 225 options and KOSPI 200 options are 54.05%, 21.19% and 17.63%, respectively. For each sample year, the volatility spreads are positive except for the years 2000 and 2002 for the S&P 500 options, and for the year 2000 for the KOSPI 200 options. The frequencies at which the volatility spreads are positive are greater than 50%. Overall, these results are consistent with those of Bollerslev et al. (2009), Carr and Wu (2009), and Todorov (2009), which theoretically and empirically prove that the positive volatility (variance) spread reflects the compensation for bearing variance risks that are not diversifiable, and denoted it as a variance risk premium.

2 -distribution with degree of freedom equal to the number of orthogonality conditions minus the number of estimated parameters. If the hypothesis that T  JT is statistically zero is rejected, the representation of volatility spreads may be specified incorrectly. For example, the assumption of the constant relative risk aversion preference may be misspecified or fourth and higher moments may affect the volatility spreads. An important empirical issue is the choice of sampling windows when estimating physical moments. A narrow sample window results in the underestimation of higher moments such as skewness and kurtosis, and a wide sample window results in the overestimation. Thus an inappropriate estimation window can lead to erroneous higher moments, which tend to wrongly reject the theoretical restriction above. To avoid this, we use both 3-month and 6-month returns to estimate skewness and kurtosis, following Bakshi and Madan (2006). Additionally, we adopt two variable sets as instrumental variables:

IV. Empirical Results

(i) Set 1: a constant and risk-neutral volatility lagged by one period q ðtÞ. (ii) Set 2: a constant, the risk-neutral volatilities lagged by one and two periods, q ðtÞ, q ðt  1Þ.

Volatility spreads and risk aversions Following Bakshi and Madan (2006), we construct the following error term: "ðt þ 1Þ ¼

q2 ðt þ 1Þ  p2 ðt þ 1Þ

p2 ðt þ 1Þ  1=2  p ðt þ 1Þ þ  p2 ðt þ 1Þ 

 

2  p2 ðt þ 1Þ  p ðt þ 1Þ  3 2

ð14Þ

T 1X "ðt þ 1Þ  Zt T t¼1

ð15Þ

As shown in previous sections, the volatility spreads are determined by the skewness and kurtosis of the physical density and investor risk aversion. Thus, if the volatility spreads, skewness and kurtosis are calculated, the risk aversion can be estimated using the Generalized Method of Moments (GMM) (Hansen, 1982). Based on the orthogonality conditions of the error with respect to the instrumental variables Zt , the following moments converge to zero as the sample size T increases: gT ½  

Next, we construct the GMM estimator by minimizing the criterion function JT  min g0T WT gT

ð16Þ

where WT is the inverse of variance–covariance matrix of gT . Here T  JT asymptotically follows a

Table 5 shows the risk aversions of the S&P 500, Nikkei 225 and KOSPI 200 index options. Table 5 contains the risk aversion values for the two sets of instrumental variables, using higher-order physical moments based on both returns lagged by 60 trading days and returns lagged by 120 trading days. We also consider conditional heteroscedasticity and autocorrelation when estimating the weighting matrix, WT . Here, the lag in the moving average is set to three. For all samples, restriction (16) is not rejected based on the p-value of JT statistics. Additionally, most of the estimates are statistically significant regardless of the selection of instrumental variables and maturities. For a maturity of 1 month, the risk aversion values estimated from the S&P 500, Nikkei 225 and KOSPI 200 options are about 10.64, 6.18 and 2.74, respectively. For maturities of 2 months, their values are 6.29, 5.06 and 1.56, respectively, which are slightly small compared to those with a maturity of 1 month, but the pattern is very similar. The negative relation between implied risk aversion and maturity can be interpreted as follows. Since options with a long maturity are useful to hedge underlying asset movements for a long time without rollover, they are preferred more by hedgers than by speculators. Hedgers appear to be more risk averse than others, and thus the risk aversion implied for options with a longer maturity appears to be high.

66

S.-J. Yoon and S. J. Byun

Table 5. Implied risk aversions 60-day sample

S&P 500

Horizon

IV

1 month

Set Set Set Set Set Set Set Set Set Set Set Set

2 month Nikkei 225

1 month 2 month

KOSPI 200

1 month 2 month

1 2 1 2 1 2 1 2 1 2 1 2

120-day sample

df

tð Þ

JT

p-value

tð Þ

JT

p-value

1 2 1 2 1 2 1 2 1 2 1 2

10.64 11.05 6.29 6.03 6.18 5.89 5.06 4.91 2.74 2.72 1.56 1.48

2.87 2.94 1.23 1.16 4.54 4.61 3.62 3.53 3.22 3.26 1.86 1.76

3.70 3.96 2.33 2.61 0.27 0.91 0.69 2.77 1.62 3.20 1.21 2.49

0.15 0.26 0.12 0.27 0.60 0.63 0.40 0.25 0.20 0.20 0.27 0.28

8.59 8.35 4.02 5.02 4.54 3.72 3.63 2.92 2.16 2.08 1.12 1.18

2.82 2.75 1.10 2.06 5.20 5.65 4.15 3.81 3.13 3.08 1.66 1.80

2.63 2.85 2.38 2.69 0.01 3.18 0.29 4.54 2.15 3.24 2.17 2.89

0.11 0.23 0.13 0.44 0.94 0.20 0.58 0.10 0.14 0.20 0.14 0.23

Notes: This table presents the implied risk aversions derived from the volatility spreads of the S&P 500, Nikkei 225 and KOSPI 200 index options with maturities of 20 and 40 trading days. The methodology applied to the empirical test is supported by the work of Bakshi and Madan (2006). According to these authors, the relation E½"ðt þ 1ÞjZðtÞ ¼ 0 should be satisfied for a set of information variables ZðtÞ, as

"ðt þ 1Þ ¼

2 rn ðt, Þ  p2 ðt, Þ

p2 ðt, Þ

 1=2  

2 þ  p2 ðt, Þ p ðt, Þ   p2 ðt, Þ  p ðt, Þ  3 2

where p and p are the conditional skewness and kurtosis of the physical index distribution, respectively. The risk aversion is estimated using the GMM. The reported p-value is based on the minimized value of the GMM criterion function, JT . Here, T  JT follows the 2 -distribution with degrees of freedom, df, which is the number of instrumental variables minus one. Two sets of instrumental variables are considered: Set 1 contains a constant and rn ðtÞ; and Set 2 contains a constant, rn ðtÞ, and rn ðt  1Þ. The skewness and kurtosis are calculated from returns lagged by 60 trading days.

This pattern is consistent with the results of Bliss and Panigirtzoglou (2004), in which the risk aversion is 9.52, 5.38, 6.85 and 4.08 for options with maturities of 1 week, 2 weeks, 3 weeks and 4 weeks, respectively. The right part of Table 5 shows risk aversion estimated using returns lagged by 120 days when we calculate the higher moments of physical distribution. Since the sample window is longer, the risk aversion is lower. This is because the use of longer sample window results in an increase in skewness and kurtosis, thereby underestimating the risk aversion that satisfies the restriction of volatility spreads. Delta-hedged gains This subsection documents the empirical and statistical properties of delta-hedged gains linked to volatility risk premiums. Since the volatility risk premiums correspond to hedging demands against market declines, the differences in degrees of risk aversion that we formerly investigate may affect the volatility risk premiums. We adopt the nonparametric approach of Bakshi and Kapadia (2003), in which discrete delta-hedged gains can be used to test for the effects of volatility risk premiums.

This approach does not impose any specification on the pricing kernel or the volatility process, thereby freeing the results from misspecification errors. We use a portfolio of a long position in a call, dailyhedged by a short position in the underlying asset equal to the option delta, @C=@S, until the maturity date. The portfolio is not hedged for any risk factors other than market risk, and the gains on the portfolio are termed ‘delta-hedged gains’ (Bakshi and Kapadia, 2003). For tractability, a delta-hedge ratio Dtn is implemented as the BS hedge ratio, Nðd1 Þ, where NðÞ is the cumulative normal distribution and   2 u lnðS=K Þ þ r þ 0:5t,tþ d1 ¼ ð17Þ pffiffiffiffi t,tþ u

Under the allowance of time-varying volatility, BS delta will bias the delta-hedged gains if the volatility is correlated with the stock returns. As shown by Bakshi and Kapadia’s (2003) simulation, however, the bias resulting from the BS delta is negligible. Thus we use the BS delta-hedge ratio for simplicity. To construct a daily hedge ratio in Equation 17, we have to estimate the return volatility. For robustness,

67

Implied risk aversion and volatility risk premiums Table 6. Delta-hedged gains for S&P 500, Nikkei 225 and KOSPI 200 index call options 1 month (in %) Moneyness (S/K1)%

N

=S

S&P 500 options

10 to 7.5% 7.5 to 5% 5 to 2.5% 2.5 to 0% 0 to 2.5% 2.5 to 5% 5 to 7.5% 7.5 to 10%

158 303 439 454 398 278 154 85

Nikkei 225 options

10 to 7.5% 7.5 to 5% 5 to 2.5% 2.5 to 0% 0 to 2.5% 2.5 to 5% 5 to 7.5% 7.5 to 10%

71 83 66 68 68 67 50 33

0.00 0.01 0.02 0.05 0.05 0.05 0.01 0.01

KOSPI 200 options

10 to 7.5% 7.5 to 5% 5 to 2.5% 2.5 to 0% 0 to 2.5% 2.5 to 5% 5 to 7.5% 7.5 to 10%

94 120 128 117 116 96 95 61

0.15 0.18 0.23 0.21 0.23 0.10 0.16 0.22

0.29 0.38 0.24 0.27 0.04 0.07 0.03 0.15

2 months (in %) 150

N

=S

(14.78) (11.90) (6.50) (2.13) (0.93) (0.57) (0.51) (0.51)

67.08 59.07 53.53 59.25 57.78 62.23 53.24 45.88

124 210 268 305 211 111 70 34

0.07 0.10 0.13 0.16 0.10 0.07 0.11 0.10

(0.04) (0.03) (0.03) (0.03) (0.05) (0.06) (0.06) (0.09)

(38.41) (11.44) (9.19) (4.76) (2.43) (1.75) (1.34) (1.23)

71.83 67.47 66.67 72.05 70.58 58.20 70.00 72.72

75 75 75 75 58 57 47 33

(0.08) (0.10) (0.10) (0.10) (0.10) (0.14) (0.16) (0.17)

(27.23) (16.98) (6.70) (2.75) (2.18) (1.74) (1.34) (1.25)

65.95 68.33 64.06 64.95 55.17 56.25 47.36 36.06

92 116 121 107 107 74 47 32

0.12 0.27 0.37 0.20 0.23 0.13 0.28 0.38

=C (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.04) (0.04) (0.05) (0.06) (0.08) (0.07) (0.07) (0.09) (0.09) (0.11) (0.10) (0.10) (0.07) (0.09) (0.09) (0.10) (0.11) (0.11)

31.96 37.43 13.79 5.03 1.38 0.85 0.10 0.04

85.77 45.25 22.75 11.17 6.79 1.20 2.13 2.55

26.09 36.28 20.40 7.49 1.22 0.00 0.96 2.13

0.54 0.43 0.30 0.22 0.28 0.09 0.47 0.25

150

=C

(0.11) (0.10) (0.10) (0.12) (0.16) (0.16) (0.21) (0.26)

(23.42) (13.81) (4.39) (1.77) (1.27) (1.08) (0.81) (0.90)

58.87 55.23 62.68 68.19 63.50 59.45 65.71 47.05

(15.36) (9.33) (5.92) (3.42) (2.26) (2.51) (2.03) (1.69)

69.33 66.67 78.67 62.67 72.41 54.38 63.82 60.60

39.24 (12.34) 9.41 (7.30) 4.99 (3.80) 0.97 (3.48) 0.34 (2.42) 3.96 (2.31) 6.77 (2.68) 3.39 (2.47)

70.65 64.65 58.67 58.87 57.94 48.64 36.17 40.62

31.27 34.48 11.14 5.20 2.07 1.07 1.22 0.92

21.04 25.07 18.93 5.48 4.61 0.68 2.45 3.41

Notes: This table presents the delta-hedged gains on a portfolio of a long position in a call, hedged by a short position in the underlying asset. The options delta is computed as the BS hedge ratio. The portfolio is rebalanced daily. We report the deltahedged gains normalized by the index, =S, and the delta-hedged gains normalized by the option price, =C. The moneyness is defined as the underlying asset over the strike price, S=K. The SEs are shown in parentheses. Here, 150 is the proportion of delta-hedged gains with  5 0 and N is the number of options.

the return volatilities are calculated using two methods: Generalized Autoregressive Conditional Heteroscedasticity(1,1) (GARCH(1,1)) and the sample SD. The GARCH(1,1) model is adopted over the entire range of the sample period Rt1,t ¼ c þ et

2 t2 ¼ !0 þ e2t1 þ t1

ð18Þ

where the -period return is defined as Rt,tþ ¼ lnðStþ =St Þ and the -period GARCH volatility is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 250 X VOLG ð19Þ ^ 2 t ¼  n¼t n with the fitted value from GARCH estimation (^n2 ). The other volatility estimate is the sample SD of daily returns, as in Equation 8. Both volatility estimates provide very similar results and thus to save space, we document only the results based on the GARCH estimate.

As shown in Equation 12, delta-hedged gains tell us two important facts about volatility risk premiums: the sign of delta-hedged gains coincides with that of the volatility risk premium, and the volatility risk premium mainly influences the sign of delta-hedged gains on At-The-Money (ATM) options, since their vegas are higher than those of in-the-money and outof-the-money options. Using these relations, Bakshi and Kapadia (2003) adopt a cross-sectional and timeseries analysis to support the existence of a volatility risk premium in the S&P 500 index options market. Our article also applies a method similar to that of Bakshi and Kapadia (2003) in the three options markets, but our main concern is only to compare the magnitude of their delta-hedged gains. Thus we focus on the cross-sectional pattern of delta-hedged gains and the hedged gains on ATM options, given fixed maturities. First, we calculate the delta-hedged gains to understand their general pattern, classified by moneyness and maturity. Table 6 includes the delta-hedged gains on an index call option portfolio in the

68 S&P 500, Nikkei 225 and KOSPI 200 options markets. The reported numbers are the average values scaled by the index level, =S, and the average values scaled by the call price, =C. To guarantee that the numbers are not contaminated by extreme values, we include the frequency of negative delta-hedged gains in the last column. The cross-sectional pattern of delta-hedged gains of the three options markets is as follows. For all three options markets, the delta-hedged strategy loses money over most ranges of moneyness and maturities. For S&P 500 options, it loses money by a maximum of 0.05% and 0.16% of index levels for 1- and 2-month samples, respectively. For Nikkei 225 options, it loses by 0.23% and 0.38% of index levels for 1- and 2-month samples, respectively. Finally, for KOSPI 200 options, it loses money by 0.38% and 0.54% of index levels for 1- and 2-month samples, respectively. Only options with few ranges of moneyness earn money through this strategy. That is, Nikkei 225 options and KOSPI 200 options have higher negative delta-hedged gains to the index level, =S, than S&P 500 options. However, their cross-sectional pattern that coincides with the negative volatility risk premium is shown only in S&P 500 options, the hedged gains of which are proportional to the option vega. The hedged gains of Nikkei 225 options are similar over the entire range of moneyness and those of KOSPI 200 options are highest in the out-of-the-money range. That is, their hedged gains are inconsistent with the presence of a negative volatility risk premium. For the three index put options, the delta-hedged gains have a similar pattern, except that their magnitude is slightly higher. Thus it appears that the negative volatility risk premium is prominent in the S&P 500 index options market. Since the negative volatility risk premium is expected as the hedging demand for market declines, this result is consistent with the pattern of risk aversion among the three options markets. To more directly compare the size of the volatility risk premium, we focus on the delta-hedged gains of ATM options. According to Equation 12, ATM options are most sensitive to the volatility risk premium and thus their delta-hedged gains reflect the sign and magnitude of the volatility risk premium, given a fixed vega. To fix the vega, we employ only fixed maturities of ATM options. If the volatility risk requires a risk premium, the delta-hedged gains on ATM options should differ from zero and their sign should be same with that of the volatility risk premium. Table 7 shows the mean delta-hedged gains and statistics for ATM calls with maturities of 20 and 40 trading days. Here, ATM is defined as the range of moneyness(S/K  1) from 2.5% to 2.5%.

S.-J. Yoon and S. J. Byun Table 7. Mean delta-hedged gains and statistics for ATM calls with maturities of 20 and 40 trading days ATMs (2.5 to 2.5%)

Panel A: S&P 500 index options Number of options Mean delta-hedged gains (in dollars) Standard t-statistics Bertsimas et al.’s (2000) t-statistics (Historical value:  ¼ 2:36,  ¼ 17:51) Panel B: Nikkei 225 index options Number of options Mean delta-hedged gains (in yen) Standard t-statistics Bertsimas et al.’s (2000) t-statistics (Historical value:  ¼ 3:25,  ¼ 22:13) Panel C: KOSPI 200 index options Number of options Mean delta-hedged gains (in won) Standard t-statistics Bertsimas et al.’s (2000) t-statistics (Historical value:  ¼ 14:62,  ¼ 32:05)

1 month

2 months

852 0.59

516 1.28

2.73 0.35

3.54 0.50

136 28.40

133 27.27

3.01 0.33

2.63 -0.36

233 0.15

214 0.26

2.35 0.22

2.58 0.35

Notes: This table presents the statistics for ATM calls with the maturities of 20 and 40 trading days. Panel A, B and C contain the average hedged gains and the t-statistics of the S&P 500, Nikkei 225 and KOSPI 200 options, respectively. The physical volatility required for calculating hedge ratios is measured using the GARCH(1,1) model.

For 20- and 40-day S&P 500 calls, the mean gains are 0.59 and 1.28, which correspond to t-statistic values of 2.73 and 3.54, respectively. For 20- and 40-day Nikkei 225 calls, these are 28.40 and 27.27, which correspond to t-statistic values of 3.01 and 2.63. Lastly, for 20- and 40-day KOSPI 200 calls, these are 0.15 and 0.26, respectively, which indicate t-statistics of 2.35 and 2.58, respectively. Accoridng to Table 7, the overall t-statistics of S&P 500 options are higher than the others. Only Nikkei 225 options with a maturity of 1 month have higher t-values than the S&P 500 options, but the difference is not much and, furthermore, the cross-sectional result for the Nikkei 225 options in Table 6 do not support the existence of a negative volatility risk premium. In addition, the t-statistics of the KOSPI 200 options are the lowest among the three options markets, which is consistent with the cross-sectional pattern observed in Table 6. Additionally, we adopt an alternative method that uses the SD of discrete delta-hedged gains. Under a

Implied risk aversion and volatility risk premiums

69

BS economy in which the expected return and volatility are constant, the SD of delta-hedged gains is found by Bertsimas et al. (2000). ^ t,tþ is calculated by standardizing each hedged gain by the corresponding pSD ffiffiffiffi and the t-statistics is computed as P ^ t,tþ = N, where N is the number of observations. These results are also documented in Table 7. For S&P 500 options, the t-statistics of 1- and 2-month samples are 0.35 and 0.50, respectively, with the historical values of  ¼ 0:0236,  ¼ 0:1751; for Nikkei 225 options, the t-statistics are 0.33 and 0.36 with historical values of  ¼ 0:0325,  ¼ 0:2213, respectively; and for KOSPI 200 options, the t-statistics are 0.22 and 0.35 with the historical values of  ¼ 0:1462,  ¼ 0:3205, respectively. It is important to note that this method starts with the assumption of a BS economy, and thus these results may be inconsistent for the true process with time-varying volatilities. So, if the daily-updated GARCH volatilities are used in calculating delta-hedged gains, the bias will decrease and the results will be reliable (Bakshi and Kapadia, 2003).

markets is in contrast to the participation share of individual investors. Generally, individual investors are considered speculators without relevant information, and they prefer trades with which they can earn a large amount of money using a small outlay of cash. This is a typical characteristic of investors with low risk aversion. The implied risk aversion estimated in this article is the combination of all investors’ risk aversions and the weight of individual investors with low risk aversion can influence the magnitude of total risk aversion negatively. As a result, the participation of individual investors can be a possible reason behind the discrepancy of implied risk aversion. For reference, earlier literature provides the following information regarding individual investors’ trading behaviours in derivatives markets. Lakonishok et al. (2007) and Kang and Park (2008) show that, for index options markets, individual investors prefer trades of out-of-the-money options and support the direction-learning-hypothesis, such that an investor trades based on forecasts for future index changes. These trades require smaller amounts of money than trades of in- and ATM options, and thus can have a leverage effect. Similarly, Barber and Odean (1999, 2000, 2008) and Barber et al. (2004) report that individual investors have a tendency to make trades dependent on changes in past returns, using technical analysis. Such traders typically experience persistent losses on their trades.

Interpretations So far, we have compared the implied risk aversion and delta-hedged gains across the S&P 500, Nikkei 225 and KOSPI 200 options markets and found a close relation between the two. S&P 500 options are mostly affected by the volatility risk premium, followed by the Nikkei 225 and KOSPI 200 options, which is in contrast to the pattern of implied risk aversion. This phenomenon can be interpreted as follows. Because the negative volatility risk premium is comprehended as hedging demands for market declines, the more risk-averse investors pay greater premiums for hedging volatility risk, thereby making volatility risk premiums different across options markets. However, we cannot answer why the investors of the three options markets have different level of risk aversion. This could be an interesting research issue, but is beyond the scope of this article. Instead, we propose a possible rationale behind the different risk aversions. The background of this rationale is based on the behaviour of individual investors. The Bank for International Settlements (BIS) Quarterly Report (BIS Monetary and Economic Department, 2005) announces the trading shares of individual investors of the three options markets: the share in KOSPI 200 options is greater than 50%, that in Nikkei 225 options is approximately 10%, and that in S&P 500 options is smaller still. That is, the sequence of investor risk aversion in the three options

V. Conclusion This article compares investor risk aversions on the S&P 500, Nikkei 225 and KOSPI 200 index options markets. Since risk aversion determines a premium for bearing risk, this comparison provides information on the different structures of risk premiums across financial markets. According to the results, the implied risk aversion of the S&P 500, Nikkei 225 and KOSPI 200 options is about 10.64, 6.18 and 2.74, respectively, for a maturity of 1 month. This implies that S&P 500 option investors require more compensation than other option investors for bearing the same risk. Furthermore, using delta-hedged gains, we confirm the effect of risk aversion on the volatility risk premium. Because more risk-averse investors are willing to pay greater premiums for hedging volatility risk, a higher degree of risk aversion can result in a severe negative volatility risk premium, which is usually understood as hedging demands against the underlying assets’ downward movement. Our results show that S&P 500 option investors

70 with higher risk aversion pay more premiums for hedging volatility risk. This study is, however, not free from limitations. Although we provide evidence on the difference in risk aversion and its impact on volatility risk premiums, we cannot answer why the degrees of risk aversion are so different across options markets. To answer this question rigorously, we need a sophisticatedly designed survey, which is beyond the scope of this article. In addition, this study is restricted to the assumption of the power preference of the representative agent. Many recent studies explain the behaviour of investors by assuming other classes of preferences, and thus such an assumption may improve the results.

Acknowledgement This research was supported by Hallym University Research Fund, 2011(HRF-201109-039).

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Journal of Risk ResearchAquatic Insects Vol. 14, No. 10, November 2011, 1161–1176

Internal loss data collection implementation: evidence from a large UK financial institution Cormac Bryce*, Robert Webb and Jennifer Adams Glasgow Caledonian University, Glasgow, UK (Received 25 February 2011; final version received 23 May 2011) The paper conducts a critical analysis of internal loss data collection implementation in a UK financial institution. We use elite semi-structured interviews, with a sample of 15 operational risk consultants from a leading international financial institution. Using content analysis, the data covers a wide range of business areas, with particular attention drawn towards the development of internal loss collection and operational risk management. The results suggest that the development of operational risk management as a function stems from external compliance (Basel II) and the internal pressure to add value to the business portfolio. This need for compliance was augmented as a driver of internal loss data collection; however, participants also recognised that the function of loss data collection is a tool of solid internal risk management and enhances managerial decision-making. The research also highlights the problems in cleansing data in order to ensure that all information implemented in the capital allocation model is valid and reliable. Keywords: loss data collection; operational risk; Basel II; regulatory capital allocation

1. Introduction It has long been established that an economy’s banking sector plays a key role in economic development and growth. Traditionally, this has been achieved by a process of intermediation, but more recently, over the past few decades, banks have undertaken increasing amounts of off-balance sheet business that has increased the income generated by non-interest areas of the banking firm. It has also become ever more noticeable that banking (and bank performance) is heavily reliant on risk measurement and consequently risk management. As Cebenoyan and Strahan (2004) suggest, it is difficult to imagine as many risks being managed jointly as in banking. Given the level of risk in bank operations, and the importance of the sector to the efficient functioning of the economy, it is no coincidence that domestic supervisory regulation has shaped the way in which UK financial institutions conduct their day-to-day operations. However, due to the global nature of banking, domestic

*Corresponding author. Email: [email protected] ISSN 1366-9877 print/ISSN 1466-4461 online Ó 2011 Taylor & Francis http://dx.doi.org/10.1080/13669877.2011.591501 http://www.tandfonline.com

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regulation works alongside international agreements (where agreement can be found) with the Basel II Accord approved by the European Parliament being an example. The Accord aims to provide stronger incentives for improved risk management whilst developing a greater understanding of current industry trends and practices for the managing of risk. Power (2005) highlights the extended regulatory requirements of Basel II along with the Sarbanes Oxley Act (2002) as predisposing factors that have initiated the rapid development of operational risk management within banks – both in the UK and worldwide. Hoyte (2005) and Power (2005) further suggest that Basel II reflects a general climate of regulatory attention to organisational control systems and cultures of control, making operational risk a key component of global banking regulation. The framework comprises three pillars that include: (Pillar 1) minimum capital requirements, (Pillar 2) supervisory review process and (Pillar 3) market discipline. These pillars are developed in a bid to reflect the way institutions are really managed with the aim being not to raise or lower the overall level of regulatory capital held by financial institutions, but to enable regulatory requirements to be assessed on a more risk sensitive basis (see Stirk and Rew 2001 or Lastra 2004). This riskbased capital approach supplements existing fixed minimum capital standards with a fluid, formula-based minimum capital standard, which rises and falls appropriately to a financial institution’s portfolio (Barth 2001). One of the main justifications for regulating bank capital is to mitigate the potential of failures to create systemic risk throughout an economy (see Santos 2002; Rowe, Jovic, and Reeves 2004; Kroszner 1998). In addition, Pillar 1 includes the risk categories of credit risk and market risk, which has been further enhanced by the introduction of operational risk. Justification of which can be underpinned by the difficulties at Northern Rock and the continued difficulties in world markets as witnessed by the fall of Lehmann Brothers or Bradford and Bingley. For the purposes of our research, we utilise the Basel Committee’s definition of operational risk, which they believe is ‘the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events’ (1998, 3) considers that unchecked operational risks can be fatal for financial institutions of all sizes, which would then inevitably have consequences for the economy and the wider society as losses attributed to operational risk may result in contaminating the domestic (and possibly the international) financial environment. Examples of the so called ‘domino effect’ have been much in the news recently with the widespread credit problems in the United States affecting institutions around the globe. However, there have been other major operational risk events in banking over the past decade which the authorities have managed in the interests of the economy to limit in effect, examples being Allied Irish, National Australia Bank, or Socitete Generale to name a few. The Basel Committee (2003) argues that such risk events are amplified by a number of contributing factors such as deregulation, globalisation and the growing sophistication and complexity of many financial products. Such complexity has made many off balance sheet instruments difficult to value, and consequently made financial institution risk profiles difficult to measure. The paper begins with a look at operational risk and Basel II in more detail, and then introduces the methodology before presenting the results. The conclusions follow.

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2. Operational risk and Basel II Power (2005) describes the process of capital regulation as logic of prudential banking, which requires individual financial institutions to maintain a buffer adequate to cover unexpected losses. In the international financial environment, capital adequacy was first recognised formally by the Basel 1 Capital Accord (1988), which made important contributions to the prominence of bank capital regulation and promoting and improving international convergence of capital standards (Santos 2002). However, it must be recognised that the body responsible for the first capital accord, the Basel Committee for Banking Supervision (BCBS) has no national regulatory powers, but has de facto power via the implementation of its recommendations by national supervisory bodies, and within the UK, this is conducted by the FSA (Power 2005). Influenced by the shortcomings of Basel I (see for example Jobst 2007; Petrou 2002; Vieten 1996; Underhill 1991; Jones 2000; Rowe, Jovic, and Reeves 2004; Santos 2002; Power 2005) the Basel Committee began to develop a new strategy that recognised for the first time the calculation of regulatory capital based in part on the risk models and systems of the banks themselves (Power 2005). One area that had previously been ignored by Basel, but had emerged as an important issue in recent risk events, was the need for more cohesive operational risk measurement and management. As a result, Pillar 1 of the Basel II capital accord focused for the first time upon operational risk, and as Power (2005, 583) states: the category of operational risk is significant not merely as an assembly point for existing risk management practices, but also as a category of attention and visibility for threats which were either ignored by banks, or made insufficiently explicit in their management systems.

This view is further supported by de Fontnouvelle et al. (2006) and Hiwatashi (2002) who suggest that financial institutions have now started treating operational risk as a homogenous risk category that can be both managed and measured – but, this has most likely arisen after initial soundings coming from Basel. The Basel II framework provides three approaches of increasing sophistication to calculate risk-based capital within the area of operational risk. Under the most sophisticated approach formally known as the Advanced Measurement Approach (AMA), there is a firm specific calculation of the capital requirement based on the firms’ internal risk measurement models (see Netter and Poulsen 2003 or Moody’s Investor Service 2002). These models use the institution’s own metrics to measure operational risk including internal and external loss data, scenario analysis and risk mitigation techniques to set up its capital requirements (Alexander 2003; de Fontnouvelle et al. 2006; Netter and Poulsen 2003). This approach imposes a number of modelling assumptions upon the financial institution that have to be made as the accuracy in predicting future loss values depends on the volume and quality of historical data (Embrechts, Furrer, and Kaufman 2003). As a result, some institutions are adopting the Loss Distribution Approach (LDA) where capital calculations are based on an historical database of operational loss events involving frequency and severity of events. Using this approach, a financial institution details its loss experience via ‘internal loss data’, with institutions gathering data from the general ledger or from other pre-existing sources for this purpose (see Embrechts, Furrer, and Kaufman 2003; de Fontnouvelle et al. 2006; Haubenstock and Hardin 2003).

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Alexander (2003) and Moosa (2007) highlight two ways of implementing the internal LDA. First, the ‘bottom-up’ approach is based on an analysis of loss events in individual business processes, and attempts to identify each type of risk at that level. In contrast, the ‘top down’ approach calculates losses at a firm level and then attempts are made to disseminate and allocate losses to the various business units. There is little evidence to suggest which approach best suits the process of data collection – with various opinions expressed in the literature for both (see Haubenstock and Hardin 2003; Gelderman, Klassen, and van Lelyveld 2006; Allen and Bali 2004; Currie 2004). Furthermore, de Fontnouvelle et al. (2003) argue that little modelling of operational risk has occurred in the past due to the lack of a structured internal loss data collection process. As a result, financial institutions have tended to allocate operational risk capital via a top-down approach. Problems are, however, further exacerbated because both approaches show flaws due to the data within the LDA only covering short periods of time and thus containing few really large operational losses (de Fontnouvelle et al. 2006). Pezier (2003) and Allen and Bali (2004) believe that these losses are ‘tail events’ that are low in frequency yet high in severity making them particularly elusive to capture, and crucial to the stability of a financial institution. Unfortunately, the LDA has been attacked for having further limitations. Alexander (2003) for example believes that the approach is too data intensive with a minimum of three years of historical loss data required in order to comply with Basel requirements. Second, the approach is potentially backward looking, which, in markets that evolve rapidly, may mean that the data will not fully reflect future risks (Di Noia and Di Giorgio 1999; Briault 2002; Jordan 2001; Claessens, Glaessner, and Klingbel 2002; Aue and Kalkbrener 2007). Furthermore, Power (2005) suggests that the collection of loss data may be misconstrued as people concentrate on historic losses whilst ignoring critical events such as near misses and potential losses. The exclusion of such events may be detrimental to financial institutions as Muerman and Oktem (2002) and Phimster et al. (2003) believe that crises and accidents are generally preceded by a build up of clear signs and anomalies. Consequently, Goodhart (2001) and Wei (2007) argue that because of these ambiguities about data collection and use, and also because significant single event operational losses are rare, databases are ‘shaky and fragile’. Although Wilson (2001) believes that given the difficulty of quantifying aspects of operational risk, the reliance on a single number (capital allocation) may itself be an operational risk, leading authors such as Andres and Van der Brink (2004) and Kuhn and Neu (2005) to discuss alternative approaches to fulfilling AMA in the form of scorecard and scenariobased approaches. In contrast, it has been argued that a comprehensive and reliable internal lossevent database will not only aid a financial institution in addressing the Basel II requirements but will also create a solid base for an effective operational risk-management approach (Moody’s Investor Service 2002). Harris (2002) further believes that despite all the perceived weaknesses, the loss-event database is necessary for the effectiveness of sound risk management and capital allocation. Given the lack of agreement on the approach and effectiveness of the LDA, most would concur that a sound operational risk management framework has numerous advantages to a financial institution. Kalita (2003) believes that this strengthens security and safety not only in the financial institution but also in the financial system as a whole. The Basel II framework is developed through a process

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of identifying, measuring, reporting and controlling risks with the fundamental structure of the framework being developed from the ten qualitative principles within Basel II (Haubenstock 2002). The Basel Committee (1998) stated that institutions throughout the 1990s had started to devote substantial time to reviewing and revamping new policies and procedures – signifying the importance of policy on an institutions operational risk framework. Although this framework is directly related to the operational risk within a bank, it does not take into account the importance or holistic nature of operational risk to the synergy of risk management within the entire institution (Meagher and O’Neill 2000). Such an approach would create an organisational risk profile which is malleable – allowing for the organisation to retain some risks whilst shedding others (Meulbroek 2000). It is this inherent ability to shed and retain risk which allows an organisation to create a ‘natural hedge’ that avoids the duplication of risk management, which is crucial to the LDA and collection of internal loss data. This also creates a co-ordinated and consistent approach to risk management, which the companies have recognised as an important way to avoid individual departments being overlooked and improve reporting structures (Kleffner, Lee, and Mc Gannon 2003). The context of our current paper is therefore based on a financial institution’s ability to provide a risk-based capital measure of operational risk using the AMA whilst implementing the internal loss data collection (LDC) process. It is the implementation of this process that will provide the focus of the research as institutions come to terms with what is required in order to fully integrate such a process within their business lines. 3. Methodology We conduct a critical analysis of internal LDC implementation in a large UK financial institution. To the authors’ knowledge (and in agreement with Wahlstrom 2006; Schrand and Elliott 1998) there is a dearth of previous primary research in this area – which can be explained by the lack of any available data. As a result, our approach takes a primarily qualitative, inductive approach in order to provide crucial insights (see Yin 1994). This method is considered one of the most appropriate ways from which to gather valid data on processes and decision-making within companies (Gummesson 2000). A semi-structured approach was selected as the depth and intimacy of interviewing, on a one-to-one basis, provides a platform from which to maximise the research result (see for example Lincoln and Guba 1985; Denzin and Lincoln 1994; O’Loughlin, Szmigin, and Turnbull 2004). The method meant selecting a non-probability judgement sample that allows the hand picking of respondents for the interviews. Undertaking a non-probability judgement approach was essential (given the dearth of past research and expertise in the area being researched), as it ensured a sample which could contribute most directly and appropriately to our investigation. Glaser and Strauss (1967) argue that this approach focuses the investigation on those individuals for whom the processes being studied are most likely to occur – thus tapping into their expertise. The approach is therefore the optimal way to establish the present and future practices of operational risk data management in light of Basel II, in addition to determining the impact of LDC on a major financial institution’s business operations. The interviews aimed to uncover the ‘whys’ and ‘hows’ as the basis of qualitative material.

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In line with the chosen approach, it was decided that a sample of 15 individuals from the operational risk policy implementation team of a major UK financial institution would be selected for the semi-structured elite interview. Within the institution, each participant had major responsibility and exposure to the Basel II LDA and LDC implementation. With over 180 years of collective experience in risk management within the sample, and a minimum of eight years for selection, all participants were familiar with the before and after effect of Basel II and would therefore provide useful insight into the topic. Sykes (1990) and O’Loughlin, Szmigin, and Turnbull (2004) highlight that although normally small in size, purposive sampling provides the advantage of allowing flexibility and manipulation of the sample of particular interest to the research question. The selection of these participants was through the use of the so called snowballing technique (Burgess 1982; Patton 1990; Goodman 1961). As such, a ‘gatekeeper’ was identified to bring the researcher into contact with other participants within the bank. The Head of Group Operational Risk at the institution was identified as the ‘gatekeeper’, providing buy-in and oversight from senior management. This creates the foundations of research and sample validity as the accuracy of the data, and its degree of correspondence to reality will be determined by the participants given their prerequisite eight years’ knowledge. Furthermore, due to the nature of the study, a large amount of non-standard data is predicted to be produced – making data analysis problematic. In order to overcome this limitation of qualitative data, information gathered was analysed using common themes in the participant’s descriptions (Barritt 1986). The remaining sections of the article are organised as follows. First, the themes as derived from the interviews are outlined; this is then followed by an analysis of these themes. The final section summarises the research and provides an explanation of the critical factors involved in LDA implementation.

4. Results and discussion 4.1. The emergence of the operational risk function The first theme emerging from the data concerned the development of the operational risk function – both internally within the participating institution and externally through compliance with the regulatory authorities. Thus, there were indications that financial institutions were moving towards expanding their operational risk function, but this has been encouraged and extended by an external regulatory push. For example, all our respondents suggested that the function was currently an area of expansion within the organisational structure, with one participant stating ‘in the old retail world we had two, maybe three bodies, whereas as now we have many’. Another contributor suggested that the operational risk function ‘sprung up due to a serious fraud within the organisation, and this highlighted the need to get better at identifying these sorts of things’. This supports the work of Beans (2003) who considers that unchecked operational risks can be disastrous for financial institutions and alludes to the notion that financial institutions may be slowly evolving their perceptions of what an effective operational risk management strategy can achieve. For example, Hoyte (2005) and PWC (2007) suggest that there is now an ongoing re-evaluation of the value of the operational risk function within financial institutions with a shift of focus away from Pillar 1 towards an agenda of embedded ‘use’ in the form of Pillars 2 and 3.

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However, there has been a clear external push from regulatory bodies over the past decade and all participants in our study considered this to be key to the development of the operational risk function in their institution. This concurs with Garrity (2007) and Power (2005) who both suggest that the extended regulatory requirements of Basel II, and Sarbanes Oxley, have both acted as catalysts towards the further development of the operational risk function. However, evidence from our interviews indicates that those within financial institutions may have evolved their viewpoint, for example, it was stated that ‘I think at first the main driver was the external need to comply with regulation, but I think that has evolved now’. This indicates that the risk function may initially have been developed as a tool of compliance to appease the regulators, but is now being seen as a value-enhancing strategy in its own right. Evidence of this further emerged when discussing the pressure to add value to the financial institution. All participants perceived their role as a ‘value-adding function’ and expressed the importance of internal operational risk management to achieve this function. For example, it was stated that: the mantra we work to at operational risk is a no surprises environment and making sure that all of the processes we develop enhance that, we must ensure we are adding value, improving the business, and saving money. We have to quantify our existence because we are not an income generating part of the business.

Participants perceived the operational risk management function as more than a risk mitigation tool, because it exists to ‘identify threats as well as opportunities to the business’. From this response, it seems that participants were aware that they might not necessarily create a profit for the business, but could add value in other, less transparent ways (supporting the previous work of Buchmuller et al. 2006; Lewis and Lantsman 2005; and Alexander 2003). Importantly, they must transmit these ways to key personnel within the institution. However, Power (2005) suggests that operational risk has had difficulties attracting attention to itself and in the past visible threats have either been ignored by banks, or made ‘insufficiently explicit in their management systems’. One way of achieving this, forwarded in our study, is to highlight that the management of operational risk can offer cost-reducing possibilities that may be as important as the revenue-generating functions of other key risk areas such as market and credit risk (Alexander 2003). Again, there is evidence that institutions are becoming swayed by arguments suggesting that the operational risk function can indeed add value. For example, Hiwatashi (2002) and de Fontnouvelle et al. (2006) both suggest that financial institutions are now treating operational risk as a homogenous risk category that can be managed and measured. Wahlstrom (2006) corroborates this by suggesting that a quantitative figure can make it easier to obtain sustainable resources for improving levels of staff development. Outwith the need to comply with regulation and internal support, one participant also mentioned that it is important: that in the industry, you provide evidence to regulators and shareholders, demonstrating that you have a robust control environment, you know what your risks are and you know how to manage them effectively, this can then be used positively in investments and acquisitions.

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Operational risk practitioners now believe that operational risk management can be considered an important synergetic function within a financial institution, a function that is analysed by competitors and can be attributed to confidence within a market (this is also corroborated by Webb, Marshall, and Drennan 2005). 4.2. Drivers of the loss data collection process The second theme emerging from the primary data was that of the drivers and requirements of loss data collection. These included the need to compile three years of data to comply with the AMA as interviewees suggested that the collection of loss data can create an audit trail and best practice for ensuring risks, process and controls are above the risk appetite of the firm. Throughout all of the interviews, it was clear that participants were aware that loss data collection was a fundamental requirement of the AMA to capital allocation. However, the participants were divided in relation to the organisational requirement for the data. One participant stated that it was ‘purely to get numbers for the capital model’ and highlighted the fact that they would like to see the data inform decision-making further in order to be ‘pro-active’ as discussed by PWC (2007). It was suggested that the pressure to progress through the Basel II project had hindered this to date – as the institution concentrated efforts on collecting, rather than analysing the data. In spite of this, one participant suggested that the collection of internal loss data was indeed part of the risk management process, stating that ‘being able to know what went wrong and how much this has cost is vital because past experience can help future understanding’. This attitude concurs with Harris (2002) who recommended that loss databases are necessary for the effectiveness of sound risk management and maximising profit – although such advantages may only be realised with the interpretation and analysis of such databases. Interestingly, this seems to have led to ambiguities in responses as to the purpose of LDC within the organisation – as some participants feel it is strictly for modelling use, whereas others see it as best practice for risk management. This contrary picture could be directly related to the fact that the project was not yet fully complete and the habitual modelling of the data had not begun within the organisation. de Fontnouvelle et al. (2003) highlight this situation by stating that little modelling has been done within the market due to the lack of a structured internal loss data collection process, and this would be a fair representation of the organisation with which we undertook our investigation. As a tool of risk management, it is obvious that the information provided by the loss data is pertinent to the good decision-making of senior managers, as outlined by Dickinson (2001) when discussing the organisational issues in developing ‘Enterprise Wide Risk Management’ (EWRM). In relation to the process moving forward, the primary data was unanimous in supporting the collection of internal loss data as a method of best practice for operational risk management. Thus, the database could be utilised as a quantifiable performance indicator as to the efficacy of the organisation in managing losses/processes/controls in an enterprise-wide manner, given the expanse of operational risks across the institutions’ business lines. Furthermore, Wahlstrom (2006) suggests that the disclosure of operational risks can, at some stage in the future, be transferred to other areas of the business, adding further support to this enterprise-wide argument. As such, operational risk management, and the practices of internal loss data collection within the institution could

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be used as a fertile breeding ground for sound risk management policies that can then be disseminated to other areas of the business. This dissemination will also determine the future of operational risk data management, which may have been considered a contentious issue, as it may question the future of the participants’ jobs within the operational risk data collection function. However, all participants highlighted the importance of ‘creating a seamless operation’, which will eventually become ‘streamlined and a function of business as usual at the lowest levels within the organisation’. This suggests that loss data collection will become the daily duty of everyone within the organisation, with a move away from project-orientated risk management processing to a business as usual operation, aided by the implementation of a new risk processing tool. This tool will disseminate the risk control and process responsibilities away from the operational risk team towards those who ‘own direct responsibility of these risks’. Dickinson (2001) highlights in research related to EWRM that this delegation of responsibility will empower those closest to risk events allowing for swift rectification of incidents. One participant highlighted the future of their job in relation to this tool by suggesting that there would be no need for them to collect the data and they went further to explain that this would therefore allow them to analyse and create more pro-active uses for the data, as their time would not be spent setting up the collection process as it is now. This view is broadly in line with that of Dickinson (2001), whereas Alexander (2003) suggests that the process of internal loss data collection is labour intensive.1

4.3. Challenges of LDC The participants all expressed the view that prior to the (external) requirements of Basel II, the organisation already looked upon operational risk with a triangular strategy: the traditional heart of it in the organisation has always been around business continuity, allied to that you would have issue escalation and new product development.

It was clear that the organisation had invested in forward looking, pro-active and mitigating operational risk management strategies. The responses we received supported the work of Buchelt and Unteregger (2004) and Hoffman (2002), who also argued that long before Basel II, financial institutions had various control mechanisms. In relation to the introduction of internal loss data collection to the already active processes mentioned above, one participant concluded that: the loss data challenge was dipping into your general ledger process to suck out what’s happened, whilst ensuring that you weren’t sucking out anything more than you had to.

This clearly suggests that there are many events within the cost base/ledger that are not attributed to operational failure, and therefore it is important to ensure that the data that are being collected for the loss data submission are relevant and valid. Prior to internal loss data collection being calculated within the capital allocation model, it is critical that all loss data events are valid at the point of entry. Garrity

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(2007) highlights this point by stating that management must be confident of the loss data’s integrity before such data can be used to make decisions. This process is known as data cleansing, and acts as a filter to ensure that only relevant loss data collection issues are included in the loss data collection submission. This problem was considered by Vaughan (2005) and Harris (2002), who suggest that how organisations ‘cleanse’ errors should be part of its ritual of management. That is, the process of data cleansing should be considered as pertinent to the ability to perform good risk management. In addition, it must be remembered that operational risk within the sector is constantly evolving, as highlighted by one participant who said ‘there hasn’t been a year in the last decade that something hasn’t happened whether it be criminal fraud, twin towers or Basel II’. This is critical to the area of internal loss data collection because, if the definitions are changing over time, the event inclusions will change – making it difficult to compare data as the science evolves. This is reiterated by one participant who stated that the organisation has now tried to develop its outlook by involving the triangular process of operational risk management whilst becoming ‘more modular data driven, rather than just mental and experience driven’. This evolution suggests a move by the organisation from qualitative risk control to quantitative risk accountability. However, this move towards quantitative management methods has been outlined by Wilson (2001) who suggested that reliance on a single number can be a risk in itself, whilst Wahlstrom (2006) identified that operational risk is a phenomenon that, in practice, is hard to reveal, making a move towards quantification by the organisation more complicated (see also Allen and Bali 2004; Crouchy 2001; Mercer Oliver Wyman 2004). Nonetheless, the interviews conveyed the fact that the process of data cleansing has affected the way in which the business operates. This is due to the fact that the organisation employs a ‘bottom-up approach’, a process which Alexander (2003) and Garrity (2007) have highlighted as applicable to requirements of the AMA approach. The ‘bottom-up’ approach has offered the organisation the ability to ‘cleanse’ and filter its data streams from the point at which the data loss event occurs. This allows those who have a full working knowledge of the event or process failure to detail its shortcomings and convey this message vertically to the operational risk management function, again supporting the empowerment of staff as outlined by Dickinson (2001). This localised knowledge can then be used by those within the operational risk management function to address the event whilst reducing the workload placed upon retrieving information – as the event has already been cleansed for them. This in turn allows for a quicker response time to the issue by the operational risk team, which could be valuable if the event was a high frequency/high severity issue. However, this comes at a cost as outlined by Moosa (2007) and Power (2007) as these bottom-up models rely on information provided by employees that may not be forthcoming particularly if it was their fault or if a capital charge to a particular unit (namely their own) will increase due to the report. 4.4. Organisation-wide responsibility to LDC Paramount to the operational risk process operating effectively is the raising of awareness throughout the institution as to the importance of loss data collection as a Basel II process. As with all policy implementation within an organisation, it is

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critical to ensure that all the major stakeholders within the organisation are aware and understand its core objectives. One participant highlighted that for loss data collection to succeed, it becomes intrinsically dependent upon business level staff highlighting to their risk representative that a loss has occurred. In order for this to occur, the participant suggested that: we have become a lot more pro-active in the collection of issues outside an operational environment, so we engage with marketing, commercial, finance.

In an attempt to recalibrate the organisational routines and standards within other areas of the institution, the LDC team have engaged in organisational learning through the process of education, based on experience of the LDC process. Wahlstrom (2006) suggested the importance of this process, as it can contribute to a corporate culture that, to a much greater extent, is tolerant of human error, thus eradicating the problems discussed by Moosa (2007) and Power (2007). Our responses indicated that the cultures within areas of the business that are non-risk orientated such as marketing, are now ‘less likely to hide risks’, stating: what would have happened prior to the cascading of LDC was that marketing would have taken the loss out of their budget which means it would have been hidden as a loss.

However, now it is reported due to an open and honest communication culture through the assurance that they will not be blamed for reporting the loss. Research by Hoffman (2002) and Buchelt and Unteregger (2004) reiterates the point conveyed within our findings, that is, operational risk is not a new phenomenon; however, the focused, structured and disciplined approach outlined by Basel II could be considered innovative. Therefore, one of the toughest tasks for management effecting change is to motivate employees throughout the entire institution to be adaptive in how they see their role (Heifetz and Laurie 2001). Therefore, one of the biggest issues for financial institutions may not be the collecting and cleansing of data, or the modelling of capital requirements. It may actually be the implementation of such an organisationally wide project in order to ensure all those involved within the business realise their role in the LDC process. A view was highlighted by Acharyya and Johnson (2006) when investigating the development of ERM within the insurance sector. Key to this is the instilling of a strong operational risk culture (Waring and Glendon 2001; Chang 2001; Rao and Dev 2006), as it will act as a sense-making mechanism that guides, shapes and moulds the values, behaviours and attitudes of employees (O’Reilly and Chatman 1996). One participant gave the anecdotal example that ‘call centre managers couldn’t make any sense of the idea’, as they work as a client facing lower tier levels of the organisation with little operational risk experience. It could therefore be assumed that the implementation of the LDA has had an impact and raised awareness through communication within the organisation, as highlighted by the example of the call centre manager (Acharyya and Johnson 2006). This point was summarised by Power (2007) when he suggested that new forms of data collection are always a behavioural challenge; for the process to succeed ‘every worker needs to be a knowledge worker’ (Wang and Ahmed 2003) and communication is critical to

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this knowledge transfer (Acharyya and Johnson 2006), whether it be due to operational risk specialists engaging outside their own environment as evidenced in this research, or through the implementation of an operational risk framework based on the ten qualitative principles of Basel II (Basel 2003) as argued by Bolton and Berkey (2005). 5. Summary of findings The opportunities offered by the implementation of the Basel II AMA to operational risk capital allocation far exceed the development of an international regulatory alliance on the issue, as expected by the Bank for International Settlements. This approach not only offers a financial institution the ability to develop its own risk metrics and modelling capability but also presents the institution with the ability to continually improve its operational risk management techniques in an attempt to reduce the requirement to hold capital for this function, thus creating an opportunity investment in the reallocation of this capital whilst rewarding good risk management techniques. The overall aim of this article was to conduct a critical analysis of internal loss data collection implementation within a UK financial institution. The implementation of this process is critical to an institution’s ability to produce the above mentioned risk metrics and models as the historical loss data collected will provide the foundation data set from which to develop future pro-active modelling techniques. Furthermore, the research identified that the process of loss data collection is perceived to be a pertinent tool within holistic operational risk management practices. However, it is important to note that this tool is not without its limitations. There were signs that operational risk was beginning to take shape within financial institutions, but regulatory compliance has acted as a key driver. It is therefore no surprise that the institution under investigation reallocated considerable resources to meet the regulatory demands placed upon them as they identify weaknesses in both controls and processes during the loss data collection task. As the institution employs a ‘bottom-up’ approach, it allows them to filter relevant data at the point of failure and escalate the relevant information to the relevant operational risk department in a timely, standardised format allowing for rapid recovery and repair of events or issues. However, this process relies on information provided by employees at the lowest levels of the organisation. This may be particularly challenging due to the fact that employees may not be forthcoming with information, particularly if their department may inherit an increased capital charge due to the event or if they are directly responsible for the event occurring. On a more personal level, employees who are subject to performance-related pay may be financially discouraged in the reporting of events. Another disadvantage of such an approach is the difficulty of these lower level staff understanding the importance of loss data collection in the larger scheme of capital allocation for the institution. In order to mitigate against the above mentioned ambiguities with the ‘bottomup’ approach, the institution adopted a pro-active organisational learning strategy with business functions around the institution who may not necessarily have been exposed to the processes required prior to LDC implementation. This gave the operational risk teams the ability to create a better understanding of the process for these business areas whilst ensuring that they realise the importance of the LDA function and therefore understand that open and honest communication of losses is considered

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best business practice. The participants also highlighted that this engagement with other areas of the business helped to improve the risk culture within areas of the business, which until that point may not necessarily have considered themselves as a risk-orientated business unit in the first place. The emphasis on an embedded risk culture was highlighted throughout both the literature and in the primary research, with the participants describing the hiding of risks as an obstacle that must be overcome. However, the culture and improved understanding of the process will provide the foundations from which all other risk management processes will follow and therefore should not be overlooked in an attempt to comply with regulators. One such process that is reliant on this embedded risk culture is that of data cleansing. This is pertinent to the LDA, as it will ensure the integrity and validity of the data that will eventually be developed for operational risk modelling and management decision-making. As the institution employed the’ bottom-up’ approach, it enables the cleansing of data at the point at which the event occurs, therefore improving the amount of data rich information collected, relevant to each event. However, as the institution was seeking to compile three years of data for regulatory compliance, difficulties were encountered when looking at previous losses. This is further exacerbated by the heterogeneous nature and definition of operational risk that was evident in the replies from all participants. This process of data cleansing will improve over time as business units become more aware of the requirements of the model, to the point whereby the collection process will become a business as usual task conducted by all business units within the institution. This will disseminate the responsibility of the LDC to those with the most explicit knowledge within that particular area of the business, allowing for the operational risk practitioners to concentrate on the analysis of the data, a view which in this study was considered by some participants to be overlooked at this moment in time as resources were concentrated in the area of compliance. Although the institution under investigation already had pro-active risk management techniques in place prior to Basel II, the drive for compliance initiated a move towards enhanced operational risk governance. This consequently gave the operational risk management function a formal identity, which in turn allowed them to ‘add value’ outwith the generation of revenue through the identification of threats as well as opportunities. Unlike that of market and credit risk, the operational risk dilemma is faced by every member of every department within an institution. This makes the implementation of the loss data collection process more complicated. It is therefore imperative that the LDA foundations from which all other risk management techniques can be placed are solid, whether it be data modelling, scenario analysis or key risk indicators. Note 1. As previously mentioned, it is important to note at the primary data collection stage participants were still in the process of developing the loss data collection project, and therefore are not judging the process on its finished product.

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North American Actuarial Journal, 18(1), 14–21, 2014 C Society of Actuaries Copyright  ISSN: 1092-0277 print / 2325-0453 online DOI: 10.1080/10920277.2013.872552

Keeping Some Skin in the Game: How to Start a Capital Market in Longevity Risk Transfers Enrico Biffis1 and David Blake2 1

Imperial College Business School, Imperial College London, London, United Kingdom Pensions Institute, Cass Business School, City University London, London, United Kingdom

2

The recent activity in pension buyouts and bespoke longevity swaps suggests that a significant process of aggregation of longevity exposures is under way, led by major insurers, investment banks, and buyout firms with the support of leading reinsurers. As regulatory capital charges and limited reinsurance capacity constrain the scope for market growth, there is now an opportunity for institutions that are pooling longevity exposures to issue securities that appeal to capital market investors, thereby broadening the sharing of longevity risk and increasing market capacity. For this to happen, longevity exposures need to be suitably pooled and tranched to maximize diversification benefits offered to investors and to address asymmetric information issues. We argue that a natural way for longevity risk to be transferred is through suitably designed principal-at-risk bonds.

1. INTRODUCTION The total amount of pension-related longevity risk exposure in private-sector corporations is globally estimated at $25 trillion.1 In 2006 a market started in the United Kingdom to transfer longevity risk from the pension plans of U.K. corporations. From 2010 this market became international, with transfers taking place in the Netherlands, the United States, and Canada. However, the total value of the transfers since the market started has been little more than $100 billion, a small fraction of the total exposure. In addition, most of the transfers have ended up with insurance and reinsurance companies.2 A small proportion of the total has involved investment banks that have passed the risk exposure on to capital markets investors. This rate of transfer to the capital markets needs to grow significantly, because capital in the insurance and reinsurance industry is insufficient to absorb the total exposure. Furthermore, long-term capital markets investors need to be persuaded to hold longevity risk exposure in a form in which they are comfortable and with a suitable longevity risk premium that reflects the true level of risk they are assuming. Long-term investors, such as sovereign wealth funds, endowments and family offices, should find a longevity-linked asset an attractive one to hold in a diversified portfolio, because its return will have low correlation with the returns on other asset classes, such as financial assets, real estate and commodities. However, such investors are currently wary of longevity-linked assets. If experts in longevity risk, such as insurance companies, are trying to sell this risk on to them, is this just because they see this simply as an opportunity to gain some capital relief from their regulator, a perfectly legitimate business practice? Or is it because they have become aware that the risk is greater than they initially believed and are trying to offload it on to unsuspecting investors before the true extent of the exposure becomes more widely known? A relevant problem here is asymmetric information. One party to the transaction knows more about what is being offered for sale than the other party. In the extreme case, the potential buyer in the transaction might be so suspicious of the potential seller’s intention that they will not transact at any price: In this case, there will be no market. For a market to exist in the presence of asymmetric information, the potential seller needs to find a way of communicating to the potential buyer the true extent of the information that they have acquired about the item up for sale. In other words, they have to provide an appropriate signal to the potential buyer that they are not being sold a “lemon.”3 Address correspondence to David Blake, Pensions Institute, Cass Business School, City University London, 106 Bunhill Row, London, EC1Y 8TZ, United Kingdom. E-mail: [email protected] 1The

Life and Longevity Markets Association. market was started by newly established monoline insurers known as buyout companies. Later, larger traditional insurance companies and reinsurance companies and investment banks entered the market. See LCP (2012) for an overview of market participants and recent transactions. 3A “lemon” is something, such as a second-hand car, that is revealed to be faulty only after it has been purchased (e.g., Akerlof 1970). 2The

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FIGURE 1. Participants in a Market for Longevity Risk Transfers.

In this article we outline how a market in longevity risk transfers may grow out of the recent activity in pension liability transfers. Figure 1 provides a stylized representation of the potential participants in such a market. There are some holders of longevity risk exposures, which we may regard as “informed.” An example is a large pension plan that has undertaken a longevity study of its members and beneficiaries and therefore has acquired private information about its own longevity exposure. There are then holders of longevity exposures that (oversimplifying) we may regard as “uninformed.” An example is a small pension plan that does not have the resources to undertake a longevity study of its members. There are also informed intermediaries, such as insurance companies and investment banks, that have conducted their own longevity studies of the members of pension plans, but are considering selling on their acquired longevity risk exposure either directly to investors or to intermediaries in insurance-linked securities (ILSs). Finally, there are end investors (and other intermediaries), whom we may regard as “uninformed,” in the sense that they have not conducted any longevity studies of their own, possibly because they do not have access to the plan data needed to conduct such a study. They therefore have less information about the true extent of the longevity risk in which they are being invited to invest than those selling it, a classic case of asymmetric information. An effective way of overcoming the problem of asymmetric information for an informed seller is to disclose information about the relevant risk characteristics of the exposures being transferred (e.g., gender, age, medical status, or similar details on potential beneficiaries of any survivor benefits) and to signal their quality by retaining part of the exposures. Another effective tool is to structure the risk transfer vehicle so as to minimize the sensitivity of the vehicle’s value to the issuer’s private information. This would typically entail pooling and tranching the exposure, as we outline below. Since the bulk of global longevity risk exposures is represented by the liabilities of defined benefit pension plans, the market for pension buyouts and longevity swaps is an important origination market that could support the development of a market in longevity-linked securities. In particular, buyout firms act as aggregators of the pension liabilities of small companies into larger pools and are de facto the natural candidates to intermediate the transfer of longevity exposures originating from the pension buyout market to capital market investors. Because of capital requirements, and the returns that can be generated by deploying resources in the buyout market, buyout firms have an incentive to securitize their exposures to diversify risk and free up capital. Regulators hold an important role in this process, because they may provide incentives to disclose and use detailed information from the very same internal models used to demonstrate the capital resilience of (re)insurers and buyout firms in the pension buyout market. 2. LESSONS LEARNED SO FAR, AND THE WAY FORWARD During the last decade, several attempts have been made to launch standardized longevity-linked securities. The longevity bond designed by the European Investment Bank (EIB) in 2004, and immediately withdrawn because of insufficient demand, is probably the most widely discussed example, due to the important lessons that can be learned from its failure to launch. The EIB instrument was a 25-year bond with an issue price of £540 million and coupons linked to a cohort survivor index based on the realized mortality rates of English and Welsh males aged 65 in 2002. The initial coupon was set equal to £50 million, and the subsequent coupons would have decreased in line with the realized mortality of the reference cohort (see Blake et al. 2006 for additional details). Hence, the higher the number of survivors in the population each year, the higher the coupons paid to investors,

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meaning that the instrument should have been appealing to pension plans and annuity providers. There are two main reasons why the EIB bond did not launch:4 • Design issues: As a hedging instrument, the EIB bond did not offer sufficient flexibility. The hedge was bundled up within a conventional bond and provided no leverage opportunities. The bond format meant that a considerable upfront payment was required to access the longevity hedge component of the instrument, represented by a longevity swap that paid the longevity-linked coupons. Furthermore, the basis risk in the bond was considered to be too great: The bond’s mortality index covered just a single cohort of 65-year-old males from the national population of England and Wales, representing only a fraction of the average pension plan’s and annuity provider’s exposure to longevity risk.5 • Transparency issues: The EIB bond’s projected cash flows depended on projections of the future mortality of 65-year-old males from England and Wales prepared by the U.K. Government Actuary’s Department. The forecasting model used for the projections is not published, and the projections themselves are a result of adjustments made to the baseline forecasts in response to expert opinion. This represented a major barrier to investors and hedgers either not familiar with longevity risk or with strong views on specific mortality projection models. On the pricing side, the longevity risk premium built into the initial price of the EIB bond was set at 20 basis points (Cairns et al. 2005): In the absence of agreement on a baseline best estimate, investors had no real feeling as to how appropriate this figure was. In the wake of the EIB bond withdrawal, considerable innovation has taken the form of longevity indices and mortality derivatives, for example. Despite several attempts, so far standardized solutions have not been as successful as those following the more traditional insurance paradigm, where the hedger is indemnified (fully hedged) against changes in longevity risk. Successful longevity risk transfers have taken the form of pension buyouts, pension buy-ins, and longevity swaps. Pension buyouts involve the transfer of a pension plan’s assets and liabilities to a regulated life insurer. The transaction allows the employer to off-load the pension liabilities from its balance sheet, or to replace pension assets and liabilities with a regular loan in case buyout costs are financed by borrowing.6 In a pension buy-in, a bulk annuity contract is purchased as a plan asset to match some or all of the pension plan’s liabilities, meaning that interest rate, inflation, and longevity risk are all fully hedged, while the liabilities remain in the pension plan. Longevity swaps allow a hedger to receive longevity-linked payments (the floating leg) in exchange for a stream of fixed payments (the fixed leg). The most popular transactions to date involve bespoke, or customized, swaps, where the floating leg is linked to the specific mortality experience of the hedger. These forms of longevity risk transfer have successfully addressed the design issues mentioned above, by removing the basis risk originating from differences in the mortality experience of the hedger and the evolution of a mortality index. Pension buyouts and buy-ins still require substantial upfront costs, which longevity swaps avoid. The latter have limited upfront costs, mainly associated with set-up fees and initial margins in case of collateralization.7 The limited take-up of standardized hedges such as indexed longevity swaps is probably due to (perceived) difficulties in managing longevity basis risk within broader liability-driven investment (LDI) strategies.8 There is no reason to believe, however, that the success of standardized instruments should depend exclusively on the demand for indexed hedges by pension plans and annuity providers. The huge capital inflows that have recently targeted the reinsurance and the ILS space, as well as the natural process of aggregation of longevity exposures carried out by reinsurers and buyout firms in the pension buyout market, suggest that customized hedges may well coexist with standardized instruments appealing to capital market investors. It is reinsurers and buyout firms, rather than smaller pension plans and annuity providers, that are in a natural position to originate instruments written on large pools of exposures and to identify payoff structures that may appeal to ILS investors. The next sections aim at giving an idea of how the development of a “life market” may come about through this channel. 3. THE OPTIMAL LEVEL OF SECURITIZATION OF LONGEVITY EXPOSURES Securitization, in general, involves the bundling together of a set of illiquid assets or liabilities (in our case mainly the latter) with a similar set of risk exposures, and then the sale of this bundled package to investors.9 The purpose of such a bundling exercise is to reduce the idiosyncratic (or diversifiable) risk that is contained in each of the individual assets or liabilities when they are considered in isolation, while leaving the systematic (or nondiversifiable) risk to be borne by the investor. In our case, the 4See

Blake et al. (2006), Biffis and Blake (2010b), and Blake et al. (2013) for further details. typical longevity exposure of a pension plan spans different cohorts of active and retired members, not to mention the fact that a large portion of pensions paid by pension funds and annuity providers are indexed to inflation. 6In the case of a plan deficit, a company borrows the amount necessary to pay an insurer to buy out its pension liabilities in full. 7See Biffis et al. (2012) for details on the structuring of longevity swaps, including collateral arrangements. 8Longevity risk management in LDI is discussed, for example, in Aro and Pennanen (2013). 9See Cowley and Cummins (2005) for an overview of insurance securitization. 5The

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idiosyncratic risk relates to the uncertainty surrounding the length of life of individual plan members, whereas the systematic risk relates to the uncertainty surrounding the average length of life of the members of the group taken as a whole. We call these two types of risk idiosyncratic and systematic longevity risk, respectively. We would not expect investors to be interested in holding idiosyncratic longevity risk, any more than they would be interested in being exposed to the idiosyncratic risk from holding just a single company’s shares or bonds in their investment portfolio. This is why longevity risk needs to be pooled before securitization. However, investors would be willing to invest in a particular asset class if they are appropriately rewarded for the true level of systematic longevity risk that they are assuming. So for investors to become interested in buying into this particular asset class, they need to have confidence that the premium they are being paid by buyout firms, reinsurers, and pension plans reflects the true level of systematic longevity risk underlying the transaction. A natural way for informed sellers to persuade potential investors that this is the case is to retain some of the exposure themselves. If the seller keeps some “skin in the game,” this provides an important as well as a credible signal to investors, because longevity exposures are very capital intensive—in terms of both regulatory and economic capital requirements—for the insurance companies operating in longevity space. Investors will also realize that the larger the securitization fraction (γ )—the proportion of the total exposure that the seller seeks to offload—the greater the severity of the longevity risk exposure indicated by the seller’s own assessment of longevity risk, and the lower the price the investor will offer. Investors will try to neutralize the seller’s informational advantage through a downward sloping demand curve for the longevity exposure transferred, and so the seller will try to securitize a fraction of exposure that is optimal, in the sense that it minimizes the cost to the seller from offloading the risk exposure. If the chosen fraction is too low, an insufficient amount of the longevity risk will be transferred to investors, and the seller will have to bear higher retention costs (in terms of higher capital requirements than desired). If the fraction is too high, investors will demand too high a risk premium, thus reducing the benefits of capital relief from securitization. The seller thus faces two costs, retention costs (from the capital requirements) and liquidity costs (from the downward sloping demand curve), and will wish to minimize the sum of these costs.10 4. THE OPTIMAL TRANCHING OF LONGEVITY EXPOSURES How should the securitized longevity exposure be transferred to investors? The seller of the exposure wants to transfer a liability, whereas the buyer of the exposure wants to hold an asset. How can the sale of a liability by an informed holder or intermediary become an asset to an uninformed investor? Further, we can assume that the investor will not want to be responsible for making the pension payments once the securitization transaction has taken place. To answer these questions, note that the holders of the longevity exposure are really looking for a longevity risk hedge for a fraction (γ ) of their current holdings. The holders need to persuade the investors to provide that hedge in exchange for receiving an appropriate longevity risk premium. When sellers are more informed, a possible way to do this is for the holders of the longevity exposure to issue a principal-at-risk bond that the investors will buy. The return on the bond will incorporate a longevity risk premium11 in addition to the usual credit risk premium associated with the issuer.12 By issuing such a bond, it is the pure longevity risk that is transferred, not the exposure itself or the obligation to make the pension payments.13 The securitization of longevity risk is therefore different from other forms of longevity risk transfers (such as pension buyouts) in which assets and liabilities are bundled up and transferred in their entirety. Note that the bondlike structure discussed here is not to be confused with capital intensive hedging instruments such as the EIB longevity bond. What we have in mind is the format of catastrophe bonds,14 where the premiums paid by investors to acquire longevity-linked notes are used to purchase securities that are held in a special purpose vehicle and used as collateral for the bond’s payments.15 In particular, the principal-at-risk bond will repay the principal in full if the proportion of the members from the population underlying the risk transfer who have survived between the bond’s issue and maturity dates is below a preset level. However, if the proportion who have survived (i.e., the survival rate) is higher than this preset level (known as the point of attachment), the amount of principal returned to the investor will be reduced according to some preset formula. If the survival rate exceeds another preset level (known as the point of exhaustion), the investor gets no principal back. The point of attachment will be determined to minimize the sensitivity of the bond’s value to private information, and hence the cost to the holder of the exposure from issuing the bond.16 Multiple attachment points can be used to meet the demand of investors with different risk preferences. 10This

situation is formalized in the model of DeMarzo and Duffie (1999), for example. the model of Biffis and Blake (2010b), this premium is zero because both supply and demand are risk-neutral. 12The bond would typically be collateralized and issued via a special purpose vehicle, a procedure that would influence the size of the credit risk premium. 13Should the holder of the exposure wish to do so, this responsibility could be transferred to a third party administrator who would make the pension payments. 14See Grace et al. (2001) and Lakdawalla and Zanjani (2012), for example. 15In the baseline model of Biffis and Blake (2010a), for example, one unit of collateral is used to collateralize a representative survival probability S. The resulting payoff from the instrument is therefore a death rate D = 1 − S. 16It is a central tenet of corporate finance that debt-like instruments can minimize the adverse effects of information asymmetry when raising capital through the issuance of securities (e.g., Tirole 2005). 11In

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In financial engineering terms, the principal-at-risk bond is a combination of a standard bond paying (say) 1 at maturity and a call option that has been sold by the investors to the bond’s issuer (in the case of a single point of attachment). The option will expire in-the-money if, on the bond’s maturity date, the survival rate from the reference population (S) exceeds the preset level (S ∗ ). The principal repayment to investors will be reduced in this case. The point of attachment is therefore the same as the strike price of the option. The option will expire worthless if S is below S ∗ . We can therefore write the formula for the repayment of principal as B = 1 − max(0, S − S ∗ ), where max(0, S − S ∗ ) means the larger of 0 and (S − S ∗ ) and is the formula for the value of a call option on its expiry date. The seller of the bond therefore has an effective hedge against systematic longevity risk for survival rates S that lie above S ∗ , but bears all the longevity risk below S ∗ . Another way to think of this is that we have tranched the exposure at S ∗ with the current holder of the exposure bearing the longevity risk below S ∗ and investors bearing the longevity risk above S ∗ . The optimal tranching level will be the one that minimizes the cost to the holder of transferring the exposure, as shown in Figure 2. If S ∗ is set too low, the longevity risk premium demanded by investors will be excessive, and the price that they will agree to pay for the bond will be too low. If S ∗ is set too high, the holder of the exposure bears too much of the longevity risk. A rough measure of the ex post cost of the hedge is B − P , the difference between the principal actually repaid and the initial sale price of the bond. The bond issuer’s objective is to choose the optimal tranching level (or, equivalently, the strike price or point of attachment) to minimize the expected cost of the longevity hedge to the issuer. A key issue to address is the choice of reference population that determines the payments on the bond. So far, we have assumed that the bond issuer would prefer it if the reference population were the same as its own population of pension plan members and annuitants. This would allow the bond issuer to establish a very effective hedge against its own exposure. But in a world of asymmetric information, investors who buy the bond might not feel comfortable about this, because the bond issuer will be determining the principal repayment in a manner that might be perceived to lack full transparency. Investors, however, might be more comfortable if the reference population were the same as the national population (or other publicly monitored population) because, in this case, information on survival and mortality rates are published by the country’s independent official statistical agency. By doing this, asymmetric information issues are less of a concern, but a new problem emerges: Using a national population

FIGURE 2.

The Optimal Tranching Level S ∗ . (color figure available online)

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FIGURE 3. Tranching with and without Pooling. (color figure available online)

index to hedge the longevity risk of a subgroup of the population introduces basis risk in the hedge. Basis risk arises because the mortality experience of the national population might differ from the subgroup being hedged. In particular, we know that members of pension plans tend to live longer than members of the population as a whole, so the national index might involve underestimating the longevity risk of the pensioner subgroup being hedged. Although basis risk certainly exists, studies have shown that it can also be hedged quite effectively, and so hedges and principal-at-risk bonds based on national indices can provide very good, if not 100%, hedge effectiveness (see, e.g., Coughlan et al. 2011). Hedge effectiveness can be further increased if exposures are pooled or aggregated. 5. POOLING OR AGGREGATING EXPOSURES If multiple small exposures, such as those associated with a number of small pension plans, are pooled or aggregated together,17 there might be further diversification benefits,18 although any informational advantage concerning the individual longevity trends within the separate exposures will be destroyed.19 A principal-at-risk bond with n diversifiable exposures will have the following repayment value: B = 1 − max(0, Sn − Sn∗ ), where Sn∗ is the optimal tranching level (or, equivalently, strike price or point of attachment) and Sn is the average exposure when there are a total of n exposures. Figure 3 shows the diversification benefits from pooling: The expected cost to the issuer of the bond and the strike price are both lower compared with the case where the exposures are tranched separately. Pooling and tranching are more valuable when the private information on the severity of longevity risk affects the entire pool rather than being specific to each individual exposure. This would be the case when the survival rates relate to different cohorts of pension plan members, but the underlying characteristics are similar; for example, they come from the same

17Buyout and other insurance and reinsurance companies have already been acting as aggregators and may find themselves as naturally positioned to take on this intermediary role. 18See Subrahmanyam (1991) and Gorton and Pennacchi (1993), for example. 19See DeMarzo (2005), for example.

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FIGURE 4. Market for Longevity Risk Transfers.

geographical area, social class, gender, or age grouping (e.g., men in their 70s). Pooling on the basis of such factors can reduce the costs associated with asymmetric information. 6. CONCLUSIONS We argue that a natural format for transferring longevity risk to end investors is a principal-at-risk bond. The bond’s payments need to be structured in a way that deals with the concerns that investors will have about the seller of the bond having more information about the true level of longevity risk underlying the exposure than the investor has. This problem of asymmetric information can be managed by suitably pooling and securitizing the exposures in tranches. To give comfort to the investor that they are not being sold a “lemon,” the longevity risk hedger needs to keep some “skin in the game,” by retaining some of the exposure. To minimize the cost of the hedge, the hedger should hedge only the extreme longevity risk, that is, the longevity risk above a threshold level that we have called the optimal tranching level (or, equivalently, strike price or point of attachment). To further reduce the cost to the hedger, they should pool or aggregate their longevity exposures as much as possible. This reduces the information advantage of the seller about individual longevity trends, but it increases the degree of diversification and, in turn, might reduce the basis risk faced by hedger. Figure 4 shows the risk transfers that might characterize the stylized longevity risk market discussed in this article. Uninformed holders of longevity risk exposures, such as small pension plans, should pool and transfer their full exposures to informed intermediaries, because transferring exposures separately would expose them to “cherry picking” from more informed counterparties.20 Informed holders, such as large pension plans, will retain the fraction (1 − γ1∗ ) of their exposure and securitize the fraction γ1∗ by transferring it to (less) informed intermediaries. They could also deal directly with end investors by retaining a larger fraction (1 − γ2∗ ) of their exposure and securitizing a smaller fraction γ2∗ , possibly tranching it at an optimal level S ∗ . Finally, informed intermediaries would deal with end investors by pooling their (n, say) exposures and tranching them at level Sn∗ .21 The issuance of instruments linked to public longevity indices would side step asymmetric information issues but expose hedgers to basis risk. Even in this case, however, the principal-at-risk format would appear a natural solution, given the familiarity of investors in ILSs with catastrophe bonds with payoffs linked to indices or parametric triggers. It so happens that the principal-at-risk format was precisely the format chosen by Swiss Re when it issued its Vita bonds22 to hedge the extreme mortality risk it was hedging. A similar format was adopted for Kortis by Swiss Re in 2010. Indexed on the England and Wales and U.S. populations, Kortis was the first ever longevity trend bond: It would reduce payments to investors in the case of a large divergence between the mortality improvements experienced by male lives aged 75–85 in England and Wales and by male lives aged 55–65 in the United States. So we have very successful precedents in the form of mortality/longevity trend bonds that should help promote an equally successful capital market in longevity risk transfers. 20We note that Figures 1 and 4 encompass both cases of informed and uninformed sellers of longevity exposures. In the first case, the investors face the “buyer’s curse” (the risk of ending up with a “lemon”) when buying from more informed counterparties. In the second case, the uninformed holders of exposures face the “seller’s curse” (of being exposed to “cherry picking”) when selling to more informed counterparties. 21For a more formal analysis together with proofs of the propositions discussed in this article, see Biffis and Blake (2010b, 2013). 22See “Swiss Re Transfers 180 Million of Extreme Mortality Risk” (www.swissre.com/media/news releases/Swiss Re transfers USD 180 million of extreme mortality risk to the capital markets through the Vita securitization programme.html; accessed January 18, 2014).

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REFERENCES Akerlof, G. A. 1970. The Market for “Lemons”: Quality Uncertainty and the Market Mechanism. Quarterly Journal of Economics 84: 488–500. Aro, H., and T. Pennanen. 2013. Liability-Driven Investment in Longevity Risk Management. Technical report, Aalto University and King’s College London. Available at http://math.tkk.fi/˜haro/ldi.pdf. Biffis, E., and D. Blake. 2010a. Mortality-Linked Securities and Derivatives. In Optimizing the Aging, Retirement and Pensions Dilemma, ed. M. Bertocchi, S. Schwartz, and W. Ziemba, pp. 275–298. New York: John Wiley & Sons. Biffis, E., and D. Blake. 2010b. Securitizing and Tranching Longevity Exposures. Insurance: Mathematics and Economics 46: 186–197. Biffis, E., and D. Blake. 2013. Informed Intermediation of Longevity Exposures. Journal of Risk and Insurance 80: 559–584. Biffis, E., D. Blake, L. Pitotti, and A. Sun. 2012. The Cost of Counterparty Risk and Collateralization in Longevity Swaps. Technical report. Available at SSRN: http://ssrn.com/abstract=1801826. Blake, D., A. Cairns, G. Coughlan, K. Dowd, and R. MacMinn. 2013. The New Life Market. Journal of Risk and Insurance 80: 501–558. Blake, D., A. Cairns, and K. Dowd. 2006. Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal 12: 153–197. Cairns, A., D. Blake, P. Dawson, and K. Dowd. 2005. Pricing the Risk on Longevity Bonds. Life and Pensions 1: 41–44. Coughlan, G., M. Khalaf-Allah, Y. Ye, S. Kumar, A. Cairns, D. Blake, and K. Dowd. 2011. Longevity Hedging 101: A Framework for Longevity Basis Risk Analysis and Hedge Effectiveness. North American Actuarial Journal 15: 150–176. Cowley, A., and J. Cummins. 2005. Securitization of Life Insurance Assets and Liabilities. Journal of Risk and Insurance 72: 193–226. DeMarzo, P. 2005. Pooling and Tranching of Securities: A Model of Informed Intermediation. Review of Financial Studies 18: 65–99. DeMarzo, P., and D. Duffie. 1999. A Liquidity Based Model of Security Design. Econometrica 67: 65–99. Gorton, G. B., and G. G. Pennacchi. 1993. Security Baskets and Index-Linked Securities. Journal of Business 66: 1–27. Grace, M. F., R. W. Klein, and R. D. Phillips, 2001. Regulating Onshore Special Purpose Reinsurance Vehicles. Journal of Insurance Regulation 19: 551–590. Lakdawalla, D., and G. Zanjani. 2012. Catastrophe Bonds, Reinsurance, and the Optimal Collateralization of Risk Transfer. Journal of Risk and Insurance 79: 449–476. LCP. 2012. Pension Buy-ins, Buy-outs, and Longevity Swaps 2012. Technical report, Lane Clark & Picock LLP. Subrahmanyam, A. 1991. A Theory of Trading in Stock Index Futures. Review of Financial Studies 4: 17–51. Tirole, J. 2005. The Theory of Corporate Finance. Princeton: Princeton University Press.

Discussions on this article can be submitted until October 1, 2014. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Journal of Risk Research, 2014 Vol. 17, No. 8, 999–1017, http://dx.doi.org/10.1080/13669877.2013.841725

Learning to organise risk management in organisations: what future for enterprise risk management? Frank Schillera* and George Prpichb a Department of Sociology, University of Surrey, Guildford, UK; bEnvironmental Science and Technology, Cranfield University, Cranfield, UK

(Received 30 November 2012; final version received 11 July 2013) Enterprise risk management (ERM) was originally developed to manage financial risks and was later transferred to other businesses, sectors and, crucially, government. ERM aims at a maximum of comprehensiveness suggesting the integration of all risks to an organisation’s objective in a portfolio to inform organisational strategy. However, the concept suffers from unknown interdependencies between risks, implementation strategies that lack empirical validation and ambivalences and uncertainties arising from their management. It is only weakly rooted in organisational theory. Drawing on knowledge generation, theory key aspects for the empirical study of risk management in organisations are identified. These address the commensuration of risks, the comprehensiveness of the risk portfolio and the communication of explicit and tacit knowledge enabling organisational learning processes in different institutional contexts. Keywords: ERM; organisational risk management; organisational learning; public administration; commensuration;

1. Introduction The pervasiveness of risk in modern society has shifted the focus of the social sciences from primary risks to those risks created by the management of primary risks (i.e. secondary risks) (Power 2004). Initially, risk research identified inadequate regulation of primary risk as a core research theme (Irwin et al. 1997); however, more recently risk research has turned to understanding organisations and their capability to manage risk (Power 1999, 2008; Hutter and Power 2005). Today, various risk assessment and management techniques are applied across organisations and this article will focus on arguably the most influential of the last two decades, enterprise risk management (ERM). ERM was initially developed in the finance and insurance sectors to manage risks associated with investments and liability. It stands separate from other organisational risk management systems as the only one that attempts to integrate strategic, financial, hazard and operational risk into a single framework to inform an organisation’s strategic objectives. Indeed, the ERM framework suggests that even more risks can be progressively added to a central portfolio as management capability increases. It subsequently spread to other economic sectors and businesses *Corresponding author. Email: [email protected] © 2013 Taylor & Francis

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(FERMA FERMA 2002) before the UK took the lead in adopting it within the government sector (Cabinet Office 2002). The basis of ERM is the assumption that any event threatening an organisation’s objectives constitutes a risk and that these risks that may befall an organisation can be compared. A systematic assessment of these risks may then inform the formulation of an organisational strategy (COSO 2004). From this vantage, the opportunities appear unlimited as ERM might transform risk from a defensive concept into a future-oriented concept enabling organisations to seek new opportunities in their environment. This active position may explain ERM’s appeal across different types of organisations, sectors and fields. Despite differences in method and context, there exist a shared set of assumptions and rules underpinning risk management in both private and public organisations. When introduced to the business of government, ERM sought to improve the handling of risk – after several high-profile crises – by seemingly aligning the concept of rational precaution with that of strategy formulation. However, transferring organisational innovations from one context to another bears risks as well, since different contexts comprise different institutions and problems. Despite the diffusion of ERM through private business and government, it is surprising that many champions of risk management have remained sceptical of it (e.g. Hood and Rothstein 1999; Power 2009). This article elaborates on the challenges of carrying out risk management in organisations. In particular, we discuss the problems arising for organisational risk management when implementing and integrating risks to form a comprehensive management system. Recent research on risk management in organisations will serve as a starting point (Section 2) followed by an examination of the conceptual foundations of ERM and its limits (Section 3). Organisations will only alter their risk management, if they learn to integrate different risks coherently (Section 5). In the process, they may face trade-offs. When integrating risks commensuration poses a key challenge that demands special consideration (Section 4), and goes hand in hand with risk knowledge generation (Section 6). Finally, we discuss the identified problems and conclude with a discussion on future research needs (Section 7). 2. Risk management in organisations As organisations engage under different social context, they encounter different types of risks. What may appear like a truism however, is not well reflected in the realities of organisational risk management research. From an academic perspective, few empirical studies on organisation-wide risk management in either government or business exist (e.g. Bozeman and Kingsley 1998; Aabo, Fraser, and Simkins 2005; Fraser and Henry 2007; Mikes 2009; Arena, Arnaboldi, and Azzone 2011; Verbano and Venturini 2011) and instead the field is dominated by consultants who produce confidential reports for clients and selected surveys for wider audiences (e.g. FERMA, AXA, and Ernst &Young 2008; COSO 2010). The former does not permit wider dissementation of findings and subsequent advancement of explanations and the latter is largely based on subjective perceptions, e.g. of risk officers reporting of their perception of the field. Therefore, there is scope for academic studies to move beyond the selectivness and methodological weakness of current research by investigating the premises of organisational risk management approaches

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(cf. also Yaraghi and Langhe 2011), which in practice may be too simplistic to match organisational complexity. Academic research has only recently investigated the implementations of integrative organisational risk management. Verbano and Venturini (2011) distinguish seven different fields of application in business organisations. (1) (2) (3) (4) (5) (6) (7)

Financial risk management (FRM) Insurance risk management (IRM) Strategic risk management (SRM) Enterprise risk management (ERM) Project risk management (PRM) Engineering risk management (EnRM) Clinical risk management (CRM).

The authors point out that FRM, IRM seek to integrate risks by commensurating them in monetary terms. SRM also integrates risk, yet is signified by a literature with heterodox views of the relationship between organisational risk and economic return or organisational performances and competitive advantage indicating frictions between prices and objectives. The types vary considerably. PRM differs from the aformentioned concepts as it addresses risks to the delivery of projects while engineering and clinical risk management occur in specific types of organisations or divisions of organisations. This diversity of risk management schemes and their specific challenges is hardly surprising given the plurality of organisations, the judgement required in the management of risk and the complex environments, these schemes operate in. The implementation of ERM has proved to be even more varied. ERM differs from other organisational risk management approaches, since it aspire greatest comprehensiveness in the risks covered and managed of all organisational risk management schemes. It is even open to include secondary risks. This radical expansion of what risk management covers has been encouraged by voluntary standards issued by different, predominantly private, organisations (e.g. AS/ NZS/ISO 31000 2009). Yet, this expansion increases the challenge to commensurate the risks in the portfolio, to control for interdependencies between risks and to assure all risks are equally well communicated. Furthermore, existing frameworks claim that their guidance is readily transferable between different organisations and sectors and thus applies equally well in different institutional contexts (ibid.). However, scarce evidence about the performance of ERM suggests that the outcomes are wide-ranging, even in similar organisations (Fraser and Henry 2007; Mikes 2009; Arena, Arnaboldi and Azzone 2011).We suggests that the reasons for this are that ERM:
requires the reduction of risk to probabilities and impacts;
leads to a commensuration rationality that is not useful in practice;
suggests a degree of comprehensiveness that is often neither obtainable nor practical;
tries to reintegrate the unintended consequences of primary risk management by adding these as (secondary) risks to the portfolio;
is insensitive to organisational context.

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As a result, it is unlikely that ERM is implemented as intended and is more likely that what we might see in practice are arbitrary, biased and limited versions of organisational risk management. This has the potential to mislead managers, shareholders, taxpayers or voters by falsely making them believe that risks are being properly managed when the ERM framework is being applied. To understand ERM’s extraordinary flexibility and to scrutinise its claim of transferability (AS/ NZS/ISO 31000 2009) it is necessary to investigate ERM’s conceptual basis including how risks are integrated when different organisations implement this type of risk management. To do so, we draw on a communicative learning framework that allows us to highlight key challenges for safe organisational risk management. 3. The promise of risk management in enterprise One of the most influential frameworks providing guidance on ERM was put forth by the private Committee of Sponsoring Organisations of the Treadway Commission (COSO 1992, 2004). The 1992 framework suggested that:
by calling upon the risk awareness of employees an internal risk culture can be created;
risk attitudes are aligned with strategies and objectives;
hazards and opportunities are identified in relation to an organisation’s objectives; and
risks are assessed by the potential likelihood and impact of their harm. Additional management components include control activities, information and communication, and monitoring of risk management processes. This framework was expected to facilitate cross-silo thinking capable of revealing new correlations and ‘natural hedges’. Whereas, the 1992 framework re-established a control-bias focused on the delivery of the organisation’s strategic objectives with the help of internal audit (Spira and Page 2003) it was not until release of the 2004 framework, when risk management was defined as the purpose of internal control. This change is indicative of divergent orientations at the conceptual level. The framework effectively sets out two risk assessment processes, one for opportunities and another for hazards. Opportunities are thought to support the development of strategy and objectives while hazards are ignored for this end. This approach may miss crucial information for developing a robust, well-rounded strategy. In addition, the choice of risk management standards is left to the organisation, or more specifically its management board. Because of this lack of independent standards, the COSO framework has been compared with a closed system that establishes process control while ignoring outcome control (Williamson 2007). The framework has seen various adaptations across the world and in the UK is most closely associated with the Turnbull report (Turnbull 1999). The ERM framework of corporate governance has served as the blueprint for the UK government, which introduced statutory guidance owing heavily to the ERM framework (NAO 2000; SU 2002; HM Treasury 2004). As a conceptual innovation, UK government pioneered the notion of risk maturity (OGC 2007), itself an evolution of capability maturity models that represent an internal development process (e.g. software development). These models cast several organisational functions into a process-guiding framework (cf. MacGillivray et al. 2007) and although they offer

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different frameworks, most implicitly start from an organisation’s capability to control internal processes. Risk maturity models should thus help to organise the processes required for improving the management of a particular risk. The introduction of risk maturity can be seen as a practical response to the lack of control over implementation outcomes (Fraser and Henry 2007; Mikes 2009). Strutt et al. (2006) suggests that the process be initiated by defining goals followed by identifying processes – an approach that may remind more of strategic risk planning than ERM. The ERM concept has been highly unspecific about the actual integration and management of risks. Indeed, notional dominance of the portfolio (e.g. Ai et al. 2012) has been obstructing the recognition of the communicative social processes that support the portfolio with information. Integrating risks in a portfolio requires the recognition of those risks, their comparison and communication within the organisation. In a finance department, all risks might be comparatively expressed in monetary terms whereas in other departments, this might challenge existing risk management efforts. These issues have not been particularly addressed by either the COSO framework or UK governance guidance. Instead, this guidance has called upon an organisational risk culture to drive risk management implementation (COSO 2004; HM Treasury 2004). Indeed, no existing ERM framework has consideration for the differences between organisations and their diverse institutional contexts. While most organisations simply assess risks concerning their operations government departments, for instance, may additionally have legal obligations to identify risks to society and the environment at large. Organisations may generally be interested in expanding their search to investigate ‘known unknowns’ and possible ‘unknown unknowns’ (see e.g. Stirling et al. 2007) but some, including government departments, are responsible for a far greater variety of risks with more diverse cultural signatures creating substantial knock-on problems for the comprehensiveness and integration of their portfolios. ERM guidance suggests that organisations can abstract from these complex contexts by calling for risk management that only addresses risks to the strategy or objective of organisations (COSO 2004; HM Treasury 2004). 4. The challenge of commensurating risks and uncertainties Due to its financial origin, ERM assumes that all risks relevant for organisational strategy can be rendered commensurate in financial terms. This assumption may not be appropriate for organisations, whose risk portfolio is disparate, complex and touches upon the values of many stakeholders. These organisations face fundamental challenges of commensuration that tend to grow with portfolio size and have knockon effects for the institutionalisation of risk management in the organisation as well as the future coherence of risk management. Problems of commensurating primary risks are common and exist in all fields (e.g. in the environmental domain USEPA 1987; WBGU 1998; Pollard et al. 2004). These problems are constituted by competing explanatory natural and social scientific theories, autonomous institutions or both (Espeland and Stevens 1998). Kuhn used the term incommensurable to signify the epistemic gap between succeeding paradigms in the natural sciences (Kuhn 1962). Against Kuhn’s view of incommensurable scientific theories, it has been argued that ideas and concepts expressed in the language of one scheme (theory) can usually be expressed in the language of another (Davidson 2001).

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However, these epistemological debates do hardly touch the social and normative aspects of incommensurability. The explicit negation of commensurability has some recognition in the humanities. Values are said to be incommensurable, if they refer to different measures that lack an agreed standard to justify norms. Incommensurability has come to be defined as ‘the absence of a common unit of measurement across plural values’ (Martínez-Alier et al. 1998, 280; similarly Stirling et al. 1999). Each of these values might be objectively valid and imperative and each creates instructions for actions that are incompatible with the others (Berlin 1988). Any course of action might then be considered normatively wrong by some objectively valid standard. The stakes are altered if the associated processes are path-dependent. In opposition to this, commensurability implies that morally justified norms exist to judge amongst other things risks. Human rights, for instance, establish a commensurable state where all people hold equal rights even though they face different risks in different jurisdictions. Moral theory suggests that universal norms can justify plural ethical values (Habermas 2003), thus allowing for the recognition of weak incommensurability. Commensurability and comparability have been introduced to economics in a similar manner as they have in ethics but with an emphasis on comparability. Martinez-Alier, Munda and O’Neill (1998) argue that strong comparability may only imply weak commensurability if value preserving methods are used, though it may invoke strong commensurability if the range of risks being compared are compressed using value-insensitive methods. The sometimes divergent use of these terms in economics arises in part from the absence of a concept of non-action. Different from economics, non-action is central to law, e.g. as prohibition of particular actions reflecting not least incommensurable values expressed in society and politics. Unlike voluntary private standards, law binds all organisations equally. With respect to these problems of comparability and commensurability, there is no direction from ERM’s undefined concepts such as ‘holistic portfolio management’, ‘balanced approach’, ‘risk transfer’ or ‘risk appetite’ (e.g. HM Treasury 2006). Given ERM’s origin in finance, it is also not surprising that the framework has been a priori confined to risk (COSO 1992, 2004) at the expense of ignorance, uncertainty and ambivalence (Tversky and Kahneman 1974). Ignorance, uncertainty and ambivalence cannot be stated as probability times impact while the associated hazards, including those arising from risk management, may still be real and call for action. While ERM considers mitigation, acceptance or transfer of risks it ignores the possibility of incommensurability and incomparability. Known difficulties with comparability in ERM include lacking taxonomies to compare risks and uneven levels of maturity across the organisation to assess risks (Von Känel et al. 2010). Challenges of commensuration are not least related to independent institutions governing risks and the fact that these risks affect third parties. The need for public legitimacy of private (Economist Intelligence Unit 2007) and public organisations (HM Treasury 2004; OECD 2010) would be expected to take possible incommensurabilities into account from the onset (Levinthal and March 1981; Scott and Meyer 1994). Yet, empirically speaking, powerful organisational and institutional biases are at work (Thompson, Wildavsky 1986) that may prevent this from happening and may lead to strong Berlinian incommensurabilities instead. The incommensurability of risk is not a sole case of competing scientific paradigms or directly conflicting values, but instead comprises a blend of these two institutionalised across organisations. Its relevance may substantially differ between

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organisations as for instance government can draw on generalised coordination mechanisms like markets and law to define incommensurable states to which private and subordinate public organisations have to adapt. Such institutional complexity has not prevented ERM guidance to cast its framework in strongly commensurable terms (COSO 2004; HM Treasury 2004). This might be one of the reasons why implementation has come to add ever more risks to the portfolio. But can ERM be coherently integrated if underlying commensuration problems are ignored and risks including secondary ones are indiscriminatingly added to the portfolio? Or is ERM itself flawed, in that it simulates comprehensiveness and integration but instead creates the secondary risks it tries to reintegrate? In order to answer this question, we have to specify the problems associated with implementing a comprehensive and integrative organisational risk management. We will thus show next how commensuration problems and a bias for comprehensiveness may influence coherence of integration. 5. Integration of organisational risk management Implicit to the ERM framework are a series of presumptions including: (a) risks are and can be objectively and unambiguously defined and clearly distinguished from uncertainties, ignorance or ambiguity (COSO 1992, 2004); (b) risk-related information is reported without technical or behavioural frictions to the top of the organisation (i.e. management board) (COSO 1992, 2004); (c) the risks in an organisations’ portfolio directly correspond to executable tasks (COSO 1992, 2004; HM Treasury 2004); and (d) the success of executing these tasks can be phrased again in risk-analytical terms covering the internal (e.g. operations) and external dimensions (e.g. reputation) equally well (HM Treasury 2004). All this, however, encounters problems of translating actions (risky activities) accurately into systems knowledge (hazards) and back into now reformed actions (risk mitigation). In principle, an ERM system could apply decomposable actions (see Simon 2002). More specifically, once risk information is available at the top of the organisation it is up to the board to manage the risks and allocate responsibilities hierarchically downwards. This approach to ERM, whereby individuals (managers) interpret the natural and social environment of the organisation (relative to the outside world) and redefine roles, rules and behaviours within the organisation to implement risk management is representative of correspondence theory and in practice is associated with bounded and rational choices. If we understand the arguably vague idea of ‘risk appetite’ (Power 2009) as a propensity of choice (see March 1994), the decision context the board faces might be restated as a problem of integrating different preferences to decide upon a particular management option. From Arrow’s well-known impossible theorem (Arrow 1950), which states that three different options (e.g. risk management strategies) ranked on an ordinal scale and that adhere to the conditions of transitivity, unanimity and independence cannot be sorted consistently, it is apparent that within the confines of rational choice the way forward for resolving management decisions is by either relaxing the axioms or by moving to cardinal information. A straight-forward practical response is to measure all management options as prices and to apply costbenefit analysis. On these grounds, the measurement allows for development of cost functions e.g. costs associated with the specific management options. However, this is not a feasible option for all risks and all organisations since it establishes strong

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comparability and commensurability. It is not a sufficient condition to assure safe and coherently integrated risk management either (Lindblom 1959; Olsen and March 2004). Organisations may face more fundamental challenges with respect to commensuration that relate to legitimacy. As we have argued above, commensurating risks can be analytically distinguished into comparability e.g. by developing standards such as global warming potential and commensurability, which has been signified by organisations’ dependence on legitimacy. This can refer to authority outside the organisation exerted amongst others things as isomorphic pressure, e.g. when confidentiality is challenged by calls for sharing information (Bellamy et al. 2008); yet these challenges arise internally too, for instance, when comparing dissimilar risks such as extremities to lives lost. Although the threshold for conflicting management strategies might be lower, when risks are highly comparable plurality of values remains implicit in all risk portfolios (6, Bellamy, and Raab 2010). For example, even translation of marketable goods to a single monetary value may include problems of commensuration (e.g. Huault and Rainelli-Weiss 2011). Predicting prices ex-ante is, for instance, often difficult in the face of uncertain price-defining processes (Stehr, Henning, and Weiler 2006; Rona-Tas and Hiß 2011); and often enough prices are de facto defined ex-post (Fourcade 2011). Defining (weak) incommensurability qua legal standards ex-ante offers a solution. It is perhaps little surprising when incommensurability between different financial products has been identified as a necessary condition to resolve the financial crisis (Roubini and Mihm 2010). Given this reality, ERM insists that inconsistent comparisons can be avoided and that if commensuration conflicts still emerge, these can be reintegrated through constant bottom-up risk scanning exercises as secondary risks (e.g. Hillson 2003; Beasley and Frigo 2007). This resolution confronts three fundamental problems: (1) Management of a risk portfolio is constrained by the budget in which it operates. This suggests that no risk portfolio with different associated management options can provide information for an optimal resource allocation independent from the overall budget (Cox 2008). Moreover, all risks are interdependent with regard to budget. This immediately creates a hierarchical context within which any organisational unit or dependent organisation operates. Few organisations hold exclusive control over their budgets, instead negotiating them with boards or central government Government differs from business organisations in as much as it has the power to allocate or distribute adaptation or mitigation costs qua policies or regulation to risk producers or hazard takers (Coase 1960; Luhmann 1993). In particular, the missing link between risk and social welfare concepts constitutes a fundamental commensuration problem for public organisations because direct links between investment/public expenditure and social cost of e.g. environmental risks, may not be fully established ex-ante (e.g. by insurance markets) while others are defined ex-post (e.g. by court rulings). While for many risks, the outcome de facto depends on competitive markets for risks and ex-post attribution of responsibility risks like climate change may not allow ex-post attribution within existing institutions as these may loose management capability as the risks unfold. These risks are judged by severity instead of probability. (2) Cost-efficient risk management decisions based on fully quantified costs for each management option would render risk comparisons highly

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commensurable. This encapsulates two problems for the management of public risks. First, there exists a threat to legitimacy in which the complexity of risk comparisons collapses into the uniform minimalism associated with prices/costs (Martínez-Alier, Munda, and O’Neill 1998). This affects the public legitimacy of the organisation. Second, evaluation of existing risk management practice using a strongly commensurable standard would essentially turn strategy changes into zero-sum games. Both problems may come into direct conflict with an organisations’ search for safety. As we know from failing train operators and banks, depending on the (regulatory) context risk management can in fact become instrumental to open up safe operational areas for risk taking (Hutter 2005; Moss 2010). (3) Interdependencies between risks are emergent and in many instances unidentified. Most risks are interdependent because they are embedded in complex social systems or integral parts of these. When integrating risks in the implementation process ERM is actively creating interdependencies between the managed risks not least through the budget (Beasley and Frigo 2007). These display very different statistical behaviours compared to independent risks. Whereas independent risks generally follow Gaussian distributions, interdependent risks represent Paretian distributions, which represent extreme system behaviours or black swan events (Andriani and McKelvey 2007; Mandelbrot 2009). ERM guidance has not yet accounted for these alternations in risk character (COSO 2004). While strategic opportunities from merging organisational silos are regularly emphasised, ERM has been ignorant of how it creates interdependencies between risks. These limitations should have far-reaching implications for the future of organisational risk management. One question arising in this respect for ERM is whether risk maturity models can prevent it from misleading organisations into false safety believes. 6. Learning to manage risks Risk maturity models have been around for over a decade with one model explicitly drawing on organisational learning theory (Strutt et al. 2006). The basis of this idea is to identify the processes relevant for mitigating primary risks. Once this is achieved, the model suggests defining attributes of each process associated with greater safety and map out improvement steps for each to progress from one attributed safety level to the next. An organisation is thought to exhibit organisation-wide maturity of a certain level if all processes have reached or exceeded said level. Hence, the overall risk maturity level of the organisation is defined by its weakest process. While the learning perspective in this approach is apparent, it is confined to Argyris’ and Schön’s framework (1978) and to the improvement of a single risk (Strutt et al. 2006). Depending on the comprehensiveness aspired by organisational risk management, that is, the integration of multiple risks; a more complex organisational learning theory is needed. This is so because, the more comprehensive a risk portfolio becomes, the more learning processes have to be institutionalised (Fraser and Henry 2007; Mikes 2008). This is not addressed neither by the COSO framework (2004) nor the Orange Book (HM Treasury 2004) which suggest a feedback

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from iterative risk analyses to the emerging risk portfolio will do. As we will make clear next this fails to account for the emerging risk management structures let alone conceptualise the complexity at the frontier between codified and tacit knowledge generation processes and intentional and non-intentional outcomes. Amongst the abundance of theories addressing organisational learning, organisational knowledge creation theory (Nonaka 1994) stands out as the one offering the most compelling synthesis for organisational risk management. This theory overcomes ERM’s presumption of established, universal risk information since it views organisations as knowledge generating. It shares with cultural theory, the view that knowledge is generated in historically situated interactions (Nonaka and Toyama 2003; Nonaka, von Krogh, and Voelpel 2006; Nonaka and von Krogh 2009). In particular, the theories’ distinction between explicit and tacit knowledge is crucial for approaching complex risks in non-reductionist ways. Drawing on behavioural theories of organisational learning, Nonaka (1994) discerns four organisational learning processes, which are perpetuated by a constant alternation of tacit and explicit knowledge: socialisation to the organisation, externalisation of new knowledge, combination of new information and the internalisation of this information. Tacit knowledge creates the bounds to socialise new personnel to the organisational learning process. Because it is action-oriented, difficult to formalise and personally acquired through routines it surpasses mere data or information and is necessarily specialised and restricted to particular domains. As it is dispersed and subjective, it cannot easily be aggregated or generalised, e.g. three conversations between (a, b), (a, c) and (b, c) is not the same as a single conversation between (a–c). Distinct from this, explicit knowledge can be conveyed formally. This involves controlled, intentional actions as well as cognitive processes and represents the type of knowledge that ERM expects to use to inform the portfolio management system. Creating organisational knowledge includes conversions of both explicit aspects of knowledge such as formal risk assessments or project documentation, and tacit estimates such as personal expertise about particular risks or ambivalences. ERM guidance has ignored the generation and transfer of implicit and explicit knowledge (OGC and HM Treasury 2003; HM Treasury 2004, 2009; ALARM 2005, 2006; OGC 2007). Yet, because of the constant socialisation and internalisation processes in organisations both types of knowledge are necessary to successfully assess and communicate risks, uncertainties and ambivalences throughout an organisation and beyond. A tangible challenge in reorganising risk management in this respect remains to unlock relevant tacit and codified risk knowledge from existing intra- and inter-organisational networks of (sometimes independent external) actors to generate risk knowledge (Amin and Cohendet 1999; Borgatti and Cross 2003). This requires a valid empirical understanding of these processes. Nonaka and colleagues see the key challenge not in redesigning and adapting the organisation to information under conditions of maximising or bounded rationality, but in securing the fragile transmissions of knowledge between individuals in the organisation (Nonaka and Toyama 2003; Nonaka, Krogh, and Voelpel 2006). Because of its link with the roles and tasks of organisations, codified knowledge dominates this process. It allows specifying tasks ex-ante. Formal structures are however, always enacted by personal relationships and social networks in which mostly tacit knowledge is exchanged. Organisations thus have to align both types of knowledge, since without tacit knowledge accompanying codified risk

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information, organisational risk knowledge will be dangerously incomplete. Tacit knowledge is already necessary to understand the very meaning of codified knowledge and is particularly needed to communicate ambiguity, uncertainties and ignorance across epistemic communities, up and down the organisational hierarchy and across inter-organisational networks. To understand and govern these processes, organisations rely on empirical evidence. The alternation of both knowledge types thus goes beyond analytical models of information diffusion and may raise explicit truth claims in commensuration processes. This relates to the methods used as we have seen above as well as existing and emerging institutions, e.g. when in the interest of making timely decisions, practitioners overlook issues of commensurability (Prpich et al. 2012). Many organisations have moved to project risk management in response to this organisational complexity (Hillson 2003) but this has also led to subjective ad hoc assessments of perceived risks to cope with the increasing comprehensiveness (Mauelshagen et al. unpublished paper). While organisational biases are inevitable (cf. Thompson and Wildavsky 1986) subjective assessments are bound to increase these and hamper risk knowledge creation. Depending on the organisational context, contractual mechanisms and/or financial instalments have been used to influence role performance in risk management. Although incentive-based contracting has dominated in the past (cf. Dixit 2007), the difficulty is of course that bonus payments are linked to outcomes that only materialise in the future and are thus uncertain. There is also a particular asymmetry between private and public sector organisations. The latter’s capacity to incentivise staff is constrained by the existence of intrinsic and extrinsic motivation amongst staff (Le Grand 2003). In particular, intrinsic motivation is indispensable for socialisation and internalisation processes in risk knowledge creation. Intrinsic motivation is not an add-on to induced incentives (extrinsic motivation), since it is directly related to crowding-in and crowding-out effects that signify facilitating (crowding-in) or corrupting effects of rewards (crowding-out effect). Financial compensation for blood donations for instance are known to reduce contributions (Osterloh and Frey 2000). It is thus the mix of both types of motivation that organisations need to create risk management capability and, indirectly, to reduce risks to reputation. Intra-organisationally speaking various organisational studies have shown that knowledge creation processes are highly sensitive to social contexts, such as the timing of activities (Massey and Montoya-Weiss 2006), the organisation of processes (Dyck et al. 2005) and staff motivation (Osterloh and Frey 2000). Employees can have various motives for not sharing their knowledge including fear of criticism or misleading colleagues, indecision about the relevance of an individual contribution or the factual accuracy of a potential input (Ardichvili, Page, and Wentling 2003). The question is how do organisations address these influences so as to improve or refute existing concepts of organisational risk management. ‘Internal publics’ (IRGC 2005) have been suggested to institutionalise communicative feedback and may assume a similar function. In principle, they can attenuate strategic exploitation of otherwise weakly institutionalised processes (cf. also Fraser and Henry 2007). Internal publics confront possible strategic behaviour with peerreview and they may initiate internal discourses maximising cooperation amongst uneven risk mature units while supporting unbiased assessments of risks (Renn 2008, 343). In this way, they may facilitate internal risk discourses better supporting

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coherent commensuration of risk through their consensus orientation (Rayner and Cantor 1987). Risk maturity claims to captures this idea too by suggesting that organisations may establish skills that increase risk management capability by improving the coherent management of specific risks via vertical integration of all relevant organisational functions. The result may evoke organisational learning whilst indicating some kind of trajectories for individual risk management activities. It is unclear as to how robust these internal pathways are. In the first instance, it is unknown as to how far risk maturity can establish organisational path dependency (cf. Sydow, Schreyögg, and Koch 2009), e.g. with respect to the interdependency with the budget. This presents an empirical research question with great importance for organisational risk management (Mikes 2008). In the second instance, within the ERM framework models are immediately confronting a risk portfolio, which regularly includes unknown next to known interdependencies between risks. Risk maturity models may hold value if individual risks and their interdependencies are fully understood and if risk management implementation is path-dependent. Yet, if interdependencies are not known, empirically understood, or indeed ignored they may have non-intended effects. Therefore, if interdependencies between primary risks are understood and management has been established only then adding more risks to the risk register (thus increasing comprehensiveness) appears to be safe. Crucially, this is interdependent with organisational capacity and budget. Ignoring the latter interdependences would again compromise safety. ERM (COSO 2004) has abstracted from most of these difficulties assuming that organisations will strategise from objective risk assessments only, that problems of commensuration do not exist or are at most trivial, and that an emerging risk culture will be enough to institutionalise risk management. Although Government guidance (HM Treasury 2004) has added the notion of risk maturity it does not offer a concept of internal path dependency that might increase the chances of lasting organisational maturity. 7. Discussion The uptake of ERM by private organisations and governments has been tepid. We suggest that this may be due to ERM’s lack of underpinning in organisational theory and its lack of clear institutional design recommendations. As ERM nevertheless gains currency in many organisations, we observe more and more responsibilities being expressed as risks. While this may alter risk awareness amongst staff, it may also create internal risk-spirals not unlike that seen in the public policy domain (Rothstein, Huber, and Gaskell 2006). Yet, we have little empirical evidence showing what the resulting comprehensiveness is worth. What is limiting organisational risk management is the lack of a concept of risk knowledge generation, with current incarnations of ERM assuming risk information arises from within the organisation like a deus ex machina. This failure to recognise the value of internal communication ultimately results in poorly integrated and potentially incoherent risk knowledge bases and risk management systems. Moreover, ERM is bound to produce ambivalence and uncertainty over outcomes without institutions with the ability to generate knowledge. Such failure actively produces secondary risks.

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The confused perception of organisational processes and the lack of institutional design have resulted in a structural complexity that is difficult, if not impossible to manage. Under such structural complexity seemingly minor decisions, like opting for subjective instead of objective risk assessments, may lead risk management down unsafe pathways. For example, ERM may treat subjective assessments as a secondary risk and consider them as an instance of operational risk, yet this reintegration falls apart because of the time asynchrony between analysis and design. At times such problems are acknowledged in the literature, if only implicitly, e.g. in the form of the role of the risk manager. In a way risk management may be thought of as stop rules (against the board’s search for profitability) and as search engines (scanning the horizon for new risks and opportunities). This presents a conflict for which the literature does not offer resolution. There is a corresponding failure to recognise the difference between analytical and empirical reasoning in the ERM guidance and much of the practitioners’ literature. Against this background, it is not surprising that ERM has sometimes developed into a performance management system rather than strategic risk management system. In public organisations parallel management processes are on the rise, some of which cover exclusively secondary risk (Mauelshagen et al. unpublished paper). In business, ERM has shown a tendency to replace existing principle-based rationales by rule-based rationales (Martin and Power 2007). Indeed, ERM can make organisational processes so fluid that changes are implemented in the gap between analysis and design and new risks are added before valid empirical knowledge about existing risk management has been generated (ibid.). As risk management starts to lack empirical validation (Wilkinson and Ramirez 2010), ambivalence builds up and grows into full uncertainty over operational outcomes. Under these circumstances, the generation of risk information is likely to become even more exposed to various cultural influences thus further eroding the credibility of risk management to produce safety, let alone inform strategy. This raises the very principled question of whether or not interaction in organisations should be considered in risk terms at all (March and Shapira 1987; Bunge 2008). Many external relationships of organisations are defined in probabilistic terms, most commonly in the market place. For example, buyer-supplier relationships are commonly expressed and modelled as risks. Introducing the concept of risk to hierarchies, however, may turn these mixed-motive relationships (Scharpf 1994) sour. From a government department’s perspective budget negotiations might be seen as a risk. Yet, codifying and including it in a departmental risk portfolio would hamper open relationships with central government. Risk is thus a highly ambivalent concept in relationships and particularly in complex hierarchical contexts that invite ignorance (Rothstein 2003; but also Kutsch and Hall 2010) and possibly even deviant behaviour (Vaughan 1999). Stop rules might offer a fix as long as they are accepted by all as viable and enduring solutions institutionalised to ascertain accountably (6, Bellamy, and Raab 2010). There is a belief that ERM practice lags behind guidance and that public organisations lag behind private (Economist Intelligence Unit 2007; Collier and Woods 2011). Yet, this belief may well be turned around asking what empirical underpinning these frameworks actually provide to support the suggestion that they were theories and that they might work across organisations, sectors and indeed universally (e.g. AS/NZS/ISO 31000 2009)? The question is all the more important as interdependencies are increasing and responsibilities for tak-

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ing risks or endorsing safety become increasingly blurred between organisations and within. Knowledge generation theory suggests that the knowledge output is grounded in true belief seeking staff. Truth claims are generally established in any communication (Habermas 2003) but the more specific hypothesis put forward here suggests that organisational risk learning could be supported further if it is established transparently (IRGC 2005) and as a communicative feedback mechanism between communicative actors and organisational structure (Heracleous and Hendry 2000). As indicated above, internal commensuration processes might even be designed democratically involving affected parties in the organisation. In other organisations, such as government departments, commensuration processes are very much driven by legitimacy considerations that may involve claims of strong incommensurability, e.g. when no markets or legislation exist previously. Since ERM suggests that uncertainties, ambivalences and risks should all be rendered strongly commensurable it might in fact hinder identification and development of corresponding strategies. Commensuration processes deserve more empirical research in their own right, revealing for instance, what risks are added to risk registers and portfolios and when or, how some organisations succeed in establishing path-dependency in risk management. By contrast, some of the existing empirical research on organisational risk management is methodologically unprepared to explain the problems ERM has shown in practice. While ERM has been designed for fluidity, we actually have little empirical evidence that organisations better mitigate or adapt to risks by becoming more fluid (Schreyögg and Sydow 2010). The convergence of ERM towards risks of ICT and hierarchical policies according to organisational size (Collier and Woods 2011) may thus reflect an appetite for organisational certainty. The scarce empirical evidence we have suggests that the contingent factors influencing ERM implementations are closely related to existing capabilities or the adaptation of specific new ones (Woods 2009). This would suggest that ERM’s potential to integrate new risks may be much overstated. At the same time, poorly specified risk management concepts, such as risk transfers with their potentially huge welfare implication invite secondary risk. Given such consequences, it remains to be seen how defensible ERM in public organisations will be in society and courts and whether organisational risk-heuristics can assure accountability and liability. Because of the hybrid character of the uncertainties managed by public organisations (Miller, Kurunmäki, and O’Leary 2008), the outcome is hardly predictable. In this article, we emphasised the fundamental conceptual weaknesses of ERM. In this vein we stated our concern with studies of organisational risk management that apply the stringent but conceptually flawed ERM framework to the study of organisational risk management. We emphasise that ERM is not based on valid, empirically tested theories – a state that can only be overcome if we produce meaningful, valid empirical studies. Greater empirical efforts should thus be devoted to explain risk knowledge generation and commensuration processes. Acknowledgements The authors gratefully acknowledge the founding provided by the ESRC, the EPSRC, the Department for Environment, Food and Rural Affairs, the Natural Environmental Research Council and the Living with Environmental Change programme.

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Quantitative Finance Letters, 2013 Vol. 1, 55–59, http://dx.doi.org/10.1080/21649502.2013.865068

Maimonides risk parity PHILIP Z. MAYMIN∗† and ZAKHAR G. MAYMIN‡ † NYU-Polytechnic ‡ Independent

Institute, USA Researcher, USA

(Received 15 August 2013; in final form 18 October 2013 ) Drawing on and extending an estate allocation algorithm of twelfth-century philosopher Moses ben Maimon, we show how ‘Maimonides Risk Parity’ can link together the equal-weighted, market capitalisation-weighted, and risk parity portfolios in a unified, elegant, and concise theoretical framework, with only a single intuitive parameter: the portfolio risk. We also compare the empirical performance of Maimonides risk parity with standard risk parity and equal-weighted portfolios using monthly equity and bond returns for the past six decades and find that Maimonides risk parity outperforms risk parity for any value of the portfolio risk, and outperforms the equal-weighted portfolio for most values of portfolio risk. We also discuss the optimal choice of portfolio risk. The superior performance of Maimonides risk parity comes from the algorithm’s natural ability to robustly incorporate measurement error of seemingly small estimated risk. Keywords: Risk parity; Maimonides

1. Introduction Risk parity is a widely popular method of portfolio construction and asset allocation that seems to be far removed from standard portfolio construction methods such as mean–variance optimality or even a simple portfolio with weights given either equally or by market capitalisation. The reason for the apparent disconnect is that risk parity methods ignore market capitalisation, estimates of returns and, usually, correlations. Instead, it simply invests in each of a given set of assets inversely proportional to its volatility, or some other measure of risk (see a discussion in Maillard et al. 2010, including the definition of a more general equal risk portfolio). Note, however, that risk parity need not be merely considered as a specific instance of the more general equal risk portfolio, which would involve correlations as well as standard deviations. Fisher et al. (forthcoming) show that the standard risk parity portfolio, ignoring correlations, is optimal in the space of all possible portfolios. One central insight of the Maimonides approach presented in this paper is that low estimated volatilities are unstable; in a similar way, estimates of correlations are also often unstable. Furthermore, there is usually insufficient data to estimate correlations: among n assets and m time periods, we have mn observations of returns. Standard risk parity requires estimating n parameters, with m observations available for each estimate. Estimating n(n − 1)/2 correlations, on the other hand, means we only have 2m/(n − 1) distinct observations per parameter. When m is not sufficiently large, which is often the case in practice, especially with monthly returns but also in general because of the long-term nonstationarity of financial data and the possibility of short-term spurious correlations, the estimation breaks down, in the same way the estimation of low volatility breaks down. Because of these issues, and the overwhelming popularity ∗ Corresponding

of the standard risk parity approach in practice compared with other variants, in this paper we focus only on the standard risk parity approach and ignore correlations. This does not mean we necessarily assume correlations to be a constant or zero, in the same way that not using information about average returns means we are necessarily assuming all returns are a constant or zero. Without a simple parametric superclass of portfolio construction methods that encompass both risk parity and these other weighting methods of market capitalisation or equal weight, researchers and practitioners have no choice but to compare them side-by-side as if the possibilities are given exogenously (see, for example, Chaves et al. 2011). Our approach in this paper is to unify risk parity with the simple weighting strategies, provide a more general approach to unifying risk parity with other portfolio construction methods as well, and show how there is indeed an underlying relation between the various methods. The main insight comes from twelfth-century philosopher, and direct ancestor of the present authors, Moses ben Maimon, also known as Maimonides, or the Rambam. Asset allocation in general deals with the question of how to properly allocate a fixed investment amount among given assets or asset classes. Maimon (c. 1180) considered the similar problem of allocating estate holdings among debt holders. One might imagine two simple ways: divide equally among all debt holders or divide proportionally among all debts. This can be roughly viewed as analogues for plain risk parity or equal weighting. The proposed solution of Maimonides, however, is neither one nor the other, but a hybrid in between. We show how risk parity using the Maimonides algorithm differs from plain risk parity. We also extend the original Maimonides algorithm to allow, in our context, market capitalisation weightings instead of equal

author. Email: [email protected]

© 2013 The Author(s). Published by Taylor & Francis. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The moral rights of the named author(s) have been asserted.

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weightings, and explain what this means in his original estate allocation context. Finally, we report on empirical results comparing Maimonides risk parity with other standard portfolio construction methods. 2. Maimonides allocation algorithm Aumann and Maschler (1985) provide the 2000-year history of thought on questions of estate divisions among debtors, dating back to the Babylonian Talmud. They describe various allocation strategies, including the one of Maimonides, and provide game-theoretic analysis. Importantly, they also note that the modern approach of proportional division may not be the most equitable; for example, perhaps the excess debt owed to one person that exceeds the total value of the estate ought to be ignored on the principle that one cannot get more than there is. This feature of several non-proportional allocation strategies, including the one codified by Maimonides, is the source of the robustness to estimation error, a crucial feature. While in the present paper, we focus on the Maimonides allocation algorithm, similar approaches can be used for other non-proportional rules. The Maimonides allocation algorithm is as follows (Maimon, c.1180, chapter 20, section 4): How is the property divided? If when the property is divided in equal portions according to the number of creditors, the person owed the least will receive the amount owed him or less, the property is divided into that number of equal portions. If dividing the property into equal portions would give the person owed the least more than he is owed, this is what should be done: We divide the sum equally among the creditors so that the person owed the least will receive the money that he is owed. He then withdraws. The remaining creditors then divide the balance of the debtor’s resources in the following manner.

We first propose a simple way of visualising the Maimonides allocation algorithm: as water poured into jars. Imagine that all of the debt holders are jars, lined up all in a row and connected at the bottom, with each jar having a lid except the largest, and the amount of money to be allocated is water. Pouring the water into the largest jar will distribute it equally among all jars until the smallest jar is full. The lid on the smallest jar keeps the water from flowing out. As more water is poured in, it is distributed only among the remaining jars. Jars keep getting filled up in this way from smallest to largest. Whenever the water runs out, the amount in each jar represents the allocation. The Maimonides allocation strategy is not immune to collusion. Because smaller debts get paid out earlier, any large debt holder can split the debt among many individuals to effectively improve seniority. In the context of modern bankruptcy and estate law, where creating new entities or splitting debts is easy, the Maimonides allocation strategy would not be useful. However, when collusion is not likely or possible, such as in the context of medieval or ancient times, when the sources of the debts were common knowledge and opportunistic debt splitting would be immediately spotted and shunned, and in the context of modern financial assets, where the securities are not going

to split or merge simply to thwart a particular portfolio manager, the Maimonides allocation strategy could work quite well. There are three reasons why the Maimonides algorithm may make sense, even in preference to the more intuitively appealing proportional rule that modern bankruptcies tend to follow. First, the reliability of the debt, and the impact of the unreliability, may grow with the size of the alleged debt. A creditor alleging a $10 debt can be considered more reliable than one alleging a $1 billion debt on the same estate. Second, a larger debtor is less sensitive to his recovery value. For the $10 creditor, the difference whether he gets back $1 or $2 is significant, but for the $1 billion creditor, the difference is negligible. Third, underwater debtors may have an easier time raising additional small amounts of debt under the Maimonides algorithm than under proportional allocation. If someone has borrowed and lost a million dollars, there is no incentive for anyone to lend him another 10 dollars, even to potentially make a 100 dollars in profit, because the vast majority of the profit would ultimately accrue to the early large bondholders rather than the later small one. Under the Maimonides environment, smaller debt takes more of a priority, and so are encouraged, even for debtors deeply underwater. Of course, the later, smaller creditor would still want to make sure the investment is sufficiently worthwhile, as he would only receive about half of the profit from the venture in this case of one pre-existing large creditor. In fact, the flexibility of the Maimonides strategy in rearranging larger debts into smaller ones, a drawback for modern bankruptcy, is a feature for modern portfolio allocation, because it allows us to split the assets in a way convenient for the portfolio manager. Specifically, as a trivial example, one could treat each dollar owed as a separate individual. In the visualisation outlined above, that would be equivalent to having a long line of equally sized jars. Thus, the Maimonides algorithm can easily accommodate proportional allocation. More generally, we are able to replace single jars with combinations of smaller jars to achieve particular portfolio goals, such as market capitalisation weighting. Applying the original or extended Maimonides allocation strategy to portfolio allocation means allocating risk among assets rather than money among heirs. The risk of an asset is usually proxied by its historical volatility, though alternative measures such as implied volatility, variance, kurtosis, or even implied expected tail loss can all be used as marginal risk contribution to the resulting portfolio (Alankar et al. 2012). In this paper, we follow the standard practice and use historical volatility as a proxy for risk. Using alternative measures would be a straightforward application of our general framework. Safety is the inverse of risk, so again following standard practice we proxy safety by the reciprocal of the volatility. Risk parity could more clearly be called safety parity. The overall safety of an investor’s portfolio can be set to the overall safety of the market at the time; equivalently, the overall risk of an investor’s portfolio can be set to the historical market volatility. In the next section, we explore

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Maimonides risk parity both the robustness of this choice as well as the optimum portfolio risk. This overall safety is then allocated among assets based on their historical safety; equivalently, the reciprocal of the portfolio risk is allocated among the reciprocals of the individual volatilities of the asset or asset classes in the investment universe. The allocation mechanism is the original or the extended Maimonides algorithm. The original Maimonides algorithm is as described above and is applied to the reciprocal volatility numbers exactly as if they were money amounts in an estate context. The reciprocal of the portfolio risk is the amount to be allocated and the reciprocals of the individual asset or asset class volatilities are the debt amounts. The extended Maimonides algorithm first reallocates the safety of the assets based on their market capitalisation, and then applies the original Maimonides algorithm to the result. The first reallocation step essentially re-expresses the risk per dollar into a risk per unit percentage ownership of the asset. The effect of this reallocation is to change the starting point from an equal-weighted portfolio to a market capitalization-weighted portfolio. In other words, when risk is large and safety is low, none of the assets receive their full risk-based allocation (none of the jars are full), so the risk/safety is shared equally among all, but because the number of jars of each asset has been extended to be proportional to their market capitalisation, an equal allocation now means a market capitalisation allocation. For the remainder of this paper, we focus on the original Maimonides algorithm and comparisons with standard risk parity and equal-weighted portfolios. The analysis with the extended Maimonides algorithm and comparisons with standard risk parity and market capitalisation-weighted portfolios would be similar.

3. Ill-posed problems Informally speaking, an ill-posed problem is a problem such that the output, or the solution of the problem, is not a continuous function with regard to the inputs. For example, solving for x in the equation x − a = 1 is not an ill-posed problem, because its solution x = a + 1 is a continuous function with respect to a. On the other hand, solving the equation a·x =1

(1)

is an ill-posed problem, because its solution x = 1/a is not continuous at the point a = 0. This could be a real problem. For example, if a = 0.01, then x = 100, but if we mistakenly estimate a as 0.001, we would think that x = 1000. So a small error in estimating may lead to a big error in the solution. Ill-posed problems appear in all fields of mathematics. The generalisation of this example is any problem that involves a matrix inverse. When the determinant of the matrix is close to zero, the calculation of the inverse matrix becomes unstable. This is why, for example, the estimation of parameters in a linear regression is, generally speaking, an ill-posed problem.

Tikhonov and Arsenin (1977) formalised an approach to deal with an ill-posed problem. Here is the idea of his approach. Let us restate the problem (1) as the problem of finding min(ax − 1)2 .

(2)

x

And let us generalise the problem (2) as the problem of finding min[(ax − 1)2 + λ ∗ (x − x0 )2 ],

λ ≥ 0.

x

(3)

For any regularisation parameter λ > 0 and any initial value x0 , the problem (3) is no longer ill-posed. Of course, the question of choosing the right value regularisation parameter and the right initial value are important questions in Tikhonov regularisation. For portfolio construction, let σi , i = 1, . . . , n, be the individual volatilities of n assets. Then the corresponding (non-normalised) weights of the risk parity portfolio are 1/σi . From our example (1), we already know that this is an ill-posed problem: a small change in a volatility that is close to zero could make a large change in the associated portfolio weight. Assume that all volatilities are sorted, σ1 ≥ · · · ≥ σn . Let σ > 0 be the portfolio risk. If σ ≥ σ1 n, then the Maimonides portfolio is the equal-weighted portfolio. If σ < σ1 n then let k = k(μ, σ1 , . . . , σn ) be the largest k, k = 1, . . . , n such that 1/σ −

k

j=1

1/σj

n−k

≤

1 σk+1

.

Then the (non-normalised) Maimonides portfolio weights are given as ⎧ 1 ⎪ ⎪ , ⎪ ⎪ σ n ⎪ ⎪ ⎨1 mi = σ , ⎪ i ⎪  ⎪ ⎪ ⎪ 1/σ − kj=1 1/σj ⎪ ⎩ , n−k

σ ≥ σ1 n, i = 1, . . . , k,

(4)

i = k + 1, . . . , n.

The portfolio risk σ is not necessarily equal to the volatility of the resulting Maimonides portfolio. It is simply an input to the Maimonides algorithm whose reciprocal represents the amount of safety available to be allocated among the individual assets in the same way that the estate value is an input to the estate allocation problem. The sum of the Maimonides weights will thus equal the reciprocal of the portfolio risk. If the portfolio weights xi , i = 1, . . . , n, are the result of Tikhonov regularisation min

x1 ,...,xn

n n   (xi − 1)2 , (σi xi − 1)2 + λ i=1

i=1

then they are xi =

σi + λ . σi2 + λ

λ ≥ 0,

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Figure 1. Rolling five-year volatility of equal weight portfolio.

Figure 2. Information ratios for entire sample 1942–2012.

Similarly to Maimonides, when λ → ∞, these portfolio weights approach equal weight, and when λ → 0, they approach risk parity. In between, they are in general not the same as Maimonides weights. The main point, however, remains: when volatilities are low, we should use risk parity weights, and when volatilities are high and safety is near zero, the estimation error is large and we should use equal weights. It is remarkable that the Maimonides algorithm accomplishes a form of such a regularisation in a robust and simple way. It is also possible that the Maimonides algorithm could be applied more generally to other ill-posed problems. This is a direction of future research.

4. Data We use a combination of equity and bond indices for an illustration of Maimonides risk parity. We use the Center for Research in Security Prices (CRSP) Value-Weighted Index for the S&P 500 Universe and the CRSP 30-Year US Treasury Index. Monthly data for both are available from November 1941 through December 2012. Figure 1 shows the five-year rolling volatility of the equally weighted portfolio as well as the overall quartiles. The median volatility was 9.3% and the lower and upper quartiles were 6.8% and 10.6%, respectively. The Maimonides risk parity allocation algorithm takes as input the individual asset volatilities, to be estimated as

Maimonides risk parity the rolling 12-month historical volatilities, and the portfolio risk parameter σ . If this portfolio risk is always chosen to be very low, then Maimonides risk parity is identical with standard risk parity. If the portfolio risk is always chosen to be very high, then Maimonides risk parity is identical with the equal-weighted portfolio. One natural choice is to set the portfolio risk equal to the historical volatility of the equal-weighted portfolio divided by the number of assets. We divide the risk by the number of assets to effectively multiply the targeted safety level by the number of assets. Thus, in the case above, this would mean we set the portfolio risk parameter σ = 0.093/2 = 0.047, half the overall historical median. For robustness, we would also check between the two quartiles so the portfolio risk parameter would range between σ = 0.068/2 = 0.034 and σ = 0.106/2 = 0.053. 5. Results Figure 2 shows the information ratio of the Maimonides risk parity portfolio over the entire sample period for a variety of portfolio risk parameter values. The darker shaded portion corresponds to the information ratios for values of the portfolio risk parameter values in the scaled interquartile historical range discussed above. (Information ratios were used instead of Sharpe ratios due to the lack of Federal Funds Effective Rate prior to 1954.) The Maimonides risk parity portfolio performed at least as well as the standard risk parity portfolio for all values of the portfolio risk parameter and performed at least as well as the equal weight portfolio for nearly all values of the portfolio risk parameter. As expected, when the portfolio risk is low, Maimonides risk parity behaves like standard risk parity, and when the portfolio risk is high, Maimonides risk parity behaves like the equal weight portfolio. In between, however, Maimonides risk parity tends to outperform, and always outperforms within the range of portfolio risk given by the historical quartiles. 6. Conclusions We introduced a novel method of portfolio allocation based on an ancient system of Maimonides for allocating an estate or a bankruptcy in an unusual but fair way, and showed that this Maimonides risk parity approach can

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unify standard risk parity, equal weight portfolios, and market capitalization-weighted portfolios in a single framework. Furthermore, we have reported empirical results that suggest that Maimonides risk parity can beat the other portfolios. The Maimonides strategy in effect takes into account the desired risk of a portfolio and attempts to allocat it among the available assets in the same unusual but fair way. The essential feature of the Maimonides algorithm is its disregard for large values. A creditor of a billion dollars on an estate valued at only 100 dollars is essentially reduced to having a claim on only a maximum of 100 dollars, on the premise that there is not any more, therefore any claim beyond that amount will never be paid, and can be ignored. This concept of a potential discontinuity at large values, or equivalently at reciprocals of small values, drives the results of Maimonides risk parity relative to standard risk parity and can potentially be applied to other aspects of finance as well, such as value portfolios or penny stock momentum, where the price-to-book ratio blows up for firms with low book values or the recent return blows up for stocks with low nominal prices. References Alankar, A., DePalma, M. and Scholes, M., An introduction to tail risk parity balancing risk to achieve downside protection, 2012. Available online at: https://alliancebernstein.com/ abcom/Segment_Homepages/Defined_Benefit/3_EMEA/ Content/pdf/Introduction-to-Tail-Risk-Parity.pdf (accessed 14 August 2013). Aumann, R.J. and Maschler, M., Game theoretic analysis of a bankruptcy problem from the Talmud. J. Econ. Theory, 1985, 36, 195–213. Chaves, D.B., Hsu, J., Li, F. and Shakernia, O., Risk parity portfolio vs. other asset allocation heuristic portfolios. J. Invest., 2011, 20, 108–118. Fisher, G.S., Maymin, P.Z. and Maymin, Z.G., Risk parity optimality. J. Portfolio Manage., forthcoming. Maillard, S., Roncalli, T. and Teiletche, J., The properties of equally weighted risk contribution portfolios. J. Portfolio Manage., 2010, 36, 60–70. Maimon, M.b. (c. 1180). Mishneh Torah. The laws of lending and borrowing. Chapter 20, Section 4. English translation by Eliyahu Touger. Available online at: http://www.chabad.org/ library/article_cdo/aid/1161176/jewish/Chapter-20.htm (accessed 26 July 2013). Tikhonov, A.N. and Arsenin, V.Y., Solutions of Ill-Posed Problems, 1977 (Winston: New York).

North American Actuarial Journal, 17(1), 13–28, 2013 C Society of Actuaries Copyright  ISSN: 1092-0277 print / 2325-0453 online DOI: 10.1080/10920277.2013.775011

Managing the Invisible: Identifying Value-Maximizing Combinations of Risk and Capital William H. Panning ERMetrics, LLC, Prospect, Kentucky

This article demonstrates the linkage—often asserted but seldom described—between Enterprise Risk Management (ERM) and maximizing a firm’s value. I show that knowing a firm’s aggregate risk exposure (via ERM), when combined with a valuation model like the one presented here, can enable the firm’s managers to identify and choose value-maximizing combinations of risk and capital. Using value maximization as the criterion for choosing a firm’s capital structure is quite distinct from rules of thumb that CFOs often use for such decisions. The valuation model shows that increasing an insurer’s surplus from an initially low level typically increases the present value of future cash flows that take into account the probability of impairment from extreme losses. In contrast to traditional literature on the risk of ruin, impairment here is taken to mean a loss of creditworthiness such that the firm’s business model is no longer sustainable, whether or not the firm is solvent. However, beyond a certain optimal level relative to a firm’s risk, further increases in surplus actually reduce a firm’s value added measured in this fashion. Sensitivity analyses presented here show how these conclusions are affected by changes in the values of crucial variables. In particular, the article shows how managers can use this model to identify specific actions that their firm can take to increase its value added, and it emphasizes the practical importance of making a firm’s value both visible and manageable.

1. INTRODUCTION The purpose of computing is insight, not numbers.—Richard Hamming, computer scientist

Enterprise Risk Management (ERM) is a body of knowledge—concepts, methods, and techniques—that enables a firm to understand, measure, and manage its overall risk so as to maximize the firm’s value to shareholders and policyholders. The purpose of this article is to demonstrate this asserted linkage between ERM on the one hand and maximizing a firm’s value on the other. Most existing literature on ERM focuses on specific concepts, methods, and techniques for measuring particular risks and constructing an aggregate risk distribution for the firm as a whole. Important issues include the selection of appropriate risk measures, techniques for measuring the distribution of particular risks, alternative ways of representing dependencies among different risks, methods for producing an aggregate measure of firm-wide risk, and whether and how to allocate capital among alternative sources of risk within a firm. Implicit in this rapidly growing body of work is the assumption that measuring its overall risk exposures will enable a firm to “better” manage its risk and that this capability will “add value.” Typically missing is any concrete explanation or demonstration of what this specifically means and how it will come about. Notably scarce, for example, are papers that describe what is meant by “adding value” and that propose specific ways that this could be implemented and measured in practice or even in principle.

1 This definition is quite similar to that of the CAS Advisory Committee on ERM: “ERM is the process by which organizations in all industries assess, control, exploit, finance, and monitor risks from all sources for the purpose of increasing the organization’s short and long term value to its stakeholders.”

The original version of this article was presented at the CAS/SOA/PRMIA Enterprise Risk Management Symposium in April 2006, where it received the first ERM Research Excellence Award, an annual prize established by The Actuarial Foundation. I am indebted to Richard Goldfarb and Richard Derrig for valuable discussions and encouragement in preparing the original version. Since then, stimulated by discussions with David Ingram, Daniel Bar Yaacov, and Chuck Thayer, and especially by an anonymous referee’s superb comments and suggestions, I have substantially extended it to encompass new results. As with all my work, SDG. Address correspondence to William H. Panning, Founder and Principal of ERMetrics, LLC, 7305 Blakemore Court, Prospect, KY 40059. E-mail: [email protected]

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What I demonstrate in this article is that there are optimal combinations of risk and capital—optimal in that they maximize the value of the firm. Specifically, for a given level of risk there is a value-maximizing amount of capital. Alternatively, if capital is the constraint, then for a given level of capital there is an optimal (value-maximizing) level of risk exposure for the firm. ERM therefore can add to a firm’s value because, by measuring the firm’s aggregate risk exposure, it enables the firm’s managers to identify and choose value-maximizing combinations of risk and capital. This demonstration takes the form of an explanatory model that presumes that the ERM group at the modeled firm has successfully measured its risk exposures and has correctly constructed from them an aggregate distribution of potential firm-wide losses. The model shows how to use this aggregate distribution, together with other financial information about the firm, to identify value-maximizing combinations of risk and capital. It is an explanatory model because it is deliberately simplified so that the virtues and defects of its fundamental logic will be readily apparent. If I have constructed this model correctly it should lead to approximately the same conclusions as an elaborated model, which includes numerous complexities found in most actual firms, or a calibrated model, which is an elaborated model with details and parameters that match those of a particular firm. In contrast to an elaborated or calibrated model, the principal purpose of an explanatory model is insight, not numbers. Not all firms are alike, and not all insights can be readily transferred from one industry to another. Here my focus will be on ERM as applied to a property-casualty insurer and on surplus as the critical component of its capital structure. Although I recognize that reinsurance, debt, and various hybrid securities can be important components of an insurer’s capital structure, I will treat them in a subsequent article rather than add to the length and complexity of this one. Despite these limitations, I hope that the model and conclusions presented here add substantially to our understanding of the “M” in ERM. 2. CAPITAL STRUCTURE IN THEORY AND PRACTICE I have become a bit disenchanted with the indiscriminate use of superrationality as the foundation for models of financial behavior.—Franco Modigliani (1988), Nobel Prize–winning economist

It may seem foolhardy to speak of “optimal capital structure” when the phrase itself is considered an oxymoron by many financial economists and their present and former students. Indeed, Modigliani and Miller were honored with Nobel prizes in part for their assertion and proof of what is sometimes called the “capital structure irrelevance theorem,” which is taught to virtually every MBA student. The theorem is indeed valid, but—and this crucial qualification is typically forgotten or ignored—only under circumstances that are rarely if ever encountered.2 When bankruptcy and its associated costs are possible, the irrelevance theorem is itself irrelevant. But the influence of M&M’s original work was so powerful and pervasive that, even today, bankruptcy is virtually ignored in many of the leading textbooks on corporate finance! Even in the recent professional literature, bankruptcy is treated as one among a number of supposed “frictional costs” that cause the real world to deviate from the conditions originally assumed by Modigliani and Miller.3 But, given the rarity of convincing attempts to clarify or quantify frictional costs, most CFOs would be hard-pressed to draw any practical conclusions from the corporate finance literature generated since the early 1980s, if not before. As a consequence, the term “oxymoron” is perhaps better suited to the phrase “applied corporate finance.”4 In the absence of useful academic guidance, CFOs necessarily adopt pragmatic principles as guides to decision making concerning capital structure. Here are a few and the difficulties inherent to each: a. Maintain roughly the same financial ratios as peer companies. This criterion shifts the task from determining the right financial ratios to determining the right peer companies, which may be different from the companies that currently are the most similar to one’s own. In practice this criterion works only because firms assume a set of peer companies to which they compare themselves. b. Maintain the financial ratios consistent with our corporate risk tolerance. This is often mentioned but virtually never specified. What is a corporate risk tolerance? How is it measured? How is such a measure related to possible financial ratios? Who is consulted in these decisions? Note that executives, policyholders, bond holders, and stockholders may disagree considerably on specifications of this criterion.5 2The real significance of these authors’ path-breaking work was its use of arguments based on arbitrage—arguments that have fundamentally affected the evolution of finance as well as of financial securities and markets. For succinct reviews of portions of the voluminous relevant literature see Rubinstein (2003), Modigliani (1988), and Giesecke and Goldberg (2004). 3Examples of this literature include Hancock, Huber, and Koch (2001), Chandra and Sherris (2007), Yow and Sherris (2007, 2008), Smith, Moran, and Walczak (2003), Exley and Smith (2006), and Major (2009a, 2009b), who further extend this emerging frictional cost framework to considerations of pricing, optimal capitalization, and firm value. 4A refreshingly frank admission is found in Copeland, Weston, and Shastri (2005), p. 611: “How Does a Practitioner Use the Theory to Determine Optimal Capital Structure? The answer to this question is the Holy Grail of corporate finance. There is no completely satisfactory answer.” 5I am indebted to Richard Goldfarb for his numerous insights on this topic.

MANAGING THE INVISIBLE

15

c. Maintain the financial ratios required to achieve or maintain a target financial rating. Using target ratings ignores the problem of choosing which rating to have, which may, in turn, depend on the clients that the firm serves or wishes to attract, the nature of the products offered by the firm (short-term versus long-term and any guarantees associated with each), and the type of distribution system that it has established (agents may be sensitive to credit quality). d. Maintain a target beta and associated target cost of capital, as defined by the capital asset pricing model (CAPM). This seems plausible except for the fact that, by assuming the validity of CAPM, it confronts and is refuted by the original Modigliani and Miller arguments. In my view, the story we tell ourselves to make “measuring relative stock market performance” equivalent to “measuring our cost of capital” requires, like many stories, a suspension of disbelief that is difficult to sustain. But rather than defending such heretical doubts, I here attempt the more constructive task of outlining part of an alternative story that may prove more useful. I propose the following decision criterion: For a given set of risk exposures, select the capital structure that maximizes shareholder or policyholder value. This restatement of the problem assumes that risk exposures are fixed and that capital structure is flexible. An alternative but equivalent criterion is as follows: For a given capital structure, select the aggregate risk exposure that maximizes the value of the firm to shareholders and policyholders. The two problems are mathematically equivalent—a fact that is analogous to the duality theorem in linear programming—but not necessary identical in practice. For this alternative criterion to be practical (i.e., one that can be implemented), we need a measure of value that can be derived from obtainable data (unlike risk tolerance), that is understandable by senior executives (unlike beta), and that can be used to explore and evaluate alternative capital structures and corporate strategies. 3. VALUING A PROPERTY-CASUALTY INSURANCE FIRM Point of view is worth 80 IQ points.—Alan Kay, a computer science pioneer

Like an unwelcome guest at an elegant dinner party, a central question in corporate finance is almost universally ignored: how to measure the value of a firm.6 In practice there are two ways of answering this question. One might be called the cross-sectional approach, since it deals with the firm at a given point in time. This approach starts with the accounting balance sheet of a firm, adjusts it for differences between book value and economic value, and uses the result as an estimate of the firm’s value. For a property-casualty insurer, this approach is implemented by estimating the market value of the firm’s assets and subtracting from that the estimated present value of the firm’s liabilities (obtained by discounting the forecast liability cash flows). The result is considered the current economic value of the firm. This approach was developed and widely adopted in the domain of asset-liability management, where the objective was to protect this economic value of the firm from potential loss due to changes in interest rates.7 An alternative, described in greater detail below, is the longitudinal or going-concern approach, in which the value of the firm is the survival-adjusted present value of its future earnings or cash flows. The cross-sectional approach ignores two important but inconvenient facts that it cannot explain. First, the market value of a firm’s equity may exceed the value of its assets. From a cross-sectional point of view such a situation would be absurd, since the value of equity is equal to the market value of the firm’s assets less the market value of its liabilities. Even if the firm had no liabilities at all, its equity value could not exceed the value of its assets. The inconvenient fact, however, is that one firm in our industry recently had $12 billion in assets, $9 billion in liabilities, and therefore $3 billion in net worth. Since the firm’s assets and liabilities were both relatively short term, their book and market values were virtually identical, so that the firm’s economic value was roughly $3 billion, according to the cross-sectional point of view. And yet this firm’s market capitalization was approximately $14 billion, some $2 billion more than its total assets! A second inconvenient fact is that, from a cross-sectional point of view, firms that directly market insurance to their clients behave irrationally; that is, in ways that appear to reduce, rather than increase, the economic value of the firm. I became acutely aware of this puzzle when, as a quantitative financial analyst for a large insurer, I was asked to help build a financial planning model for a new division that would directly market auto insurance.8 The firm’s existing model, based on GAAP (Generally Accepted Accounting Principles), showed that the division’s prospective earnings would be unacceptably low, since every new policy written would lose money and add to the deficit already created by the considerable startup expenditures. But slowing the rate of growth only postponed the day when the operation would become profitable. At first, glance, there appeared to be no way to salvage an initiative on which the CEO had staked his reputation with the board. 6In finance, the phrase “value of the firm” refers to the value of its assets, not the value of its equity. Here, by contrast, I will use the phrase to mean the value of a firm to its shareholders, as imperfectly reflected in its market capitalization—the aggregate value of its stock—either as observed or as estimated by the model presented later. 7For a critique of and correction to that approach, see Panning (1994, 2006). 8The experience described here has also been discussed in Panning (2003a).

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W. H. PANNING

The solution, it turned out, lay in thinking about the problem in an entirely different way. The key was to think about the business longitudinally—as a going concern—rather than cross-sectionally, at a given point in time. From a cross-sectional point of view it made no sense at all to spend $100 in marketing costs to sell a policy that would, apart from those costs, make, say, $50 in profit. But for a going concern considered longitudinally, such a strategy made enormous economic sense, for the original $100 in marketing costs was a one-time expenditure. Those who purchased policies had a very high probability of renewing them even in the absence of subsequent additional marketing costs and despite the lack of any contractual obligation to renew. So in the second and subsequent years, each new policy generated $50 in profits, with a renewal probability of some 90%. At the time of the original sale, the present value of these future profits far exceeded the initial $100 marketing cost. Since neither GAAP nor statutory accounting recognizes the value of future renewals (because policyholders are not legally obligated to renew), from a cross-sectional accounting perspective there was no convincing reason to sell policies at all under the circumstances just described. But from a longitudinal going-concern perspective, it made sense to sell as many policies as possible, since doing so would maximize the present value of future earnings. In the longitudinal or going-concern approach, the value of a firm is the present value of its expected future earnings or cash flows, adjusted for the probability of survival. Here we will focus on earnings, since they are easier to assess than cash flows, and particularly since insurance regulators typically permit dividends that are proportional to earnings rather than cash flows. For convenience we will focus on a firm with expected earnings E[X] at the end of the current year and every future year, absent default or crippling impairment, with constant annual survival probability p and a risk-free interest rate i. The survival-adjusted present value of future earnings is equal to E[X]p/(1 + i − p). Incorporating a modest rate of growth further explains how a firm’s market capitalization can exceed the value of its assets.9 If a firm’s annual expected earnings grow by a factor (1 + g), where g is the annual growth rate, the survival-adjusted present value of future earnings is E[X]p/(1 + i − p(1 + g)). From a longitudinal or going-concern point of view, then, the market value of a firm can considerably exceed the economically adjusted current accounting value of its net worth. This going-concern value can be called the firm’s franchise value, since it includes the present value of profits from business that the firm has not yet written but can reasonably be expected to write.10 A rough but reasonable measure of a firm’s franchise value is its market value, actual or estimated, although this measure may incorporate additional variables that may be empirically important to market participants but, from a strictly theoretical point of view, are irrelevant or tangential. Alan Kay was entirely correct: Adopting the right point of view matters enormously. The cross-sectional accounting point of view fails to explain how a firm’s equity value can exceed the value of its assets or how a firm that directly sells insurance to consumers can survive and prosper. By contrast, viewing an insurer as a going concern immediately solves both puzzles and, as we will show, has important implications for strategies to maximize the value of an insurance operation. 4. A GOING-CONCERN VALUATION MODEL Here I present a detailed valuation model that incorporates the longitudinal point of view. In this model, the value of an insurance firm is the survival-adjusted present value of its expected future earnings plus the present value of any residual assets that may remain should the firm be reorganized. The foregoing statement contains within it several important features of the model that need to be stated explicitly: 1. The principal focus of this valuation model is earnings rather than cash flow because the firm is assumed to pay dividends to its shareholders equal to its annual after-tax net income, if positive. 2. These anticipated future dividend payments are discounted, at the risk-free rate, to obtain their present value. 3. These anticipated future dividend payments are further adjusted to reflect the fact that the stream of expected dividend payments will end if the firm is reorganized or dissolved due to an extraordinary loss. Given its surplus, pricing, expenses, and other 9Growth is an issue with numerous facets that cannot be adequately treated in the space available and so will be treated in a subsequent article. One well-known issue is that high growth rates cannot be sustained indefinitely. Dealing with this issue requires a more complex model than the one presented here. 10For an earlier model of franchise value see Panning (1994), which focused on the risk to franchise value of changes in interest rates. Unfortunately, the concluding equation of that paper is marred by an egregious printing error. The correct equation is readily derived from the two that precede the final one. Panning (2006) is a briefer and more sophisticated treatment of that topic, which corrects that error but likewise excludes consideration of default risk. Hancock, Huber, and Koch (2001), Smith, Moran, and Wolczak (2003), and Exley and Smith (2006) present results very similar to some of those presented here and that are based on current financial theory and its rather strong assumptions rather than on the simpler approach adopted here. Fernandez (2002) provides a thorough survey of the huge variety of valuation approaches that have been proposed or are in use. Damodaran (2005) provides a survey of rival approaches and associated evidence. Leibowitz (2004) presents a synthesis of his earlier work on estimating franchise value, and Koller, Goedhart, and Wessels (2010) provide textbook models for valuing corporations. Avanzi (2009) provides a valuable review of the extensive literature on dividend discount models and the strategies they imply, and Dickson (2005) surveys the huge relevant actuarial literature on the risk of ruin and its implications for firm strategy and valuation. The links between corporate strategy, risk, and valuation are explored in books by Coleman (2009), Pettit (2007), Schroeck (2002), Segal (2011), and Woolley (2009). Major (2011) provides an excellent overview of the principal approaches to modeling the effect of risk on valuation. Venter and Underwood (2010), Bodoff (2011), and Ingram and Bar Yaacov (2012) extend the model presented here to strategy choice and risk hedging. Finally, articles such as Brockman and Turtle (2003) and Episcopos (2008) present models for pricing corporate securities based on a barrier option pricing framework, which has important similarities to the model presented here.

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17

parameters, in every year there is some likelihood—typically quite small—that the firm’s losses will exceed some critical amount that triggers reorganization or bankruptcy. If this occurs, the firm is essentially dissolved, and the stream of dividend income will cease permanently. Each future dividend payment is therefore multiplied by the probability that it will occur or, equivalently, the probability that reorganization has been avoided in the current and prior years. 4. By reorganization I simply mean that the firm ceases to exist as a going concern. Its assets are liquidated to pay policyholders. Any cash that remains after these are paid (i.e., any residual surplus) is distributed to shareholders. The model presented here incorporates an important simplifying assumption. I assume that every year the firm begins with a given amount of surplus and writes the same volume of business with the same expense ratio, the same expected losses, and therefore the same expected net income after tax as in prior years. Its actual losses, and therefore its actual net income, are stochastic. If the firm’s net income is positive, it dividends that amount to shareholders; if negative, it raises sufficient capital (from existing shareholders) to restore its surplus to the amount held at the beginning of the year. Consequently, it begins every year with the same surplus as it held a year earlier. Further details of this valuation model are as follows. 4.1. Underwriting 1. The firm writes 100 units of premiums every year; a unit is some fixed amount in dollars: $1 million, for example. 2. All policies are written on January 1 and take effect at 12:01 AM that day. 3. All policies have a term of one year and expire at midnight on December 31. 4. At midnight on December 31, the losses associated with these policies become known precisely and are paid immediately. As a consequence of these assumptions it follows that accident-year, policy-year, and calendar-year financials are identical for this hypothetical firm. 4.2. Cash Flow and Earnings 5. All premiums are paid when the policies are written; for example, at 12:01 on January 1. 6. All expenses are known precisely and paid immediately, at the inception of the policy. 7. The firm earns investment income, at the risk-free rate, on its cash balance during the year. This cash balance consists of written premiums, less expenses, plus surplus. 8. The firm pays taxes on positive net income and receives tax rebates on net losses. Here I assume that the firm can utilize net operating loss carryovers to obtain tax recoveries on negative pre-tax income. In reality this is not always the case. 9. The firm’s after-tax net income is known at midnight on December 31 (recall that losses are known and paid at that time, as stated in assumption 4). a. When after-tax net income is positive, the firm immediately pays a dividend in that amount to its shareholders. b. When after-tax net income is negative but the loss is less than some specified critical percentage of surplus, the firm immediately sells additional shares (to existing shareholders) to bring its surplus to the level that existed at the beginning of the year. The firm therefore has a constant surplus from one year to the next. c. If the firm’s operating loss exceeds some specified critical percentage of its beginning surplus, then a reorganization occurs that results in liquidation of the firm through bankruptcy or purchase by some third party. d. The costs of reorganization are here assumed to be zero, although this is seldom true in reality. e. If reorganization occurs, any assets remaining after losses are paid are sold at fair value and the proceeds returned to shareholders. 4.3. Losses and Enterprise Risk 10. The only stochastic feature of this firm’s operation is the value of its claims or losses (L), which are lognormally distributed with a known mean and standard deviation. (In reality, these two parameters are estimates that can be wrong.) These lognormally distributed losses are a net result of all of the risks—not just claims—that affect the earnings and cash flow of the firm. They encompass multiple lines of business as they are affected by pricing risk, credit risk from policyholders and suppliers, operational risk, catastrophe risk, and the like. Losses, then, are a random draw from an enterprise-wide distribution of potential losses, estimated by ERM staff. 11. Assuming that these aggregate losses are lognormally distributed is a convenience, not a necessity. It enables the results presented here to be calculated directly rather than through the use of simulation. Many aggregate distributions can also be closely approximated by a mixture of parametric distributions. Doing so here, where the objective is to produce a convincing explanatory model, would have made the results more complex and less transparent, with no offsetting benefit. But doing so could certainly make sense when the objective is to produce an elaborated or calibrated model.

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TABLE 1 Input Parameters and Their Initial Values A 100 50 25 6% 35% 35% 70 0.25 306.25 4.22 0.246 17.5 32.5

B

C

Symbol

Definition

100 80 25 6% 35% 35%

100 110 25 6% 35% 35%

70 0.25 306.25 4.22 0.246 28 52

70 0.25 306.25 4.22 0.246 38.5 71.5

P S0 E i t c L E[L] v V[L] μ σ

Premiums (written and earned) Surplus (initial) Expense Interest rate (risk free) Tax rate Critical percentage loss of surplus Losses: stochastic, lognormally distributed Expected loss = exp(μ + σ 2/2) Loss volatility: (Standard deviation of loss)/(Expected loss) Variance of loss = exp(2μ + σ 2)[exp(σ 2) − 1] = v2E[L]2 Mean of ln(L) Standard deviation of ln(L) Critical dollar loss of surplus = cS Critical surplus: minimum surplus required to survive

Sc

4.4. The Value of the Firm The value of the firm has two components. One is franchise value, the survival-adjusted present value of the firm’s after-tax net income, which is paid as dividends to shareholders so long as the firm survives. The other is residual value, the present value of whatever residual payments shareholders receive when the firm is terminally reorganized. In the analysis that follows I will conveniently assume that the costs of reorganization are zero and allow the reader to assume whatever alternative estimate they prefer. 4.5. Essential Input Parameters The input parameters for the model are shown in column A of Table 1. The assumed values for premiums (P), initial surplus (S0 ), expense (E), the interest rate (i), and the tax rate (t) are all raw inputs (i.e., not derived from any other values within the model), as are expected loss and loss volatility, the latter defined as the standard deviation of losses as a percentage of expected loss. The mean (μ) and standard deviation (σ ) of the logarithms of losses are calculated using the well-known standard formulas in Table 1. The tax code is complex and here represented by a simple tax rate that is applied both to net profits, which are taxed, and net losses, which result in a tax recovery. A remaining raw input is the critical percentage loss of surplus (c). The rationale for this parameter is that a firm can remain solvent but experience financial weakness that results in a ratings downgrade or abandonment by credit-sensitive clients, so that its business model is no longer viable. Under these circumstances the firm must be reorganized or sold and so will no longer exist in its earlier form. 4.6. Implications: Conditional Expected Losses and Net Income Given the input parameters just described, and given the losses it actually experiences during the year, a firm will end the year in one of three states. In state 1 its net income is such that its surplus remains above the critical level that triggers reorganization. Consequently, the firm survives to write business the following year. But before doing so it distributes a dividend or raises capital sufficient to restore its surplus to its initial value, S0 . In state 2 it experiences losses sufficiently severe to trigger reorganization but not so severe as to render it insolvent. In this case it ceases to survive, and any remaining surplus is paid as a dividend to its existing shareholders or policyholders. Finally, in state 3 its losses have rendered it insolvent, so that remaining surplus is zero but, thanks to limited liability, not negative. The firm is reorganized, but there is no residual surplus to distribute. To calculate a firm’s value we need to obtain the annual probability of each of these three states, the present value of the firm’s expected earnings during each year that it continues to survive, and the present value of its expected residual surplus after it has experienced a critical loss. A critical loss occurs when the firm’s net income N ≤ −cS. From the parameters in Table 1 we can determine lc , the smallest loss that triggers reorganization, as follows: lc = (P − E)(1 + i) + Si + cS/(1 − t) = (75)(1.06) + (50)(0.06) + (0.35)(50)/(1 − 0.35) = 109.42.

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We can likewise determine ls , the smallest loss that would reduce the firm’s surplus to zero: Ls = (P − E)(1 + i) + Si + S/(1 − t) = (75)(1.06) + (50)(0.06) + (50)/(1 − 0.35) = 159.42. For the parameters in Table 1, the firm’s annual probability of survival (state 1) is Pr[L ≤ lc ] = ((ln(lc ) − u)/σ ) = 0.9737, where  is the cumulative normal probability distribution. The firm’s probability of experiencing a loss that triggers insolvency (state 3) is Pr[ls < L] = 1 − ((ln(ls ) − u)/σ ) = 0.0003. It follows that the probability of state 2, in which the firm is reorganized but not ruined, so that there is a residual distribution to shareholders, is Pr[lc < L ≤ ls ], or 0.0261. For all firms, net income (N) after tax at the end of any single time period is N = ((P − E)(1 + i) + Si − L)(1 − t), from which it follows that L = ((P − E)(1 + i) + Si − L)(1 − t). The next several results are based on the fact11 that, for any lognormal random variable X, with parameters μ and σ 2, where E[X] = exp(μ + σ 2/2), the expected loss for losses less than M = lc are 

M 0

xfx (x)dx = e

μ+σ 2 /2



log M − μ − σ 2  σ



,

so that E[L|L ≤ lc ] = (1/Pr[L ≤ lc ]) exp(μ + σ 2 /2)((ln(lc ) − μ − σ 2 )/σ ). The firm’s expected annual loss in state 1, in which it survives, is therefore (1/0.9737) (70)((ln(109.42) − 4.21 − 0.2462 )/0.246) = 68.63, so that its expected annual net income while surviving is 9.02. The firm’s expected annual loss if it remains solvent (states 1 or 2) is 69.97, which corresponds to an expected loss in state 2 of 120.12 and an expected net income in state 2 of −24.45. It follows that in state 3 its expected loss is 169.99 but, due to limited liability, its expected net income in state 3 is limited to −50, the negative of the firm’s initial surplus. So, given the assumed parameters, there is a probability of 1− Pr [L ≤ lc ] = 0.0263 that a critical loss will occur, comprising a probability of Pr[lc < L ≤ ls ] = 0.0261 that the residual surplus dividend to shareholders will be (50 − 24.45) = 25.55 and a probability of Pr[ls < L] = 0.0003 that this dividend will be zero, since no surplus will remain. This implies that the expected dividend to shareholders sr when a critical loss occurs is ((0.0261)(25.55) + (0.0003)(0))/(0.026 + 0.0003) = 25.29. 4.7. Implications: Net Income and Valuation The value of the firm is the sum of its franchise value and its residual value. The franchise value of the firm consists of the present value of future dividends, paid annually, and discounted for time value and survival probability. The residual value of the firm is paid once, when a critical loss occurs, and consists of the probability weighted present value of the residual surplus that is paid as a final dividend to owners. To simplify notation, I will represent the probability of survival as p and the probability of reorganization as q. Dividends paid annually while the firm survives will be Ns , the firm’s expected annual income while it survives. Further, I will let d = 1/(1 + i), the time discount factor. The firm’s franchise value, as an annuity, is therefore V franchise = Ns dp/(1 − dp) = 101.66. By contrast, 11See

Boland (2007), p. 48, for details.

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TABLE 2 Key Results and Their Values A 109.42 159.42 0.9737 0.9997 0.0261 0.0003 68.63 69.97 120.12 169.99 9.02 −24.45 −50.00 −24.71 25.29 101.66 7.72 109.38 59.38

B

C

Symbol

Definition

127.38 207.38 0.9947 1.0000 0.0053 0.0000 69.64 70.00 138.08 218.27 9.53 −34.96 −80.00 −34.98 45.02 145.15 3.66 148.82 68.82

145.33 255.33 0.9990 1.0000 0.0010 0.0000 69.91 70.00 155.91 266.80 10.52 −45.38 −110.00 −45.38 64.62 172.30 1.06 173.36 63.36

lc ls Pr[S1 ≥ Sc ] Pr[S1 > 0] Pr[Sc > S1 > 0] Pr[S1 = 0] E[L|L ≤ lc ] E[L|L < ls ] E[L|lc < L < ls ] E[L|ls ≤ L] E[N|L ≤ lc ] E[N|lc < L ≤ ls ] E[N|ls < L] E[N|lc < L] sr V franchise V residual V total V added

Loss lc that produces a critical dollar loss of surplus Loss ls that triggers insolvency Probability of state 1: survival (year-end surplus S1 ≥ Sc ) Probability of state 1 or state 2 Probability of state 2 (reorganizing but still solvent) Probability of state 3 (insolvency) Expected losses in state 1 Expected losses in state 1 or state 2 Expected losses in state 2 Expected losses in state 3 Expected net income after tax in state 1 Expected net income after tax in state 2 Expected net income after tax in state 3 Expected net income after tax in state 2 or state 3 Expected remaining surplus in state 2 or state 3 Present value of future dividends while surviving Present value of expected surplus remaining after critical loss Total present value of dividends and residual surplus Value added = V total − S

the present value of the one-time payment of the firm’s residual surplus is V residual = sr dq/ /(1 − dp) = 7.72. The firm’s total value is the sum of these two quantities: V total = V franchise + V residual = 109.38. Column A in Table 2 provides a summary of the principal results of the model just described.12 4.8. Implications: Market Valuation Now that we have calculated a value for the firm by estimating the survival-adjusted present value of its future earnings and dividends, what do we do with it? This question really has two parts. First, what is the relationship between the value we have calculated and the actual market value of the firm, if it is publicly traded? Second, if there are important factors omitted from the model presented here, can a firm that improves its modeled value be nonetheless reasonably confident that doing so will improve its real-world market valuation as well? I suspect that the value of a firm, as calculated using this model, may be somewhat overestimated relative to the value of that firm’s shares in the market. This suspicion is based on the fact that yield spreads on corporate bonds typically exceed the break-even spreads necessary to compensate for the historically experienced probabilities of default for bonds of different credit qualities. This difference is usually described as a risk premium. The model presented here presumes the absence of any such risk premium, since it incorporates only a time-value discount and a discount for the probability of survival. If investors require some risk premium as well, then the resulting value of the firm will be lower than estimated by this model. Of course, this is a matter for empirical investigation rather than armchair speculation.13 Whether or not a risk premium exists and, if it does, whether it is large or small, it is nonetheless plausible to assume (pending empirical studies) a reasonably high correlation between the value of the firm as derived from this model and its value in the marketplace. If so, then actions taken to improve the modeled value of the firm should also improve the actual market value of 12One additional implication is worth pointing out. Expected losses conditional on the firm’s survival are necessarily lower than unconditional expected losses. Consequently, observed rates of return on premiums or on surplus are likely to be biased upwards. Firms that experience extremely high losses will be reorganized and disappear from view. So unless the underlying data-gathering process is extremely thorough, both the industry and the firms within it will appear to be more profitable than underlying risk exposures would warrant. A similar phenomenon occurs in the investment world, where funds that significantly underperform market averages are liquidated or merged, and statistics concerning their poor results disappear with them. Given the cyclical nature of property-casualty insurance, such upward bias could significantly distort the view and actions of both regulators and investors. Seminal papers on survival bias include Brown et al. (1992) and Brown, Goetzmann, and Ross (1995). 13See Elton et al. (2001) for an analysis of the components of a risk premia on bonds, Derrig and Orr (2004) for a comprehensive review of the empirical literature on the equity risk premium, and Eling (2012) for a compendium of post-2004 research on risk premia.

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MANAGING THE INVISIBLE

the firm. In this respect the model can be a useful guide to market-value-improving strategies without necessarily being a perfect predictor of market value.

5. MANAGING CAPITAL TO MAXIMIZE VALUE Having a valuation model enables management to ascertain the likely consequences of alternative actions or strategies. In this section I will demonstrate how the model can be used in three ways: (1) to estimate how much surplus the firm should have, (2) to estimate the consequences for the firm’s value and optimal surplus of changing various input variables, taken separately, and (3) to estimate the consequences of simultaneously changing multiple variables, as occurs in many strategic decisions. 5.1. Maximizing Value Added In Section 2 I described and critiqued four practical principles that CFOs use to determine the amount of surplus their firm should have. I also proposed an alternative principle: Choose the level of surplus that maximizes value for shareholders or policyholders. The model presented in Section 4 makes this alternative principle feasible. We must be careful how we implement that principle, however, for a reason that is subtle but important. If we use V total as the variable we wish to maximize, then adding surplus is always beneficial, for it increases the value of the firm’s assets as well as its investment income and makes the firm less likely to default or reorganize. The important question is whether the value of these benefits exceeds the dollar cost of the added surplus. Beyond some point, adding an additional dollar to surplus creates additional value of less than a dollar. In this case, shareholder value is better served by distributing the additional dollars as a dividend rather than retaining them to increase surplus. To address this important question of marginal costs and benefits we need to focus on maximizing V added , which equals V total minus surplus (S). To maximize shareholder value, a firm should add (or reduce) surplus so long as doing so increases V added . The relationship between surplus and V added is shown graphically in Figure 1 for the firm in our continuing example, still based on column A in Tables 1 and 2. This firm has an initial surplus of 50 units and an initial V added of 59. This graph shows that the firm can maximize its V added at 69 by increasing its surplus to 80. Beyond that point, however, each additional dollar of increased surplus adds less than a dollar of additional value: Adding more surplus reduces V added . Although an increase in surplus affects value added in numerous direct and indirect ways, two are especially important. One, the income effect, reduces value added. Each additional dollar of surplus increases after-tax investment income by (i)(1 − t) dollars, an amount that is adjusted downward to reflect the probability of survival and then discounted at the pre-tax risk-free rate. The present value of this additional income is necessarily less than a dollar, so that the resulting value added is lower than before. Offsetting this, however, is the survival effect: increasing surplus increases the firm’s probability of survival, which in turn increases its franchise value. When the firm’s initial survival probability is sufficiently less than one, this increase in franchise value exceeds the income effect and so produces an increase in value added. But as the firm’s survival probability approaches one, the impact of surplus additions on franchise value diminishes, so that the resulting changes in value added become negative. Columns B and C in Tables 1 and 2 illustrate the income and survival effects, along with several indirect consequences of changes in a firm’s surplus. In Table 1, surplus increases from 50 in column A, the base case, to 80 in column B and 110 in column C. The other input parameters are unchanged.

70

Value Added

60

50

40

30

20 0

25

50

75

100

125

150

Surplus

FIGURE 1.

Effect of Surplus on Value Added. (color figure available online)

175

200

22

W. H. PANNING 200

E[L] = 60

180 160

E[L] = 75

Value Added

140 120 100 80 60 40 20 0 0

20

40

60

80

100

Surplus

FIGURE 2. Effect of Surplus on Value Added for Varying Expected Losses. (color figure available online)

Table 2 shows the consequences of these surplus changes.14 In column B, adding 30 to the base case surplus in column A increases the firm’s survival probability by 2.1% to 99.47%. This in turn increases franchise value by 43.49 but reduces residual value by 4.05, for a net gain of 39.44 in total value. So value added has been increased by (39.44 – 30) = 9.44. Adding another 30 to surplus, as in column C, increases survival probability to 99.9%, a gain of only 0.43%. Consequently, franchise value increases by only 27.15 while residual value is reduced by 2.60, for a net gain of 24.54 in total value. This is less than the addition to surplus, so value added is reduced by 5.46.15 5.2. The Effect of Other Variables: Sensitivity Tests and Strategic Alternatives A simple but effective way to determine the precision of a result is to perform a sensitivity test: How much would the result change if the value of another variable in the model is changed from its current value? But the answer to that question has another use as well: it can inform managers concerning the effect on optimal surplus and value added of deliberately changing other variables—for example, by altering the firm’s mix of business to change its loss ratio, its expense ratio, or the variability of its losses. Figure 2 shows the relationship between surplus and value added for expected losses ranging from 60 (the top line) to 75 (the bottom line). As expected losses approach 75, the lines in the graph become rather flat, so that it becomes difficult to ascertain just where on the curve value added is maximized. So the results are presented somewhat differently in Figure 3, which shows, for alternative values of expected losses, the maximum value added and the amount of surplus needed to produce that maximum. As expected losses increase, the maximum value added decreases considerably, in a nearly linear fashion, as one would expect. But, surprisingly, as expected losses increase from 60 to roughly 70, optimal surplus increases and then subsequently decreases as expected losses increase further. When expected losses are about 76 or higher, optimal surplus suddenly falls to zero. Figure 3 also shows that the estimated optimal surplus is nearly constant for expected losses ranging from the upper 60s to the low 70s. This has two implications. One is that modest potential error in estimating expected loss is not terribly crucial so long as expected loss is within this range. The other is that attempts to change optimal surplus by altering the firm’s expected loss will have very little success when expected loss is in this range. Figure 4 shows how the relationship between surplus and value added is affected by different levels of expenses, ranging from 15 to 31 in increments of 1. As one would anticipate, the results are very similar, although not identical, to those obtained for different levels of expected loss. Figure 5 shows that as expenses increase from 15 to 25 or so, optimal surplus remains virtually constant at 80, while value added (with optimal surplus) decreases, as one would anticipate. However, when expenses increase beyond that level, optimal surplus decreases and suddenly falls to zero as expenses reach 31.

14The numerical precision of the results shown in Table 2 is misleading, since the precision of a model’s results depends upon the correctness of the valuation model and the precision with which its inputs are measured. The several decimal places shown for the values in Tables 1 and 2 are intended to enable readers to check the model and are not indicative of the precision with which the model’s inputs can actually be estimated or its results applied to real firms. I note, however, that most of the key inputs to the model can be reasonably measured or estimated. Consequently, it would not be unreasonable to conclude that, for this hypothetical firm, optimal surplus is likely to be closer to 80 than to 50 or 110. 15Optimal surplus is not always positive. The model implies that there are certain circumstances—partly defined by combinations of expected loss and standard deviation of loss—in which optimal surplus is zero. In other words, there are boundaries beyond which risks are, in effect, economically uninsurable or expected profits are so low as to preclude a viable business model.

23

MANAGING THE INVISIBLE 200 180

Value Added with OpƟmal Surplus

160 140 120 100 80 60 40

OpƟmal Surplus, given E[L] 20 0 60

65

70

75

Expected Losses E[L]

FIGURE 3. Effect of Expected Losses on Optimal Surplus and Value Added. (color figure available online)

Figure 6 shows the relationship of surplus to value added for different levels of investment yield, ranging from 3% (the highest curve) to 13% (the lowest) in increments of half a percent. Higher yields increase investment income but nonetheless reduce value added because they reduce the present value of future income. What is especially noteworthy about Figure 6 is the point on the left side of the figure where all the curves closely converge. At that point, achieved when the firm’s surplus is about 42, the firm’s value added—and therefore its total value as well—is virtually constant regardless of potential changes in interest rates. Given that level of surplus, the firm’s value and value added is essentially immunized, provided that the other variables in the model (apart from interest rates) remain at their assumed values. Equally notable, however, is the fact that this point is suboptimal for all of the interest rates shown in Figure 6. To maximize shareholder value, optimal surplus for this firm ranges from roughly 110 when the yield is 3% to 47 when the yield is 13%. What this strongly suggests is that immunizing a firm’s value from potential changes in interest rates, whatever its merits when considered alone, may be a suboptimal strategy when considered within the broader framework of maximizing a firm’s value or value added. This raises questions that clearly warrant further investigation. Figure 7 shows that optimal surplus and value added (given optimal surplus) both decrease when investment yields increase from 3% to 13%. Figure 8 shows the relation of surplus to value added for different levels of loss volatility, here defined as the standard deviation of losses (in dollars rather than in logarithms of dollars) as a percentage of expected losses (also in dollars). The curves shown are

200

Expenses = 15 180 160 140 Value Added

Expenses = 31 120 100 80 60 40 20 0 0

20

40

60

80

Surplus

FIGURE 4. The Effect of Surplus on Value Added for Varying Expenses. (color figure available online)

100

24

W. H. PANNING 200 180

Value Added with OpƟmal Surplus 160 140 120 100 80 60 40

OpƟmal Surplus, given Expenses 20 0 15

20

25

30

35

Expenses

FIGURE 5. Effect of Expenses on Optimal Surplus and Value Added. (color figure available online)

for levels of loss volatility ranging from 20% to 50%. Consistent with Figure 9, as loss volatility increases, optimal surplus also increases, but value added, given optimal surplus, decreases. A direct implication of Figures 8 and 9 is that managing a firm’s standard deviation of losses can have an extremely important impact on its value added, potentially equivalent in importance to managing its loss and expense ratios. Reinsurance is certainly one way to accomplish this objective, but it introduces complexities that are beyond the scope of this article. 5.3. The Combined Impact of Changing Multiple Variables Strategic choices often involve changes in multiple key variables. The model presented here can assist managers in identifying combinations of changes that best achieve particular goals. This is illustrated in Figure 10 for just two key variables: loss standard deviation and expected loss. The two lines in the graph show the combinations of these two variables that produce value added of 50 (top line) and 100, respectively. Graphs of this sort (which can be extended to multiple variables) are useful in several ways. Suppose that an insurer has a current value added of 50, with a loss volatility of 40% and an expected loss of 68. Further suppose that management wants to double its value added from 50 to 100, without changing the firm’s surplus. Figure 10 shows that this can be accomplished by lowering expected losses from 68 to just under 62 or, alternatively, by lowering its loss volatility from 120

3% yield 100

Value Added

80

13% yield 60

40

20

0 0

20

40

60

80

100

Surplus

FIGURE 6. Effect of Surplus on Value Added for Varying Yields. (color figure available online)

120

25

MANAGING THE INVISIBLE 120 110 100 90

OpƟmal Surplus, given Investment Yield 80 70 60

Value Added, given OpƟmal Surplus 50 40 3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

13%

Investment Yield

FIGURE 7. Effect of Investment Yield on Optimal Surplus and Value Added. (color figure available online)

80

v = 20% 70

Value Added

60

50

40

v = 50% 30

Loss volaƟlity v is the standard deviaƟon of losses as a percentage of expected losses 20 0

20

40

60

80

100

120

140

Surplus

FIGURE 8. Effect of Surplus on Value Added for Varying Loss Volatilities. (color figure available online)

40% to 20% or by some combination of these two strategies. Further, they could consider combinations of these two strategies with other changes that affect expense ratios, reinsurance, and the like. The point here is that the model results shown in this graph or others similar to it can assist managers in clarifying strategic alternatives and quantifying their impact on value added. 6. IMPLICATIONS: MANAGING THE INVISIBLE Ultimately, managers effectively manage only what they can see and measure.16 Visibility and quantification are both crucial. Things that are invisible typically fail to win managerial attention, always a scarce resource. Moreover, without quantification, effective management becomes nearly impossible, for a manager cannot know whether his actions have brought about intended improvements or even whether conditions are improving or deteriorating. What is visible at virtually every insurance firm is what has to be reported in the firm’s statutory and/or GAAP financial statements. But these financial statements are essentially cross-sectional and focus only on income statement and balance sheet 16Some

of the ideas stated here draw on Panning (2003a, 2003b).

26

W. H. PANNING 140

OpƟmal Surplus, given Loss VolaƟlity 120

100

80

60

40

Value Added, given OpƟmal Surplus 20

0 20%

30%

40 %

50%

Loss VolaƟlity

FIGURE 9. Effect of Loss Volatility on Optimal Surplus and Value Added. (color figure available online)

values that reflect business already written. While changes in these accounting values from one period to the next can indicate potential trends that may need attention, there is little, apart from claims handling, that managers can do about business already written. Ironically, then, the numbers that managers typically and most frequently see pertain to matters over which they have very little influence. Whatever the value of GAAP and statutory financial statements for audiences outside the firm, their value for managing a firm’s future is somewhat dubious. Consider, by contrast, the potential value to managers of the valuation model presented in Section 4 and of the sensitivity and strategy analyses presented in Section 5. A firm’s value added as calculated in the model is derived from a longitudinal valuation of the firm as a going concern, and the analyses focus managerial attention on the variables with the greatest potential impact on value added and other key statistics. Analyses like these have several important consequences. First, because the model focuses on the survival-adjusted present value of future income, it addresses the future of the firm, which can be managed, in contrast to accounting values that reflect the results of past decisions. Second, the model provides managers with a visible and quantitative estimate of the firm’s franchise value and its sensitivity to the various significant variables that management can influence. It

75

Value Added = 50

70

65

Value Added = 100 60

55 15%

20%

25%

30%

35%

40%

45%

50%

Loss VolaƟlity

FIGURE 10.

Constant Value-Added Combinations of Expected Loss and Loss Volatility. (color figure available online)

MANAGING THE INVISIBLE

27

therefore provides managers with a sound basis for decisions concerning capital structure, in contrast to the typically used rules of thumb described in Section 2. But the third and, to me, most compelling feature of the model presented here is that it provides an understandable dollar-based measure of the cost of risk. The classical problem with risk-return analysis is that it provides no compelling reason for choosing a particular level of risk exposure, since risk and return are incommensurable. This is why appeals to risk tolerance or other rules of thumb are thought to be essential. By contrast, in the model presented here there is a quantifiable cost of risk, measured in dollars. Specifically, if we change the mean or standard deviation (as a percentage of the mean) of the firm’s aggregate loss distribution, we can use the model to calculate the consequences of doing so, in dollars, for the firm’s value and value added. This point may be somewhat obscured by the way in which I have approached the problem—as one of optimizing surplus relative to a given level of risk. As I emphasized at the beginning, this problem is mathematically equivalent to one in which surplus is given and the level of risk is the variable to be managed in order to maximize value added. My rationale for approaching the problem in the way I have is that most insurers will find it easier to change their level of surplus (or reinsurance, for that matter) than to alter their portfolio of business and its characteristics. The latter strategy depends on costly changes to an existing infrastructure (including a distribution system and internal underwriting, pricing, and reserving capabilities) geared toward an existing mix of business. I could equally well have presented the model as one in which surplus was fixed (as for a mutual insurer) but premium volume was variable (since the crucial assumptions are all stated relative to written premiums). In that case the key objective would be to identify the optimum premium-to-surplus ratio. In the absence of a model like the one presented here, or an equivalent model in which surplus is fixed and risk exposure is variable, the consequences of changing a firm’s capital structure are matters of guesswork and are therefore in practice typically ignored, in favor of actions that change the anticipated distribution of aggregate losses, a domain where consequences are considered more visible and quantifiable. There are numerous ways of extending the model presented here: (a) by incorporating a sophisticated model of reorganization costs; (b) by incorporating more sophisticated assumptions concerning the tax status of the firm; (c) by taking into account the fact that dividends are taxed differently from ordinary income and so may be more valuable to shareholders than is taken into account here; (d) by incorporating customer preferences, and willingness to pay higher premiums, for policies from firms with low probabilities of default; (e) by incorporating various reinsurance strategies; (f) by incorporating a more sophisticated approach to the effects of growth on the value of a firm; (g) by incorporating stochastic interest rates; (h) by taking into account strategies for coping with an underwriting cycle; (i) by incorporating more flexible ways of representing the firm’s aggregate loss distribution, and so on. Although I could suggest even more ways to amend and improve the valuation model presented here, I hope that this brief list indicates my recognition that there is much potentially valuable work yet to be done. Above all, I hope that the model and analysis presented here will have a significant impact on the goals and practice of Enterprise Risk Management. ERM is, in my view, potentially the most significant development in risk analysis and risk management in decades. My greatest concern about the future of ERM is that it will be a victim of excessive hype, based on an implicit assumption that its benefits will somehow become evident to senior management. I strongly believe that what ERM needs to hedge this risk is a compelling value proposition—an answer to the question “What are the benefits to my firm of embracing ERM?” The answer I have proposed and (I hope) demonstrated here is that ERM, done properly, does two things. First, it makes visible a firm’s franchise value and so stimulates and enables managers to focus attention on managing the survival-adjusted present value of the firm’s future income. Second, it enables managers to anticipate and measure the consequences of alternative actions and strategies intended to maximize shareholder value, as estimated by a pragmatic valuation model whose parameters are observable or reasonably estimated. In essence, value-focused ERM, as proposed and demonstrated here, can provide managers with reliable tools that can increase the scope and effectiveness of their decisions. Is this ambitious goal indeed practical, or is it merely wishful thinking? Two facts are strongly encouraging. One is that models similar to the one presented here, although considerably more complex, are already pervasive in significant areas of the capital markets. A second is that managers who attempt to increase shareholder value cannot avoid reliance on a model but instead face an inevitable choice between an implicit and impressionistic mental model or, as I propose here, an explicit and empirically verifiable model subject to professional scrutiny and improvement. If Enterprise Risk Management is to be more than a passing fad, it must, in my view, accept responsibility for making this second alternative a reality and so provide managers with the tools that they need to quantify and manage what is now invisible to them. REFERENCES Avanzi, B. 2009. Strategies for Dividend Distribution: A Review. North American Actuarial Journal 13(2): 217–251. Bodoff, N. M. 2012. Sustainability of Earnings: A Framework for Quantitative Modeling of Strategy, Risk, and Value. Casualty Actuarial Society E-Forum. http://www.casact.org/pubs/forum/12sumforum/Bodoff.pdf. Boland, P. J. 2007. Statistical and Probabilistic Methods in Actuarial Science. Boca Raton: Chapman and Hall/CRC.

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Brockman, P., and H. J. Turtle. 2003. A Barrier Option Framework for Corporate Security Valuation. Journal of Financial Economics 67(3): 511–529. Brown, S. J., W. Goetzmann, R. G. Ibbotson, and S. A. Ross. 1992. Survivorship Bias in Performance Studies. Review of Financial Studies 5(4): 553–580. Brown, S. J., W. N. Goetzmann, and S. A. Ross. 1995. Survival. Journal of Finance 50(3): 853–873. Chandra, V., and M. Sherris. 2007. Capital Management and Frictional Costs in Insurance. Australian Actuarial Journal 12(4): 399–447. Coleman, L. 2009. Risk Strategies: Dialling Up Optimum Firm Risk. Burlington, VT: Gower Publishing Company. Copeland, T. E., J. F. Weston, and K. Shastri. 2005. Financial Theory and Corporate Policy. 4th ed. Boston: Pearson Addison Wesley. Damodaran, A. 2005. Valuation Approaches and Metrics: A Survey of the Theory and Evidence. Foundations and Trends in Finance 1(8): 693–784. Derrig, R. A., and E. D. Orr. 2004. Equity Risk Premium: Expectations Great and Small. North American Actuarial Journal 8(1): 45–69. Dickson, D. C. M. 2005. Insurance Risk and Ruin. Cambridge: Cambridge University Press. Eling, M. 2012. RPP II: The Risk Premium Project Update. CAS, Actuarial Review. http://www.casact.org/newsletter/index.cfm?fa=viewart&id=6215. See also http://www.casact.org/research/rpp2/. Elton, E. J., M. J. Gruber, D. Agrawal, and C. Mann. Explaining the Rate Spread on Corporate Bonds. 2001. Journal of Finance 56(1): 247–278. Episcopos, A. 2008. Bank Capital Regulation in a Barrier Options Framework. Journal of Banking and Finance 32(8): 1677–1686. Exley, C. J., and A. D. Smith. 2006. The Cost of Capital for Financial Firms. British Actuarial Journal 12(1): 229–283. Fernandez, P. 2002. Valuation Methods and Shareholder Value Creation. New York: Academic Press. Giesecke, K., and L. R. Goldberg. 2004. In Search of a Modigliani-Miller Economy. Journal of Investment Management 2(3): 1–6. Hancock, J., P. Huber, and P. Koch. 2001. The Economics of Insurance: How Insurers Create Value for Shareholders. Zurich: Swiss Re Insurance Company. Ingram, D., and D. Bar Yaacov. 2012. Trifurcation. http://www.soa.org/library/monographs/other-monographs/2012/april/mono-2012-as12-1-ingram.aspx. Koller, T., M. Goedhart, and D. Wessels. 2010. Valuation: Measuring and Managing the Value of Companies. New York: John Wiley and Sons. Leibowitz, M. L. 2004. Franchise Value: A Modern Approach to Security Analysis. New York: John Wiley and Sons. Major, J. 2009a. The Cost of Risk: A COTOR-VALCON Discussion. Casualty Actuarial Society E-Forum, Spring. Major, J. 2009b. The Firm-Value Risk Model. Paper presented at the American Risk and Insurance Association 2009 Annual Meeting. http://www.cb.wsu. edu/aria2009/ARIA2009Papers/Full%20Papers/session1D Major.pdf. Major, J. 2011. Risk Valuation for Property-Casualty Insurers. Variance 5(2): 124–140. Modigliani, F. 1988. MM – Past, Present, Future. Journal of Economic Perspectives 2(4): 149–158. Panning, W. H. 1994. Asset-Liability Management for a Going Concern. In Financial Dynamics of the Insurance Industry, ed. E. Altman and I. Vanderhoof, pp. 257–291. New York: Dow Jones-Irwin. Panning, W. H. 2003a. Managing the Invisible. Best’s Review 104(2): 87. Panning, W. H. 2003b. Focusing on Value. Best’s Review 104(5): 81. Panning, W. H. 2006. Managing Interest Rate Risk: ALM, Solvency, and Franchise Value. Paper presented to the International Conference of Actuaries, Paris, May. http://www.casact.org/library/studynotes/panning.pdf. Pettit, J. 2007. Strategic Corporate Finance: Applications in Valuation and Capital Structure. New York: John Wiley and Sons. Rubinstein, M. 2003. Great Moments in Financial Economics: II. Modigliani-Miller Theorem. Journal of Investment Management 1(2): 7–13. Schroeck, G. 2002. Risk Management and Value Creation in Financial Institutions. New York: John Wiley and Sons. Segal, S. 2011. Corporate Value of Enterprise Risk Management: The Next Step in Business Management. New York: John Wiley and Sons. Smith, A., I. Moran, and D. Walczak. 2003. Why Can Financial Firms Charge for Diversifiable Risk? Paper presented at the Thomas P. Bowles Jr. Symposium, Atlanta, Georgia. http://www.casact.org/education/specsem/sp2003/papers/. Venter, G. G., and A. Underwood. 2010. Value of Risk Reduction. CAS Study Note. http://www.casact.org/library/studynotes/Venter-Underwood Value of Risk.pdf Woolley, S. 2009. Sources of Value: A Practical Guide to the Art and Science of Valuation. Cambridge: Cambridge University Press. Yow, S., and M. Sherris. 2007. Enterprise Risk Management, Insurance Pricing and Capital Allocation. Geneva Papers on Risk and Insurance: Issues and Practice (July): 34–62. Yow, S., and M. Sherris. 2008. Enterprise Risk Management, Insurer Value Maximisation, and Market Frictions. ASTIN Bulletin 38(1): 293–339.

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Quantitative Finance, Vol. 12, No. 10, October 2012, 1547–1556

Market risks in asset management companies BERND SCHERER* Morgan Stanley, 25 Cabot Square, London E14 4QA, UK (Received 6 February 2011; in final form 13 December 2011) This paper shows that revenues from a sample of publicly traded US asset management companies carry substantial market risks. Not only does this challenge the academic risk management literature about the predominance of operative risks in asset management, it is also at odds with current practice in asset management firms. Asset managers do not hedge market risks even though these risks are systematically built into the revenue generation process. This is surprising as shareholders would not optimally choose asset management companies as their source of market beta. They rather prefer to participate in alpha generation and fund gathering expertise of investment managers as financial intermediaries. At the very minimum, asset managers need to monitor their ‘fees at risk’ to understand what impact product design, benchmark choice and fee contract design have on revenue volatility. This calls for a much wider interpretation of the risk management function that too narrowly focuses on client risks. Keywords: Applied econometrics; Bayesian statistics; Risk management; Asset management JEL Classification: C1, C2, C11, C23, G0

1. Introduction The textbook view on risks in asset management companies is summarized by Hull (2007, p. 372): ‘‘For an asset manager the greatest risk is operational risk.’’y This view was seriously challenged by the events of 2008. Asset management companies came under severe profitability pressure from the market, not operational risks. Responsible for this is the direct market exposure built into asset management fee models. Asset management fees are usually paid as a percentage of assets under management. A 50 basis points asset-based fee on 10 billion US equity assets translates into a directional market exposure of 50 million USD. What has been seen as an annuity stream (‘annuity business view’) that was thought to expose asset management firms to little or no earnings risk, materialized as directional stock market exposure combined with high operational leverage (high ratio of fixed to variable costs). While operational leverage leads to what has been praised as a scalable business (low costs of taking on additional business) in

good times it creates the potential for large losses when revenues fade. As client benchmarks went down, so did asset-based fees (percentage fee applied on average assets under management within a year) and hence revenues. In accordance with Stulz (2003) it can be argued that asset management companies should hedge their ‘fees at risk’. The gist of this argument is that while the beta components in asset-based fees are zero net present value (NPV) projects, they still create revenue risks that in a world with capital market friction and taxes are costly. Removing those risks from the P&L account of an asset management company would free up risk capital for new ventures, increase the observability of effort by senior management and reduce frictional bankruptcy costs.z Our objective is to test for the direct relationship between asset management revenues and market returns. The estimated sensitivity is a measure of market risks inherent in the revenues of an asset management firm. If different from zero it needs to be incorporated into the risk management process of an asset management firm.x So far the academic literature exclusively focuses on

*Email: [email protected] ySee Scherer (2009) for a review of this topic. zSee Basu (2010) for a review on firms’ diversification and refocusing in the last three decades. xCommunications with risk managers from leading Wall Street firms as well as personal experience confirm the author’s view that these risks remain unmanaged. We find evidence for this in our data set. Section 4 gives an account of the most popular argument against hedging ‘fees at risk’ and why they fail in the face of modern finance. Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2012 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/14697688.2011.650185

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Table 1. Statistical properties of revenue growth data from 2004 Q2 to 2010 Q1. The table provides the mean (and p-value under the null hypothesis of zero revenue growth), volatility, skew, kurtosis, and the Jarque/Bera normality test (and p-value under the null hypothesis of normality) for the quarterly revenue growth of 17 US asset management companies from Q2 to 2010 Q1. Appendix C explains the abbreviations for each asset manager and discusses the data sources. All data provided by Klimek Advisors. Asset manager

Mean

p-value

Volatility

Skew

Kurtosis

JB test

p-value

AB AMG BAM BEN BLK DHIL EV FIL GBL GROW IVZ JNS LM SMHG TROW WDR WHG

0.65 2.70 9.49 2.72 15.19 19.08 2.55 0.35 1.60 12.69 21.16 0.19 3.06 3.92 2.85 2.58 6.42

0.80 0.23 0.15 0.18 0.11 0.02 0.08 0.74 0.71 0.13 0.29 0.93 0.43 0.38 0.09 0.10 0.16

12.64 10.67 31.00 9.64 44.89 35.72 6.88 4.98 20.51 39.29 94.90 10.74 18.74 21.37 7.98 7.32 21.79

0.37 0.97 1.03 1.32 4.14 2.11 1.03 0.62 0.92 0.18 3.05 1.49 0.75 0.64 1.89 1.34 0.50

0.97 0.55 2.32 2.71 18.91 5.61 1.77 0.89 0.36 0.53 8.67 4.42 3.03 0.77 5.68 3.40 5.64

1.50 4.10 9.62 14.25 426.09 49.23 7.37 2.34 3.50 0.41 112.36 28.39 11.42 2.21 46.63 18.72 32.83

0.47 0.13 0.01 0.00 0.00 0.00 0.03 0.31 0.17 0.81 0.00 0.00 0.00 0.33 0.00 0.00 0.00

managing client risks not business risks. In the view of this paper, this needs addressing as the volatility of asset markets has a first-order effect on asset management revenues P&L, while it gets virtually no mention in the risk management literature.y The only paper close to ours is Scherer (2010). He first argues that market risks have a dominant effect on revenue volatility for large asset management companies with a well diversified client base.z For a sample of nine asset management firms and for a limited number of data points (eight annual data observations from 2000 to 2007 per firm), he uses various panel regression techniques to show that the revenue beta of asset management revenues is indeed one as initially conjectured. We extend the above analysis in various ways. First, we look at a panel that is both longer and broader. Our data consist of 24 quarterly observations of 17 pure asset managers. Our dataset is higher in frequency and contains more asset management firms. In the language of panel data analysis, this is a broader and longer panel. Second, contrary to Scherer (2010), we include the crisis years 2008 and 2009. This should help us to investigate any potential asymmetry in the sensitivity of asset management revenues to severe down markets. Third, we now distinguish between up and down market betas. Fourth, we employ a Bayesian random coefficient model that allows us to more flexibly deal with the data without making strong structural assumptions. The data and our prior views will lead us to the correct model, which will be reassuringly simple once we conclude our research. Moreover our research confirms the results of

Scherer for a different data set, time period and econometric methodology. Section 2 outlines the model and data used in this paper in combination with some initial data analysis. Section 3 presents our empirical results for the Bayesian random coefficients model, while sections 4 and 5 interpret these results from a corporate finance perspective.

2. Model and data 2.1. Manager data Our focus on listed asset management companies arises from the fact that revenue data are simply not available for private partnerships or bank-owned asset management units. Table 1 provides a summary of the statistical properties for our data set. The publicly listed asset management companies in our sample cover active as well as passive investments, asset management infrastructure and private equity alternatives. First, observe that unconditional average revenue growth varies considerably across asset management firms. Invesco (IVZ) shows the largest average revenue growth for our sample, driven by external asset under management growth. None of the unconditional means is statistically significant at the 99% confidence level, even though anecdotal evidence tells us that asset management firms benefitted from strong net inflows over the sample period. Given the limited number of data points for each individual mean combined with the high volatility of revenue growth this is

yMarket risk in asset management revenues can be hedged using Asian forwards (stacked standard forwards with different maturities), which better represent the averaging process in asset-based fees. zSmall asset management companies with few clients and one star fund manager (who could leave to a competitor) will be exposed to a lot more idiosyncratic risk. However, these risks are hardly operative in nature and as such the common view that the biggest risk in asset management is operative risk might be misplaced.

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Market risks in asset management companies

2.2. Market data Next, we relate revenue risk to market risk. The Russell 1000 in USD is used as a proxy for stock market returns.x However, instead of calculating a simple quarter end return r ¼ ðR1000t =R1000t1 Þ  1, we use the average value of the Russell 100 over the quarter as numerator, i.e. the quarterly percentage changes in average price relative to prices at the start of the quarter, rav, t . We conjecture the latter to better represent the nature of asset-based fees as asset-based fees are calculated as a fraction of average assets under management for a given period. Appendix A proves that the regression beta of year end stock market returns against revenue growth will lead to misleading stock market sensitivities. Under fairly standard assumptions, fee sensitivities rev in an OLS regression will converge against half the asset class beta, , as averaging becomes continuous (n ! 1), i.e. rev ¼

ð1 þ ðn  1=2ÞÞm2 n ð1 þ ðn  1=2ÞÞ  ¼ , ¼ n!1 2 m2 n2 n ð1Þ

0.225

0.200

0.175

0.150 Average revenue growth

hardly surprising. It is also prima facie evidence for the potential advantage of pooling data with hierarchical panel data models to arrive at more precise parameter estimates. Quarterly revenue volatility shows exceptionally high levels for an industry that has been perceived as an annuity business. This supports our prior that asset management risk contains a large fraction of market risk, directly built into fee contracts.y The relationship between average revenue growth and revenue volatility is linear and exceptionally strong with an adjusted R2 of 0.76 and a t-value of 6.2 on volatility when regressed against mean returns.z Figure 1 provides support that a common factor (generating the same return per unit of risk) runs through the asset management industry and different firms exhibit different exposure (leverage) to this factor. Revenue risk is highly non-normal for almost all firms with large negative skew (similar to market risks). In combination with high operative leverage (high fixed costs relative to variable costs), this risk characteristic creates a risky position most asset managers have boxed themselves into. If financial regulation would force investment management firms to set regulatory capital aside to cover potential losses from market risk, we should see more careful use of economic capital.

0.125

0.100

0.075

0.050

0.025

0.000 0.00

0.25

0.50 Revenue volatility

0.75

1.00

Figure 1. Average revenue growth versus revenue volatility. The figure plots average revenue growth versus the volatility of revenue growth for the data in table 1. A regression of average revenue growth versus revenue volatility suggests an adjusted R2 of 0.76.

where m2 denotes benchmark volatility. In other words, the averaging process for fee income hides market exposure. Low betas result from the averaging process and are not an indication of low market exposure. The averaging process effectively decouples asset returns and fees even though there is a deterministic one-to-one relationship. Finally, we need to allow for the possibility that revenue streams react differently to up and down markets. Bear markets lead to falling revenues as assets under management fall both due to market impact or to client rebalancing to less risky (cash, fixed income) but less profitable investment products. Anecdotal evidence suggests that this rebalancing on the downside was particularly painful in 2008. The reduced ability to take risks by institutional investors also meant that clients did not restock their portfolios of risky assets, leading to an asymmetric response. We therefore estimate dn revi,t ¼ i,0 þ i,up rup av, t þ i,dn rav, t þ "i,t :

ð2Þ

pffiffiffi yWe can approximate the revenue volatility of asset-based fees using = 3, where  is the volatility of benchmark returns. This approximation is based on a geometric average that will be close to the arithmetic average (average AUM) that underlies fee calculations. Given that the MSCI WORLD return in this period was about 20%, it is clear that additional (and possibly correlated) factors are at work. In other words, asset manager specific risk is considerable. Clients might change from risky high fee business to safe low fee assets, client groups might disappear or particular products become out of favor. zRunning a robust regression (MAD) instead, the t-value increases to almost eight with an almost identical slope. However, given that our data include both outliers as well as influential observations (the leverage value for IVZ is six times larger than the average leverage value), just robustifying against outliers might not be sufficient. In any case, removing IVZ as most likely the wackiest data point will not change our results. xThis choice is likely to underestimate the amount of beta exposure in revenue streams as the business mixes of asset management firms differ. It again makes our result very conservative as we could easily find higher systematic market exposure by data mining for the best in sample fitting market benchmarks.

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B. Scherer

Table 2. Individual OLS regressions. Revenues for each asset management firm are regressed against up- and down-side average market returns according to (2). We perform significance tests using robust standard errors (HAC adjustment with three autocorrelation lags). Asset manager

^i,0

t-value

^i,up

t-value

^i,dn

t-value

H0 : ^i,up ¼ ^i,dn ð p-valueÞ

R 2 ð%Þ

AB AMG BAM BEN BLK DHIL EV FIL GBL GROW IVZ JNS LM SMHG TROW WDR WHG

0.01 0.05 0.06 0.06 0.16 0.22 0.05 0.00 0.01 0.19 0.25 0.00 0.07 0.03 0.04 0.03 0.00

(0.45) 2.47 0.68 2.86 1.27 2.21 3.38 0.25 0.25 1.64 0.89 0.23 1.24 0.50 5.41 2.89 0.04

2.18 0.65 3.26 0.46 1.68 1.17 0.13 0.01 1.57 1.50 1.59 1.45 1.01 1.92 1.06 0.78 1.88

(3.29) 1.24 1.43 0.77 0.48 0.44 0.34 0.02 1.06 0.48 0.21 3.09 0.69 1.25 6.09 2.50 1.12

1.49 2.06 1.56 1.70 2.70 3.12 1.32 0.00 1.72 2.06 4.18 1.83 1.06 1.62 1.65 1.38 1.53

2.95 5.15 0.90 3.70 1.02 1.53 4.36 0.01 1.52 0.87 0.73 5.12 0.95 1.38 12.41 5.81 1.20

0.48 0.07 0.61 0.02 0.84 0.62 0.02 0.98 0.94 0.43 0.81 0.57 0.34 0.89 0.03 0.19 0.17

57.37 62.80 16.11 39.72 7.69 13.43 48.98 0.00 18.93 3.71 3.44 70.59 4.87 19.22 92.61 72.07 8.97

up Here, rdn av, t ¼ minðrav, t , 0Þ and rav, t ¼ maxðrav, t , 0Þ. We can use this formulation to test HO : i,up ¼ i,dn , which is the specification used by Scherer (2010). Table 2 summarizes the results. There is a clear indication that ^i,up 5 ^i,dn (for 12 out of 17 asset managers), although again none of the tests show statistical significance at the 99% confidence level. Only results for TROW, BEN and EV are significant at the 95% confidence level. More interestingly, adjusted R 2 values show a large variation. The explanatory power of the model is largest for stable (no recent M&A activity) and diversified asset management firms like TROW, JNS and AB. For those firms, market risks are the dominant source of revenue volatility, questioning the conventional wisdom that the biggest risks in asset management are operative risks. An R 2 of 93%, as in the case of TROW, leaves little room for interpretation. Only 7% of revenue volatility could potentially arise from operative risks. On the other side, smaller and mid-size firms do not explain risks sufficiently, and some negative up-market betas are difficult to understand (for example, FIL, WHG and GROW).

2.3. Econometric model Our objective is to estimate market risks implicit in asset management revenues. In extension to Scherer (2010), we distinguish between up- and down-market exposures (betas). Initial calculations in the previous section have shown that the small sample (24 observations, 21 degrees of freedom) combined with volatile data proves to be a challenge for the statistical significance of each separate regression model. While a fully pooled model assumes that all asset management firms scatter around the same regression line, the nature of our data set might require us

to relax this assumption. Firms differ in size, client set, product mix, brand and asset gathering facilities. A natural way to deal with this heterogeneity with great flexibility (without imposing a solution) is to extend with a hierarchical prior for each regression coefficient. This also allows us to take advantage of a larger data set. To be more precise, I assume 2 3 3 02 1 i,0 0 ð3Þ i ¼ 4 i,up 5  N@4 up 5,  A ¼ Nð,  Þ: i,dn dn

Individual exposures, i , are normally distributed around  (3  1 vector) with covariance  .y If the upper left-hand element in  is large (small), then allows much (little) variation around the first element of  (autonomous revenue growth) across individuals. The above specification results from the following hyperpriors (all prior information entering the model is indicated by a superscript 0:   Nð0 , 0 Þ,

ð4Þ

1 ¼ Wðv0 , 1,0

  Þ,

ð5Þ

where W indicates the Wishart distribution and v0 is often used as a sample equivalent. If v0 ¼ 100, this would mean has the same precision as if it was that our prior 1,0  calculated from a (hypothetical) observed sample of 100 data points. Finally, I assume a Gamma prior on the precision h (1/variance of residuals), h  Gðs2,0 , Þ,

ð6Þ

for the product of all N individual likelihood functions (with individual parameter vectors i ). In essence, states that the individual parameters for each firm are

yNote that it does not matter whether this is really the case. All that is required is that the decision maker does not possess additional information.

Market risks in asset management companies (constrained to be) drawn from a common pool. This pool in turn is governed by (4) and (5). Computationally, I use Gibbs sampling to derive the distribution for each i as well as the pool mean . All calculations are detailed in Appendix B. In effect, our model allows every asset management company to be different. To impose some common structure (otherwise we could run individual regressions), our hierarchical priors control the way firms are allowed to differ. At the higher level I assume that individual betas are best described by a multivariate normal distribution, i.e. a vector of means and a matrix of covariates. At the lower level, I assume that, given individual betas, revenue growth is driven by a linear regression. Initial pooling estimates are estimated for each asset management unit to represent a starting point for the Gibbs sampler. The strength of this Bayesian random coefficients model with hierarchical prior modely is its flexibility, both in estimation and testing. It allows us to impose different views (beliefs) on how similar or dissimilar the investigated asset management companies are, i.e. on the poolability of our data set. In addition, aggregate estimation models confound heterogeneity and noise by modeling individuals rather than an average relation. The separation of signal (heterogeneity) from noise leads to more stable models. Draws (replicates) for each firm provide a rich source of information for more accurately conducting statistical tests.

3. Estimation results This section discusses our estimation results. I set the prior mean and precision for the mean of the pooled density as uninformative. More precisely, 31 02 3 2 0 1 0 0 B6 7 6 7C ð7Þ   N@4 1 5, 4 0 1 0 5A: 1 0 0 1 In other words, I do not impose any structure on the true industry-wide model for revenue risks in (3). This is the relationship we finally want to estimate. It should therefore not be heavily influenced by the choice of a specific prior. However, we need to impose some structure to benefit from our Bayesian approach. If we set all priors to be uninformative, our results would only be philosophically different from a simple random coefficient model. Hence, we set the parameters for the (inverse) Wishart prior on the variance of the pool from which the individual coefficients are drawn in to 2 3 100 0 0 6 7 0 5: ð8Þ 0 ¼ 4 0 0:25 0

0

0:25

1551

Our rationale is to allow estimates for the autonomous revenue growth to vary with a standard deviation of 10 (square root of 100) across units, which is still quite uninformative given the empirical distribution of autonomous growth in table 2. For individual betas, we impose a somewhat tighter prior. Whatever the mean of the pooled density looks like, my prior belief is that individual betas should—with 95% confidence—differ from their ‘hierarchical’ beta by one ( 1:96  2). This is a crucial model input. The tighter we set this prior, the closer will our results resemble a pooling solution. The reverse is true for uninformative priors. In the case of totally uninformative priors we revert back to separate OLS regressions. First, we present results for the individual regression models when confronted with a hierarchical prior in table 3. The Gibbs sampler is started from the individual regression estimates. Splitting our sample of 20,000 samplings into two halves (after removing the first 4000 burns in observations) the Geweke z-score is well below 1. Given the ‘flaky’ relationships (insignificant regressions with low explanatory power) from single regressions it is of little surprise that our random coefficient model suggests shrinkage towards the hierarchical model. Second, we show the simulated posterior industry-wide revenue ‘betas’ in figure 2. We use the term ‘industry wide’ as our model assumes that individual up- and downbetas are random draws from latent industry betas given in (3). While up-market betas are persistently lower than down-market betas, up-market betas also exhibit markedly higher volatility in their simulations. This is the reason why we do not find a significant difference in market exposure for asset management firms in up- and down-markets in figure 3. The above results so far point us in the direction of a simple pooling model as individual differences are hardly significant. Despite leaving considerable freedom as to how individual betas might vary across asset management units, the individual estimates scatter narrowly around common values. For comparative purposes, I therefore estimate a simple pooled OLS regression dn revt ¼ 0 þ up rup av, t þ dn rav, t þ "t :

ð9Þ

All data are stacked on top of each other (individual subscript i is dropped) and a single regression equation is run, essentially enforcing the same parameters across all units. The results are summarized in table 4. All estimates are remarkably close to our first stage hierarchical relation described by and shown in the last row (denoted by all firms) of table 3. While the statistical significance for up- and down-market betas is similar, it is significantly higher for autonomous revenue growth. The reason for this lies in the very uninformative prior for 0 , which implies a large uncertainty about the true beta. OLS ignores this and hence arrives at a high t-value.

yThe random coefficient model was motivated by Swamy (1970) and is essentially a matrix weighted GLS estimator that will weight regression coefficients depending on the quality (residual variance) of each regression and is usually applied when the data appear to be non-poolable. This is similar to practitioners that weight regression coefficients based on their t-values.

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B. Scherer

Table 3. Bayesian random coefficient model. This table reports the results for our Bayesian random coefficient model using Gibbs sampling. Individual beta estimates, t-values and p-values directly follow from the posterior distribution of regression coefficients. Asset manager

^i,0

t-value

^i,up

t-value

^i,dn

t-value

H0 : ^i,up ¼ ^i,dn ð pvalueÞ

AB AMG BAM BEN BLK DHIL EV FIL GBL GROW IVZ JNS LM SMHG TROW WDR WHG All firms

0.01 0.04 0.10 0.04 0.16 0.20 0.03 0.01 0.02 0.14 0.22 0.01 0.04 0.05 0.04 0.03 0.07 0.07

0.25 0.67 1.84 0.68 2.88 3.63 0.62 0.21 0.45 2.48 4.02 0.19 0.73 0.85 0.66 0.61 1.21 0.37

1.04 0.94 1.11 0.85 1.03 1.01 0.86 0.84 0.99 0.79 1.06 0.99 0.79 1.02 0.95 0.93 0.93 0.95

1.49 1.34 1.84 1.22 1.46 1.44 1.23 1.20 1.42 1.12 1.52 1.42 1.12 1.46 1.36 1.34 1.33 1.85

1.65 1.68 1.69 1.61 1.79 1.82 1.57 1.42 1.67 1.62 1.95 1.68 1.53 1.67 1.64 1.60 1.29 1.64

2.76 2.79 2.83 2.70 2.98 3.00 2.60 2.35 2.79 2.71 3.19 2.81 2.53 2.77 2.74 2.67 2.16 4.31

0.25 0.21 0.26 0.21 0.21 0.19 0.22 0.26 0.24 0.18 0.17 0.23 0.22 0.25 0.23 0.23 0.34 0.22

4

Down market beta

Simulated values from GIBBS sampler

3

2

1

0

Up market beta

-1

-2 2170 2480 2790 3100 3410 3720 4030 4340 4650 4960 5270 5580

Simulation run

Figure 2. Hierarchical up- and down-market beta from Gibbs sampling for the Bayesian random coefficient model. The graph shows 6000 random draws from Gibbs sampling (after discarding the first 4000 observations) for our Bayesian random coefficient model described in appendix B.

We can use this setup to test the assumptions of homogeneity across asset management firms. This is usually addressed by a Chow F-test of the formy RSSCP  RSSSR dfSR F¼ ¼ FðdfCP  dfSR , dfSR Þ, RSSSR dfCP  dfSR ð10Þ ySee Baltagi (2005).

where RSSCP is the residual sum of squares for the pooling model, RSSSR is the residual sum of squares from separate regressions (sum of each regression’s residual risk multiplied by the number of observations) and dfCP , dfSR are the degrees of freedom for each model. The test statistic is 0.18 and distributed as F(31,374).

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Market risks in asset management companies

Table 4. Pooling regression model. The table shows the estimates of a pooled OLS regression. The test statistic for a Chow for poolability versus a random coefficients model takes a value of 0.18 and is distributed as F(31,374).

0.5

0.4

Variable OLS 0 OLS up OLS dn

Density

0.3

Estimate

Standard error

t-statistic

p-value

0.07 0.94 1.64

0.02 0.57 1.64

3.32 1.64 3.72

0.00 0.10 0.00

0.2

Table 5. Simplified pooling model. The table shows the estimates of a pooled OLS regression where no attempt has been made to distinguish between up- and down-market betas. We cannot reject the null hypothesis that H0 : OLS av ¼ 1 (the corresponding t-test shows a p-value of only 0.15).

0.1

0.0 –5.0

–2.5

0.0

2.5

5.0

Difference in up and down market beta from GIBBS sampler

Figure 3. Distribution of up- and down-market beta for 20,000 Gibbs samplings. This graph shows a histogram of the differences between up- and down-market betas for 20,000 Gibbs samplings. While there is evidence of higher down- than upmarket beta (78% of the distribution of posterior returns lie below zero), it is not significant at conventional confidence levels.

The zero hypothesis of the classical pooling model is not rejected.y While the Bayesian random coefficient model is conceptually much more general than a naive pooled regression, a researcher aiming for less sophistication (Occams razor) might further simplify the estimation process. In fact, we cannot reject the null hypothesis that OLS OLS (p-value of 0.3 for a standard F-test). This up ¼ dn leads to an even simpler model specification where we make no difference between up- and down-market betas: revt ¼ OLS þ OLS 0 av rav,t þ "t :

ð11Þ

Table 5 summarizes these results. Our results show that asset management companies do not hedge market risks despite their being mechanically built into their revenue stream. This is surprising given that market risks arise from the choice of benchmark of

Variable OLS 0 OLS av

Estimate

Standard error

t-statistic

p-value

0.05 1.36

0.01 0.25

3.91 5.29

0.00 0.00

the client. Asset management companies have little control over this. In fact, those revenue risks are incidental to the asset managers production process, which consists of the creation of value added relative to a client chosen benchmark.z Why should (or should not) asset management firms hedge their fees at risk and what drives them to ignore this? This is addressed in the next two sections.

4. The case for asset managers hedging beta risk Market or beta risk exposures create a number of potential costs for asset management companies that represent benefits from hedging these risks.x Market risks directly affect asset manager fees through their impact on returns. Both asset- and performance-based fees are affected and they correspond roughly to beta (general economic and market exposure) and alpha (outperformance versus a risk-adjusted benchmark) risks.{ Committing capital with the aim of producing alpha is one of the core competencies of an asset management

yAlternatively, one could estimate a SUR model for all firms and test the coefficient restrictions using a Wald test. zIdeally, one would want to show in a cross-sectional regression that firms that are hedging their fees at risk receive higher market valuations (after controlling for factors that make firms different). However, given that virtually all asset management firms do not hedge, there is no variation in the independent variable (degree of hedging). xA general review of the benefits of risk management can be found in Doherty (2000). {How can we hedge asset management fees? The easiest way to hedge asset-based fees is not to offer them. This follows the idea of duality in risk management. We can either root out the cause (variability in markets) or the effect (offer fixed fees). There are, however, limits to this argument. First, this simply shifts risks to clients. If assets fall, percentage fees rise. Second, operative risks are still proportional to assets under management. A pure flat fee would not reflect this. Also, flat fees are not suitable where limited capacity exists (which is less an issue for pure beta exposure). How do we practically implement a hedge program aimed at insulating an asset managers P&L from market-induced variations of its average assets under management for a given time period? A simple way would be to sell futures with one year maturity on the underlying assets with a notional   A, where  represents the asset-based percentage fee and A the assets under management at the beginning of the period. If assets increase in value, the hedge (ignoring carry) creates a loss of   DA, while asset management fees rise by an offsetting þ  DA. In other words, how do we hedge a 50 bps fee on a 100 million USD mandate? We go short 500,000 USD in the futures market. If the market goes up 10%, you will lose 50,000 USD on this hedge, but you will get an equal amount back from rising fees (50 bps now on 110 million USD will provide 550,000 USD in fee income). In total, you locked in 500,000 USD in fees at the beginning of the year.

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B. Scherer

firm and, to the extent the firm is confident in its own capabilities, represents a positive NPV investment. For most long-only asset managers, however—at least those that do not profess to have market-timing skills—the taking of beta risk is generally an incidental consequence of taking alpha risk. In the attempt to produce alpha, most long-only asset managers, whether they acknowledge it or not, end up bearing considerable amounts of beta risk. Beta, or broad stock market, risks affect asset managers not only directly by their potential to reduce AUM, but also by resulting in systematic capital outflows from the company across a range of products. During a severe equity down market, retail investors will shift their asset allocation out of fee-intensive equity funds and into money market funds or government guaranteed deposits. At the same time, institutional client redemptions could be motivated by their own financial distress; for example, the client may need to raise cash or de-risk their asset allocation, or regulatory constraints may be binding in the case of insurance companies. Compounding the redemption problem for fund managers is that it is at precisely these times when bank funding also dries up. In other words, there is a high correlation between revenue (or capital market) risks and funding risks.y Hedging the beta risk exposure of its P&L could benefit asset managers, first of all by preserving the liquidity necessary to finance new projects—including new products, people, or IT platformsz—at a time when losses have reduced internal capital and external capital has become expensive. Hedging protects costly liquidity and debt capacity.x If an asset management firm chooses not to hedge its asset-based fees, it will be necessary to hold additional cash in reserve against P&L risks. But perhaps equally important, the hedging of a longonly asset manager’s beta risk should also provide the firm’s current and prospective clients with a clearer signal of whether its managers are succeeding in the firm’s mission of generating alpha. What is more, by isolating manager performance, hedging can facilitate more effective incentive compensation linked to shareholder value creation. This has two additional benefits: (1) it should motivate greater effort to generate alpha because performance is rewarded on a relative basis; and (2) it should help attract skilled talent more capable of generating alpha (while encouraging those managers who implicitly relied on beta exposure to leave).

5. Why do asset managers not hedge their P&Ls? Few companies in the asset management industry hedge market risk, which means that almost all suffered a large

loss in revenue because of the 50% drop in stock prices during the worst of the financial crisis. Long-only asset managers usually give a few general reasons for not hedging. The most common is that, by hedging, asset managers would forgo the expected increase in fee income that comes from a rising market. In other words, it would eliminate the ‘upside’ option that such fees represent for a conventional asset management firm. But unless an asset manager has a distinctive advantage in market timing, and cites that advantage when raising capital from investors, it is not clear that investors are willing to pay asset managers just for bearing beta risk. And if one evaluates an asset manager as a long-run economic enterprise, the question becomes: Do the benefits of bearing risk (mainly in the form of higher fees during up periods) outweigh the costs of trouble during the down periods? The answer provided by most economists would be no. While fund managers may benefit from the upside option, the value of an asset management firm in theory reflects its long-run prospects in creating value for its client—and value is not created simply by riding a rising market.{ Viewed in that light, those long-only asset managers that choose to limit their market exposure as part of their investment strategy— much like market-neutral hedge funds—are making a very different value proposition both to their investor clients and, in cases where the asset manager is publicly traded, to their stockholders. The other common objection to hedging beta risk is the difficulty of explaining hedging losses and market underperformance during up markets. This difficulty can be managed through clarity and persistence in communicating the firm’s objectives and market-neutral stance to its investor clients and, again, in the case of publicly traded asset managers, the firm’s stockholders.

6. Practical implications We have shown that asset management revenues carry substantial market risks. This challenges both the academic view in the risk management literature concerning the predominance of operative risks as well as the current industry practice of not hedging market risks that are systematically built into the revenue generation process. As these risks are incidental to the production process (alpha generation, product development, asset gathering) financial theory suggests that these risks need to be eliminated. For asset management companies to return to an annuity model, this is imperative. Shareholders do not want to get exposure to market beta via holding asset

yGatzert et al. (2008) analyse strongly related risk concentration in financial firms. zUn-hedged swings in fee income will also increase the value of the tax option the government holds against the asset management company. Taxes have to be paid if profits are made, but with limited carry forwards and backwards, no equal amount is received if losses are made. The larger these swings, the higher the value of this option. This argument obviously depends on whether the tax option is at the money. xSee Mello and Pearsons (2000). {As Ross (2005, p. 71) has argued, ‘‘. . . since the fee is contingent on asset value, as a contingent claim its current value is independent of expected rates of returns.’’

1555

Market risks in asset management companies management companies, rather they want to participate in their alpha generation and fund gathering expertise as financial intermediaries. At the very minimum, asset management companies need to rethink their risk management functions. They need to move from an ex-post fiduciary risk measurement reporting approach to managing a firm’s revenue risks.

We can now work out the covariance between average portfolio prices over n days and the final market price at day n,   Xn p m 1 S , Sn Cov n i¼1 i 1 0 0 C B B ¼ Cov@n1 @Sp0 þ DSp1 þ Sp0 þ DSp1 þ DSp2 A |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Sp1

References

þ    , Sm 0 þ

Baltagi, B., Econometric Analysis of Panel Data, 3rd ed., 2005 (Wiley: New York). Basu, N., Trends in corporate diversification. J. Financ. Mkts Portfol. Mgmt, 2010, 24(1), 87–102. Chevalier, J. and Ellison, G., Risk taking by mutual funds as a response incentive. J. Polit. Econ., 1997, 105, 1167–1200. Doherty, N., Integrated Risk Management, 2000 (McGraw-Hill: New York). Gatzert, N., Schmeiser, H. and Schuckmann, S., Enterprise risk management in financial groups: Analysis of risk concentration and default risk. J. Financ. Mkts Portfol. Mgmt, 2008, 22(3), 241–258. Hull, J., Risk Management and Financial Institutions, 2007 (Prentice Hall: Engelwood Cliffs, NJ). Mello, A. and Pearson, J., Hedging and liquidity. Rev. Financ. Stud., 2000, 13(1), 127–153. Ross, S., Neoclassical Finance, 2005 (Princeton University Press: Princeton, NJ). Scherer, B., Fees at risk. EDHEC Working Paper, 2009. Scherer, B., A note on asset management and market risks. Financ. Mkts Portfol. Mgmt, 2010, 24(3), 309–320. Sirri, E. and Tufano, P., Costly search and mutual fund flows. J. Finance, 1998, 53, 1589–1622. Stulz, R., Rethinking risk management. In The Revolution in Corporate Finance, 4th ed.,edited by J. Stern and D. Chew, pp. 367–384, 2003 (Blackwell: Oxford). Swamy, P.A.V., Efficient inference in a random coefficient regression model. Econometrica, 1970, 38, 311–323. Verbeek, M., Modern Econometrics, 2004 (Wiley: New York). Zimmermann, H. and Schultz, J., Risikoanalyse schweizerischer Aktien: Stabilita¨t und Prognose von Betas. J. Financ. Mkts Portfol. Mgmt, 1989, 3(3), 196–209.

Appendix A: Market beta for asset-based fees Asset-based fees are based on average prices over a year, while market sensitivities are calculated using year-end prices, ignoring the ‘middle part’ of the stock price path over a year. How will this affect estimated stock market sensitivities? How do the returns for the market portfolio and returns on average fund prices move together? We start from our definition of market sensitivity P m Covðn1 ni¼1 Spi =Sp0 , Sm n =S0 Þ , ðA1Þ rev ¼ m m VarðSn =S0 Þ

p where Sm i (Si ) denotes the market (portfolio) level at p time i. Without loss of generality we set Sm 0 ¼ S0 ¼ 1. Assume that follow P portfolio and market Pn prices m m ¼ S þ DS , where Spn ¼ Sp0 þ ni¼1 DSpi and Sm n 0 i i¼1 VarðDSm Þ ¼ m2 n, such that X  n  m m Var Sn ¼ Var DSi ¼ nVarðDSÞ ¼ m2 n2 : ðA2Þ i¼1

Xn

i¼1



Sp2

DSm i :

ðA3Þ

DSpi

¼ DSm Let us assume further that i þ "i , Xn p m 1 Si , Sn Þ Covðn  i¼1m    ¼ Var DS1 þ ððn  1Þ=nÞVar DSm 2   ðn  ðn  1ÞÞ Var DSm þ  þ n n ¼ ð1 þ ððn  1Þ=2ÞÞm2 n: Substitute and into and we arrive at   2   1 þ n1  1 þ n1  2 m n 2 ¼ : ¼ rev ¼ 2 2 n!1 n m n 2

ðA4Þ

ðA5Þ

Even though asset-based fees in the above example are determined by market returns, regression betas will equal half the market beta as a result of the averaging process. Appendix B: Gibbs sampling for the random coefficient model with hierarchical prior We can perform the following calculations for our model to in the (arbitrary) order given below. To initialize the first calculation in a Markov chain the missing parameters need some assumptions. Depending on how lucky we are with our initial assumptions, this might mean that early simulations arrive at unlikely values. It is therefore common practice to discard the first number of draws (the so-called burn-in period). Gibbs sampling consists of a large number of cycles though the following set of equations. First, we sample the individual exposures from i  Nðpi , pi i Þ, ðB1Þ

pi i ¼ ðh½ 1

rup av

0 rdn av  ½ 1

rup av

1 1 rdn av  þ  Þ ,

ðB2Þ

0 1 ðB3Þ pi ¼ pi i ðh½ 1 rup rdn av av  revi þ  Þ: An obvious way to initialize the Markov chain would be to use equation-by-equation OLS to arrive at an initial estimate for pi , pi i . Note that all posterior values (parameters calculated repeatedly throughout the chain for statistical inference) are indicated by superscript ‘p’, while all priors (our input) are indicated by superscript ‘0’. Next, we draw the pool parameters according to our hierarchical prior structure

  Nðp , p Þ, 1  N X p 1,0 0 1 p  ¼   i þ   ,

p

¼

ðN 1 

þ

i¼1 1 1,0  Þ ,

ðB4Þ

ðB5Þ

1556

B. Scherer p p p 1

1   Wðv , ðv  Þ Þ,

 p ¼ TN þ  0 , (P up up 0 N i¼1 ðrevi,t  i,0  i,up rav, t  i,dn rav, t Þ

ðB6Þ

vp ¼ N þ v 0 ,

p ¼

N X

ði  Þði  Þ0 þ 0 :

ðB7Þ

s2,0 ¼

up 0 2,0 ðrevi,t  i,0  i,up rup av, t  i,dn rav, t Þ þ  s

p

:

ðB9Þ

i¼1

Finally, we draw the posterior for the error precision from h  Gðs2,0 ,  0 Þ,

)

ðB8Þ

All random draws only involve either a normal, gamma or Wishart distribution and as such are easy to perform with standard software.

Appendix C: Data description BLOOMBERG CODE AB:US($) AMG:US($) BAM:US ($) BEN:US($) BLK:US($) DHIL:US($) EV:US($) FII:US($) GBL:US ($) GROW:US ($) IVZ:US ($) JNS:US($) LM:US ($) SMHG:US($) TROW:US ($) WDR:US($) WHG:US($)

COMPANY

WEBSITE

AllianceBernstein Holding LP Affiliated Managers Group Inc Brookfield Asset Management Inc Franklin Resources Inc BlackRock Inc Diamond Hill Investment Group Inc Eaton Vance Corp Federated Investors Inc GAMCO Investors Inc US Global Investors Inc Invesco Ltd Janus Capital Group Inc Legg Mason Inc Sanders Morris Harris Group Inc T Rowe Price Group Inc Waddell & Reed Financial Inc Westwood Holdings Group Inc

www.alliancebernstein.com www.amg.com www.brookfield.com www.franklintempleton.com www.blackrock.com www.diamond-hill.com www.eatonvance.com www.federatedinvestors.com www.gabelli.com www.usfunds.com www.invesco.com www.janus.com www.leggmason.com www.smhgroup.com trow.client.shareholder.com www.waddell.com www.westwoodgroup.com

Data sources: Company reports and press releases, Bloomberg, Capital IQ, Google Finance, and Yahoo Finance.

MEASURING BASIS RISK LONGEVITY HEDGES

IN

Johnny Siu-Hang Li* and Mary R. Hardy†

ABSTRACT In examining basis risk in index longevity hedges, it is important not to ignore the dependence between the population underlying the hedging instrument and the population being hedged. We consider four extensions to the Lee-Carter model that incorporate such dependence: Both populations are jointly driven by the same single time-varying index, the two populations are cointegrated, the populations depend on a common age factor, and there is an augmented common factor model in which a population-specific time-varying index is added to the common factor model with the property that it will tend toward a certain constant level over time. Using data from the female populations of Canada and the United States, we show the augmented common factor model is preferred in terms of both goodness-of-fit and ex post forecasting performance. This model is then used to quantify the basis risk in a longevity hedge of 65-year old Canadian females structured using a portfolio of q-forward contracts predicated on U.S. female population mortality. The hedge effectiveness is estimated at 56% on the basis of longevity valueat-risk and 81.61% on the basis of longevity risk reduction.

1. INTRODUCTION Pension plan sponsors face a myriad of risks, one of which is longevity risk that arises from the increasing life expectancy trends among pensioners. Traditionally plan sponsors manage longevity risk by relying heavily on mortality improvement scales, such as the AA Scale in the Society of Actuaries 1994 Group Annuity Mortality Table (Society of Actuaries Group Annuity Valuation Table Task Force 1995). The scales are applied to a base mortality table to produce a mortality projection, which is then used to calculate pension liabilities. However, recent acceleration in longevity improvement has exceeded the scale factors significantly.1 This means that pensions need to be paid much longer than expected, raising the value of plan sponsors’ obligation to their members. Significant underestimates of past longevity improvements have made plan sponsors more aware of the threat of longevity risk. In sharp contrast to the random variations between lifetimes of individuals, longevity risk affects all pension policies and, therefore, cannot be diversified by increasing the size of the portfolio. Recently some plans have made attempts to transfer their longevity risk exposures to other parties. One way is to purchase bulk annuities from life insurers. In recent years the United Kingdom has seen an increasing number of bulk annuity transfers between pension plans and major insurers such as AVIVA, Legal and General, and Prudential. Another way is to rely on the capital markets by writing contracts that are linked to longevity improvements. Although the market for such contracts is still in its infancy, it is expected that it will develop and mature, because a strong demand exists

* Johnny Siu-Hang Li, PhD, FSA, holds the Fairfax Chair in Risk Management in the Department of Statistics and Actuarial Science at the University of Waterloo, Waterloo, Ontario, Canada, [email protected]. † Mary R. Hardy, PhD, FIA, FSA, CERA, holds the CIBC Chair in Financial Risk Management in the Department of Statistics and Actuarial Science at the University of Waterloo, Waterloo, Ontario, Canada, [email protected]. 1 See Continuous Mortality Investigation Bureau (1999, 2002) and Li et al. (2007).

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from pension plans and investors who are interested in securities that have little correlation with standard market risk factors. Broadly speaking, there are two types of mortality-linked contracts. The first type is contracts that are linked to the actual mortality experience of the pension plan’s own portfolio. Examples include the longevity swap agreed between Canada Life and JPMorgan in 2008 and the longevity swap agreed between Babcock International and Credit Suisse in 2009.2 With this type of contract, the hedger (the pension plan) can create an exact hedge that completely eliminates its longevity risk exposure. Nevertheless, contracts of this type are difficult to price because mortality data from an individual pension plan are often limited. The uncertainty involved in pricing makes them costly. Furthermore, as they are customized to the hedgers’ own risk characteristics, contracts of this type have poor liquidity. Hedgers will find it difficult to adjust or unwind their hedges in the future. The second type is standardized contracts that are based on the mortality experience of a certain national population. One example is a product called a q-forward, which was launched by JPMorgan in 2007. This product is a derivative involving the exchange of the realized mortality rate at some future date, in return for a fixed mortality rate agreed at inception. The reference mortality rate is determined by a LifeMetrics Index, which is derived from the demographic statistics of a national population. As of this writing, the LifeMetrics Index is available for England and Wales, the United States, the Netherlands, and Germany. Given that contracts of this type are based on broad population mortality indexes, they are likely to be less costly and more liquid, but the disadvantage is that they do not, in general, completely eliminate the hedger’s longevity risk exposures. The residual risk is called population basis risk. In more detail, population basis risk refers to the risk associated with the difference in mortality experience between the population of individuals associated with the hedging instrument and the population of individuals associated with the underlying exposure. Such risk exists due to, for example, differing profiles of socioeconomic group, lifestyle, and geography. Population basis risk does not necessarily mean the hedge is ineffective, but in designing hedging solutions it is important to ensure that any basis risk is acceptably low. As such, there is a need for a model that helps us measure the population basis risk involved in a longevity hedge. Moreover, having a sound method for measuring basis risk will encourage pension plans to use standardized longevity securities for hedging purposes. This will, in turn, stimulate a greater demand, facilitating the development of the market for standardized longevity securities. Unfortunately, existing stochastic mortality models, by and large, are designed for modeling the demographics of a single population only. Although it is technically possible to run simulations on the basis of two independent stochastic mortality models, the resulting estimate of population basis risk is likely to be too large because positive correlations are not taken into account. The primary objective of this study is to develop a stochastic model for measuring population basis risk. In particular, we consider several variants of the Lee-Carter model (Lee and Carter 1992), which has been widely used in actuarial science and other areas (see, e.g., Li et al. 2009). We illustrate the proposed basis risk model with a hedge, formed by JPMorgan’s q-forward contracts, for a hypothetical pension plan in Canada, for which a LifeMetrics Index is not available. The rest of this article is organized as follows: Section 2 gives a brief description of q-forwards and the LifeMetrics Index; Section 3 discusses different methods for modeling mortality of a group of populations; and Section 4 applies the proposed basis risk model to a hypothetical pension plan and details how the longevity hedge is formed, and examines the materiality of population basis risk to the effectiveness of the hedge. Section 5 examines, on top of population basis risk, how finite sample risk will affect the hedge effectiveness if the pension plan is not sufficiently large. Section 6 concludes

2 Information regarding these trades can be found in the following articles: ‘‘Canada Life Hedges Equitable Longevity with JPMorgan Swap,’’ Life and Pensions, Oct. 2008: 6; ‘‘JPMorgan Longevity Swap Unlocks UK Annuity Market,’’ Trading Risk, Sept. / Oct. 2008: 3; ‘‘Credit Suisse Signed Babcock Longevity Deal,’’ Reuters UK, May 13, 2009.

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the paper. The data we use for illustrative purposes are obtained from the LifeMetrics website (www.jpmorgan.com/lifemetrics) and the Human Mortality Database (2009).

2.

Q-FORWARDS AT A

GLANCE

This section briefly describes the mechanism of q-forward contracts on which illustrations in later parts of this paper are based. Most of the materials in this section are drawn from a technical document written by Coughlan et al. (2007b). A q-forward is a zero-coupon swap that exchanges on the maturity date a fixed amount, determined at time 0, for a random amount that is proportional to the LifeMetrics Index for a certain population (the reference population) in some future time (the reference year). In more detail the LifeMetrics Index on which the random payment is based is the realized single-year death probability, which has been graduated to eliminate noise from the raw data.3 The index is available by gender and by single year of age. Because there is a lag in the availability of the index data, the reference year is slightly earlier than the maturity date. On the other hand, the fixed payment is proportional to the so-called forward mortality rate for the reference population. This rate is chosen so that no payment changes hands at the inception of trade. However, at maturity, a net payment will be made by one counterparty or the other. The settlement that takes place at maturity is illustrated diagrammatically in Figure 1. A pension plan must pay out pension benefits to its members on the basis of realized mortality rates. If realized mortality rates are smaller than expected, then pension payments, on the whole, will last longer, which means larger pension liabilities. So a pension plan wishing to hedge longevity risk could enter into a portfolio of q-forward contracts in which it receives fixed mortality rates and pays realized mortality rates. At maturity, the portfolio of q-forward contracts will pay out to the pension plan an amount that increases as mortality rates fall to offset the correspondingly higher value of pension liabilities. Equivalently speaking, it stabilizes the value of pension liabilities with respect to changes in mortality rates. As in hedging other sources of risk, the hedger needs to calibrate the portfolio carefully to ensure that it includes an appropriate mixture of q-forwards that are linked to mortality rates of different ages and time horizons. We will revisit this issue in Section 5. There is no free lunch. A pension plan wishing to lay off its longevity risk exposures with q-forwards must pay the other counterparty a risk premium as a compensation for taking on the risk. The price for hedging longevity risk is reflected in the forward mortality rate determined at the inception of the contract. To attract investors (fixed-rate payers), the forward mortality rate must be smaller than the corresponding expected mortality rate, so that on average (i.e., if mortality is realized as expected), the investors will be paid. The difference between expected and forward rates, therefore, indicates the expected risk premium to the investor of taking on longevity risk.

Figure 1 Settlement of a q-Forward Contract at Maturity

3 The data sources, methodologies, algorithms, and calculations used in the production of the LifeMetrics Index are fully disclosed in the Appendix of Coughlan et al. (2007a).

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Figure 2 Expected and Forward Mortality Rates

Other than investors, life insurers with the liability to pay death benefits may also be interested in taking the position of a fixed rate payer. In this situation the investment bank can act as an intermediary, standing in the middle between life insurers and pension plans. The intermediation provided by the investment bank can enhance liquidity and simplify the management of counterparty default risk. We refer readers to Coughlan (2009) for a further discussion of intermediation. Coughlan et al. (2007b) argue that, given the required Sharpe ratio, the forward mortality rate can be determined by the following equation: q f ⫽ (1 ⫺ T ⫻ ␭ ⫻ ␴)q e, where q f is the forward mortality rate, q e is the expected mortality rate, T is the time to maturity, ␭ is the required annualized Sharpe ratio, and ␴ is the volatility (standard deviation) of changes in the mortality rate in question. In Figure 2 we plot the forward mortality rates for the cohort of U.S. females who are aged 65 in 2008, assuming that the required Sharpe ratio ␭ is 0.25. Also shown is the corresponding expected mortality rates derived from a Lee-Carter mortality projection. The widening divergence between the two curves reflects that investors demand a higher risk premium from a q-forward that is linked to the mortality rate at an older age of the cohort. The is partly because death rates at older ages are more volatile (␴ increases) and partly because predictions further into the future involve more uncertainty (T increases). Pricing q-forwards itself is a huge research area. Sophisticated pricing methods are beyond the scope of this paper. We refer readers to Cairns et al. (2006), Li and Ng (2010), and Lin and Cox (2005, 2008) for details of pricing methods that have been proposed in recent years.

3. MODELS

FOR

MEASURING BASIS RISK

To measure basis risk, what we need is a stochastic model that projects mortality of two populations simultaneously. However, most existing stochastic mortality models, including those that can be im-

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plemented with the LifeMetrics software,4 are originally designed for modeling mortality of a single population only. In this section we explore four possible solutions, which are based on the well-known Lee-Carter model (Lee and Carter 1992).

3.1 Independent Modeling It is possible to model mortality of two populations using two basic Lee-Carter models that are not related to each other. We let m(x, t, i) be the central rate of death at age x and in year t for population i. Mathematically, we may model m(x, t, i), i ⫽ 1, 2, with the following set of equations: ln(m(x, t, i)) ⫽ a(x, i) ⫹ b(x, i)k(t, i) ⫹ ε(x, t, i), i ⫽ 1, 2, where

• • • •

a(x, i) is an age-specific parameter indicating population i’s average mortality level at age x, k(t, i) is a time-varying index representing the overall speed of mortality improvement for population i, b(x, i) is an age-specific parameter indicating the sensitivity of ln(m(x, t, i)) to k(t, i) and, ε(x, t, i) is the error term that captures all remaining variations.

We estimate parameters a(x, i), b(x, i), and k(t, i) from historic data using the method of singular value decomposition (SVD). Specifically we set a(x, i) to the arithmetic average of ln(m(x, t, i)) over time and apply SVD the matrix of ln(m(x, t, i) ⫺ a(x, i)). The first left and right singular vectors resulting from the SVD give an estimate of b(x, i) and k(t, i), respectively. Alternatively the model Figure 3 Estimates of Parameters in Independent Lee-Carter Models

4 The LifeMetrics software, which is a part of the LifeMetrics toolkit, is available to aid users in calibrating, forecasting, and simulating mortality rates. The software can be downloaded from the LifeMetrics website (www.jpmorgan.com / lifemetrics).

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Figure 4 Trends of Central Death Rates for Canadian Female Population (Population 1) and U.S. Female Population (Population 2)

parameters can be estimated by the method of maximum likelihood (Wilmoth 1993; Brouhns et al. 2002) or by Markov chain Monte Carlo (Czedo et al. 2005). As an example, we fit the equations to female mortality data from Canada (population 1) and the United States (population 2). The resulting parameter estimates are shown graphically in Figure 3. To capture trends of longevity improvement, we model historic values of k(t, i), i ⫽ 1, 2, by two independent autoregressive integrated moving average (ARIMA) processes. In most cases, including our example, k(t, i) can be adequately modeled by an ARIMA(0,1,0) process: k(t, i) ⫽ c ⫹ k(t ⫺ 1, i) ⫹ ␰(t, i), where c is the drift term and {␰(t, i)} is a sequence of iid normal random variables. We assume here that ␰(t, 1) and ␰(t, 2) are independent. Note that this process is also called a random walk with drift. The ARIMA processes are extrapolated to give a forecast of future death rates. In particular, if, given the ARIMA processes, the projected mortality index in a future year s is kˆ(s, i), then the central estimate of m(x, ˆ s, i) is given by

ˆ i)k(s, ˆ i)), m(x, ˆ s, i) ⫽ exp(a(x, ˆ i) ⫹ b(x, ˆ i) are the SVD estimates of a(x, i) and b(x, i), respectively. where ˆ a(x, i) and b(x,

(1)

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This approach is straightforward to implement. However, it completely ignores the dependence between mortality rates of the two populations. In Figure 4 we depict trends of mortality rates for the female populations of Canada and the United States. These trends are roughly going in the same direction, suggesting a potential statistical dependence between them. Thus, we expect that population basis risk will be overstated if this method of mortality projection is used.

3.2 The Joint-k Model The joint-k model, introduced by Carter and Lee (1992), assumes that mortality rates of both populations are jointly driven by a single time-varying index. Mathematically, the model can be expressed as follows: ln(m(x, t, i)) ⫽ a(x, i) ⫹ b(x, i)k(t) ⫹ ε(x, t, i), i ⫽ 1, 2, where k(t) is a time-varying index that drives changes in the mortality rates for both populations. The same as the previous approach, we estimate a(x, i) by setting it to the average of m(x, t, i) over time. The other parameters can be estimated by applying SVD to the matrix that contains values of ln(m, x, i) ⫺ a(x, i) for both populations. In Figure 5 we show the estimates of b(x, i) and k(t) when the model is applied to data from the female populations of the United States and Canada. Note that the estimates of a(x, i) are just the same as those shown in Figure 3. Historic estimates of k(t) are further modeled by a suitable ARIMA process, which is then extrapolated to give a forecast of future mortality rates. Attractive statistical and demographic reasons are found for using the joint-k model. From a statistical viewpoint, a single time-varying parameter is a parsimonious way of linking the mortality trajectories of the two populations. Demographically, a single driving force may enforce greater consistency, preventing mortality rates of the two populations from changing in totally different ways. Nevertheless, when it comes to modeling population basis risk, the joint-k model may not be ideal. We expect that

Figure 5 Estimates of b(x, i ) and k(t) in Joint-k Model

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it will understate the actual level of basis risk, since the single driving force implies mortality rates of the two populations are perfectly correlated.

3.3 A Co-integrated Lee-Carter Model The main problem of the independent Lee-Carter model is that it implies no connection between mortality rates of the two populations of interest. One way to overcome this problem is, instead of using two independent processes, to treat k(t) ⫽ (k(t, 1), k(t, 2))⬘ as a vector and model it with a bivariate random walk with drift, which can be expressed as follows: k(t) ⫽ c ⫹ k(t ⫺ 1) ⫹ ␰(t),

(2)

where c is a vector of drift terms and {␰(t)} is a sequence of serially uncorrelated random vectors with mean zero and variance-covariance matrix 兺. It is assumed that 兺 is positive definite and that ␰(t) follows a bivariate normal distribution. On the basis of equation (2), we can predict future values of k(t,1) and k(t,2). Substituting these values into equation (1), we can readily obtain a projection of future mortality rates. The central projections derived from this and the independent Lee-Carter models are identical. However, this model yields more precise interval projections as it takes account of the potential dependence through the variance-covariance matrix 兺, rather than leaving mortality rates of the two populations unconnected. The process specified by equation (2) contains two unit roots (stochastic trends), one for k(t,1) and the other for k(t,2). By unit root we mean that the series {k(t,i)} is nonstationary, but its first difference, {⌬k(t,i)} is stationary.5 The intuition behind the two unit roots is that two driving forces exist that govern the longevity improvement of the two populations. It is, however, entirely possible that these two driving forces are indeed the same. Econometricians call this situation cointegration. Cointegration determines how the time-series process should be postulated. More precisely, if cointegration exists, the time-series process should consist of only one, rather than two, random walk. As the time-series process directly affects the resulting predictions of population basis risk, it must be carefully considered. Technically speaking, cointegration refers to the situation when (1) {k(t,1)} and {k(t,2)} are both nonstationary, and (2) there exists a constant ␤ such that the series {k(t,1) ⫺ ␤k(t,2)} is stationary. What we need to know is whether or not they are cointegrated, and if they are cointegrated, what is the value of ␤. Several statistical procedures are used for testing cointegration. An example is Engle and Granger’s (1987) three-step procedure, which is summarized as follows: 1. One condition of cointegration is that both {k(t,1)} and {k(t,2)} are nonstationary. This condition can be confirmed by running the augmented Dickey-Fuller test (Said and Dickey 1984) on {k(t,1)} and {k(t,2)}.6 The test is based on the following equation:

冘 ⌬k(t ⫺ j, i) ⫹ ␥k(t ⫺ 1, i) ⫹ ␰(t), p

⌬k(t, i) ⫽

j⫽1

where ⌬k(t,i) ⫽ k(t,i) ⫺ k(t ⫺ 1,i), ␰(t) denotes the error term, and p is the number of lags required to capture the autocorrelation effects in {⌬k(t)}. The null hypothesis (H0) of this test is ␥ ⫽ 0, that is, {k(t,i)} is unit root nonstationary, while the alternative (H1) is ␥ ⬍ 0. If H0 is not rejected for both series, go on to the next step.

5

Here we refer to weakly stationary. A time series {rt} is weakly stationary if both the mean of rt and the covariance between rt and rt⫺1 are time invariant, where l is an arbitrary integer. Weak stationarity implies that the time plot of data would show a fluctuation with a constant variation around a constant level. 6 This test is an extension of the original Dickey-Fuller test (Dickey and Fuller 1979).

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Figure 6 ˆ k(t,2) Stationarity of k(t,1) ⴚ ␤

2. Estimate ␤ by running the following linear regression: k(t,1) ⫽ ␣ ⫹ ␤k(t,2) ⫹ u(t),

(3)

where ␣ is a constant. Note that k(t,1) and k(t,2) are cointegrated if the series {u(t)} is stationary. 3. Examine if {u(t)} is stationary. This can be achieved by performing the augmented Dickey-Fuller test on {u(t)}. Specifically, we test for H0 ⬊ ␲ ⫽ 0 in the following equation:

冘 ⌬u(t ˆ ⫺ j) ⫹ ␲u(t ˆ ⫺ 1) ⫹ e(t), p

⌬u(t) ˆ ⫽

j⫽1

where ˆ u(t) denotes the estimate of u(t), e(t) is the error term, and p is the number of lags required to capture the autocorrelation effects in {⌬u(t)}. ˆ Let us revisit the example based on female populations of Canada and the United States. First, we apply the augmented Dickey-Fuller test to {k(t,1)} (for Canada) and {k(t,2)} (for the United States). The values of the test statistic are ⫺1.20 and ⫺0.88, respectively. Both are not greater than the critical value at 10% significance level in magnitude, indicating no evidence against the first condition for ˆ ⫽ 1.18 as an estimate of ␤ in equation cointegration. Second, we regress k(t,1) on k(t,2) and obtain ␤ (3). Finally, we apply the augmented Dickey-Fuller test on the residual series {u(t)} in equation (3). The resulting test statistic is ⫺2.58, providing evidence against H0 ⬊ ␲ ⫽ 0 at 10% level of significance. So we conclude that {u(t)} is stationary, and hence k(t,1) and k(t,2) are cointegrated. The fact that k(t,1) and k(t,2) are cointegrated can be seen graphically. From the bottom panels of Figure 3 we observe clear trends in both k(t,1) and k(t,2). The expectations of k(t,1) and k(t,2) are clearly changing with time, suggesting that both series are not (weakly) stationary. In Figure 6 we plot ˆ k(t,2). In this diagram we observe no apparent trend, indicating that this series the series of k(t,1) ⫺ ␤ is likely to be stationary, which means k(t,1) and k(t,2) are cointegrated.

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If k(t,1) and k(t,2) are cointegrated, we can no longer rely on the process specified by equation (2) to generate sample paths of future mortality rates. However, future values of k(t,1) and k(t,2) can be conveniently simulated using Phillips’s (1991) triangular representation, which can be expressed as follows: k(t,1) ⫽ c ⫹ k(t ⫺ 1,1) ⫹ ␰(t,1),

(4)

k(t,2) ⫽ ␣ ⫹ ␤k(t,1) ⫹ u(t),

(5)

where ␣, ␤, and c are constants and ␰(t,1) and u(t) are error terms, normally distributed with zero mean. The existence of cointegration has deep demographic meanings. First, cointegration means that the random walks (stochastic trends) for k(t,1) and k(t,2) cancel out each other. Hence, longevity improvements for the two populations are governed by only a single (unobserved) driving force, but not two. Second, an equilibrium relation exists between mortality rates of the two population. The equilibrium relation is specified by equation (5) in the triangular representation. Third, random departures are seen from the equilibrium relation. Such departures are determined by the random term u(t) in equation (5). The main problem of this approach is that it is sometimes not easy to determine if k(t,1) and k(t,2) is cointegrated (i.e., to decide if equation [2] or the combination of equations [4] and [5] should be used). This problem arises because the augmented Dickey-Fuller test has a rather low power. As a result, a high chance exists that the null hypothesis in step (3) of Engle and Granger’s (1987) procedure is not rejected, but that k(t,1) and k(t,2) are indeed cointegrated.

3.4 The Augmented Common Factor Model In the second half of the twentieth century, there was, in general, a global convergence in mortality levels. This phenomenon can be seen in Figure 7, in which we graph the life expectancy for various

Figure 7 Life Expectancy at Birth: Canada, France, and Japan

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countries, and is documented in White (2002), Wilson (2001), and United Nations (1998). Using two independent Lee-Carter models (our first approach) is likely to result in an increasing divergence in life expectancy in the long run, counter to this expected and observed trend toward convergence.7 The joint-k model (our second approach) does not solve the problem completely because the difference in parameter b(x,i) between two populations can still lead to diverging mortality forecasts. Li and Lee (2005) proved that the necessary conditions that the Lee-Carter forecasts of m(x, t,1) and m(x, t,2) will not diverge are 1. b(x,1) ⫽ b(x,2) for all x 2. The drift terms of the ARIMA(0,1,0) processes for k(t,1) and k(t,2) are identical. Given the above conditions, Li and Lee considered a variant of the Lee-Carter model in which b(x,1) ⫽ b(x,2) ⫽ B(x) for all x and k(t,1) ⫽ k(t,2) ⫽ K(t) for all t; that is, ln(m(x,t,i)) ⫽ a(x,i) ⫹ B(x)K(t) ⫹ ε(x,t,i), i ⫽ 1, 2,

(6)

where ε(x,t,i) is the error term. This model, which they call the common factor model, treats the two populations as a single group. Parameters B(x) and K(t) capture the central tendencies within the group of populations.8 As before, we estimate a(x, i) by setting it to the average of ln(m(x,t,i)) over time. To estimate B(x) and K(t), we apply SVD to the matrix of 兺2i⫽1 w(i)(ln(m(x,t,i)) ⫺ a(x,i)), where w(i) is the weight on population i. We can determine w(i) according to the population sizes. Historic K(t) is then modeled by a random walk with drift: K(t) ⫽ c ⫹ K(t ⫺ 1) ⫹ ␰(t), where c is a constant and ␰(t) is the error term, iid normal with zero mean. It can be shown easily that this model implies the same longevity improvements for both populations at all times. Consequently it will predict zero basis risk, which is, of course, unrealistic. Further, this model has a far more stringent structure and may therefore underfit the data significantly. To improve the model, Li and Lee introduced a population-specific factor to equation (6). The resulting model, which they call the augmented common factor model, becomes ln(m(x,t,i)) ⫽ a(x,i) ⫹ B(x)K(t) ⫹ b(x,i)k(t,i) ⫹ ε(x,t,i), i ⫽ 1, 2,

(7)

where k(t,i) is a time-varying factor, specific to population i, and b(x,i) denotes the sensitivity to k(t,i). We obtain b(x,i) and k(t,i) by using the first-order vectors in the SVD applied to the residual matrix of the common factor model (i.e., the matrix of ε(x,t,i) in equation [6]). We fit the augmented common factor model to data from the female populations of Canada and the United States. The resulting parameter estimates are depicted in Figure 8. Note that values of a(x,i) for this model are identical to those shown in Figure 3. Unlike the previous approaches, k(t,i) in equation (7) is modeled in such a way that it will tend toward a certain constant level over time. Specifically, we use a first-order autoregressive model (AR(1)), that is, k(t,i) ⫽ ␾0(i) ⫹ ␾1(i)k(t ⫺ 1,i) ⫹ ␨(t,i), where ␾0(i) and ␾1(i) are constants, and ␨(t,i) is the error term, iid normal with zero mean. We require 兩␾1(i)兩 ⬍ 1 for i ⫽ 1, 2 so that the model yields a bounded short-term trend in k(t,i). In this way the fitted model will accommodate some continuation of historical convergent or divergent trends for each population before it locks into a constant relative position in the hierarchy of long-term forecasts of group mortality.

7 Tuljapurkar et al. (2000) applied the Lee-Carter model to the G7 countries in isolation and found that over a 50-year forecast horizon, the largest gap in life expectancy among these seven countries is eight years. 8 Lee and Nault (1993) used a similar approach to forecast provincial mortality for Canada.

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Figure 8 Estimates of Parameters in the Augmented Common Factor Model

3.5 Comparing the Models We now compare the models from two perspectives: (1) How well they fit to historic data and (2) whether or not the forecasts they produce are consistent with historic data. The comparison is based on the application of the models to data from the female populations of Canada and the United States. There are a number of ways to examine the the models’ goodness-of-fit. Li and Lee (2005) use an explanation ratio, ER, which can be written as

冘 ε(x,t,i)

2

ER ⫽ 1 ⫺

冘 (ln(m(x,t,i)) ⫺ a(x,i)) . x,t,i

2

x,t,i

We can interpret ER as the proportion of variance in historic m(x,t,i) explained. A higher value of ER thus indicates a better fit to data. The values of ER for all models we discussed are shown in the second column of Table 1. In this application the highest value of ER is attained by the augmented common

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Table 1 Values of ER, l, and BIC Derived from Different Models Model

ER

l

BIC

Independent Joint-k Cointegrated Common factor Augmented common factor

0.9891 0.9884 0.9891 0.9655 0.9931

⫺36,109 ⫺40,079 ⫺36,109 ⫺58,084 ⫺26,397

⫺37,064 ⫺40,822 ⫺37,064 ⫺58,694 ⫺27,697

factor model. Note that the cointegrated model provides the same goodness-of-fit as the independent model because they have an identical structure.9 Another way is to compare the models’ log-likelihood. Assuming Poisson death counts, the loglikelihood l can be calculated by the following formula: l⫽

ˆ ˆ ⫹ D(x,t,i) ln(D(x,t,i)) ⫺ ln(D(x,t,i)!)), 冘 (⫺D(x,t,i)

x,t,i

ˆ (x,t,i) are, respectively, the actual and expected number of deaths at age x and where D(x,t,i) and D ˆ (x,t,i) can be expressed as in year t among individuals in population i. For the independent model, D ˆ ˆ ˆ D(x,t,i) ⫽ E(x,t,i) exp(a(x,i) ⫹ b(x,i)k(t,i)), ˆ where E(x,t,i) is the number of persons-at-risk at age x and in year t for population i. Values of ˆ (x,t,i) for other models can be derived in a similar manner. As in usual statistical inference, a higher D value of l indicates a better fit. The values of l for all models we discussed are shown in the third column of Table 1. We observe that the conclusions based on ER and l have no difference. Given that the augmented two-factor model has more parameters, it is not surprising that it gives a higher value of R and l than the other models do. A fairer measure of goodness-of-fit would be one that penalizes for having additional parameters. Here we consider the Bayesian Information Criterion (BIC) (Schwarz 1978), which is defined by l ⫺ 0.5j ln(n), where n is the number of data points and j is the number of parameters in the model. This selection criterion is also used by Cairns et al. (2009) in a quantitative comparison of stochastic models. It is clear from the definition of BIC that a model with a higher value of BIC is more preferable. The values of BIC are shown in the fourth column of Table 1. We observe that, even after penalizing for the extra parameters, the augmented common factor model is still ranked the highest among all models we considered. It may be possible that a model gives a good in-sample fit but produces poor ex post forecasts, that is, forecasts that are significantly different from the realized outcomes. Hence, there is a need to conduct, on top of the goodness-of-fit tests, a backtest to evaluate the ex post forecasting performance of the models we discussed. Of interest in this backtest is the models’ ability to forecast mortality differentials between two populations. Accordingly, we base the backtest10 on a metric d(t), which is defined by a(65, t,1) ⫺ a(65, t,2), where a(65,t,i) is the price of a whole life annuity due of $1 sold to an individual aged 65 in population i. The price is calculated on the basis of the period mortality rates at time t and an interest rate of 2%. The backtest is conducted as follows:

9

The models differ in the way that their time-varying indexes, k(t,1) and k(t,2), are extrapolated. This backtest belongs to the category of ‘‘expanding horizon backtests’’ defined by Dowd et al. (2010).

10

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1. Fit the models to data over a restricted sample period (1968–1995); 2. Given the fitted models, make forecasts of d(t) up to 10 years ahead; 3. Compare the forecasts against realized outcomes; the models’ performance is reflected in the degree of consistency between the realized outcomes and the prediction interval associated with each forecast. The comparison between the forecasts and the realized outcomes is shown in Figure 9. In the diagrams we show both median forecasts (solid lines) and 95% prediction intervals (dotted lines). Roughly speaking, if a model is adequate, we can be 95% confident that the prediction intervals contain the actual values. Figure 9 Actual and Predicted Values of d(t), t ⴝ 1996, . . . , 2005, on the Basis of Models Fitted to Data over the Period 1968–1995

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It is clear that the common factor model underperforms the others. The median forecast is substantially biased low, and the prediction interval is too narrow, understating the uncertainty associated with the predictions. The problem of having an overly narrow prediction interval is also found in the joint-k model. It is likely to underestimate population basis risk if these two models are used. The cointegrated model generates a prediction interval that roughly matches the variability of the realized outcomes. Nevertheless, the forecast is biased low, with about half of the realized values exceeding the upper limit of the 95% prediction interval. The forecasts derived from the remaining two models look reasonable and deserve a closer scrutiny. First, let us focus on the median forecasts. We define by xM the proportion of realized outcomes falling below a median forecast. If the median forecasts are adequate, we would expect xM to be about 50%. The indepedent model gives xM ⫽ 20%, while the augmented common factor model gives xM ⫽ 40%. In this regard the augmented common factor model is more adequate. Next, we turn to the prediction intervals. Although both intervals include all realized outcomes, the one derived from the independent model seems too conservative relative to the historic volatility of d(t). The independent model predicts that, 10 years after the ‘‘stepping off’’ year (1995), the volatility of d(t) is 27.63%, while the realized volatility of d(t) over a period of 10 years is only 8.68%. The augmented common factor model predicts 12.78%, which is more in line with the realized outcomes. Overall, the augmented common factor model outperforms the others in both the goodness-of-fit tests and the ex post evaluation. For this reason, we will rely on this model in the example discussed in the next section.

4. QUANTIFYING BASIS RISK: AN EXAMPLE 4.1 Hedging Objectives Suppose that we are charged with hedging the longevity risk associated with a pension plan that pays the pensioner $1 at the beginning of each year. We assume that the pensioner is 65 years old at time 0 and is subject to the same longevity improvements as the female Canadian population. In particular, our primary hedging objective is to stabilize the present value of the cash flows that will be made to the pensioner in 35 years. The hedging instrument we use is a portfolio of q-forward contracts, which are chosen to maximize the degree of longevity risk reduction. As of this writing, there is no LifeMetrics Index linked to Canadian population mortality. In this example, we make use of q-forwards that are based on the U.S. population, since we found that its longevity improvements are the most stably related to that of the population of Canada.11 As a shorthand, we refer to Canada and U.S. populations as populations (1) and (2), respectively. Note that our hedge is subject to population basis risk. We evaluate hedge effectiveness by examining the present value of unexpected cash flows. If the plan is unhedged, the present value of unexpected cash flows can be expressed as X ⫽ V(q) ⫺ V(E(q)), where

• • •

11

V is the time-0 value of all cash flows payable to the pensioner in 35 years; we set time-0 to beginning of year 2006; q ⫽ {q(65,2006,1), q(66,2007,1), . . . , q(99,2040,1)} is the vector of (random) death probabilities for the cohort of interest; q(x,t,i) is the probability that an individual from population i dies between ages x and x ⫹ 1 (i.e., between years t and t ⫹ 1), given that he or she has survived to age x;

This conclusion is based on the volatility of historic d(t), which we defined in the previous section.

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E(q) is the expectation of q; we derive E(q) from a simple Lee-Carter projection of female Canadian mortality.

In contrast, if there is a longevity hedge for the plan, the unexpected cash flows can be written as X* ⫽ V(q) ⫺ V(E(q)) ⫺ H(q), where H is the present value of all payoffs from the q-forwards in the hedging portfolio. Of course, both X and X* are random quantities as they depend on a random vector q. The longevity hedge is said to be effective if X* is significantly less variable than X. As such, we will base our conclusions on the following two measures: 1. Longevity value-at-risk (VaR): From simulated distributions of X and X* we calculate the VaR values, VaR(X) and VaR(X*), at a 95% confidence level. A larger difference between these two values indicates a better hedge effectiveness. 2. Longevity risk reduction: We can measure hedge effectiveness in terms of the amount of risk reduction R in the following way: R⫽1⫺

␴2(X*) , ␴2(X)

where ␴2(X) and ␴2(X*) are the variances of X and X*, respectively. A higher value of R indicates a better hedge effectiveness. We calculate the two measures above from stochastic simulations that are based on the augmented common factor model.

4.2 The Longevity Hedge The hedge portfolio should be composed in such a way that a shift in the mortality curve q will result in the same change in V, the value of the plan, and H, the value of the hedge portfolio. So how many q-forwards are needed? A single q-forward is clearly insufficient, because, like the term structure of interest rates, the mortality curve does not always shift in parallel, or even proportionately across ages. A mixture of 35 q-forwards with different terms may give a perfect match, but writing such a large number of contracts may involve substantial transaction costs. Moreover, some of the required q-forwards may not be available.12 Calibration of q-forward hedges is described in Cairns et al. (2009) and Coughlan (2009). In particular, Cairns et al. (2009) derived formulas for an approximate q-forward hedge for a survivor index, assuming that death rates at different ages are independent of one another. However, a shock to a mortality rate is often accompanied by shocks to the mortality rates at neighboring ages. This property is potentially important when we use only a handful of q-forward contracts to mimic the pension liability, which depends on the entire mortality curve.13 Li and Luo (2011) in a parallel study propose a hedging framework that exploits the property of age dependence. To explain their framework, we need to introduce some concepts. First, we define a set of key mortality rates for the cohort of interest. In this example, we consider four key mortality rates that are equally spaced: q(70,2011,1), q(75,2016,1), q(80,2021,1), and q(85,2026,1). These are the rates on which the q-forwards in the hedge portfolio are based. Other sets of key mortality rates can be defined. For instance, if a hedger decides to use two q-forwards only, then he or she can specify two key mortality rates, 10 years apart. The choice of key mortality rates also depends on the availability of the associated q-forwards because the strategy requires q-forwards that are linked to the key mortality rates chosen. 12 In the early stages of the market’s development, transactions are restricted to a limited number of standardized q-forward contracts in which liquidity can be concentrated. 13 We refer interested readers to Wills and Sherris (2008) for a further discussion on the property of age dependence.

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Figure 10 Approximation of the Shift in q as a Sum of s(x,j,␦( j)), j ⴝ 1, 2, 3, 4, Arbitrary Mortality Rates

Second, we define s(x, j,␦( j)), j ⫽ 1, 2, 3, 4, by the shift in q at age x associated with a small change ␦( j) in the jth key rate. Mathematically, we define s(x,1,␦(1)) as follows: 0,

s(x,1,␦(1)) ⫽

冦

␦(1)(x ⫺ 65) , 5 ␦(1)(75 ⫺ x) , 5 0

x ⱕ 65 65 ⬍ x ⱕ 70 . 70 ⬍ x ⬍ 75 x ⱖ 75

The shifts for j ⫽ 2, 3, 4 are defined in a similar manner. The linear interpolation is used to approximate the diminishing dependence between two mortality rates as their ages become farther apart. In effect, the shift in the whole mortality curve can be approximated by the sum of s(x, j,␦( j)), j ⫽ 1, 2, 3, 4. This approximation, which we illustrate graphically in Figure 10, allows us to use only four q-forwards to mimic the pension plan. Third, we define key q-duration, which measures the sensitivity of a price to a small change in a key mortality rate.14 Suppose that there is an infinitesimal change ␦( j) to the jth key mortality rate. This change will affect the mortality curve according to the way we define s(x, j,␦( j)). We let q and ˜ q be the original and new mortality curves, respectively. Then the jth key q-duration of a security is given by

14 The key q-duration may be regarded as an extension of q-duration, which was first published as a concept in the LifeMetrics Technical Document (Coughlan et al. 2007b).

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Table 2 Comparison between Frameworks of Ho (1992) and Li and Luo (2011) Ho (1992) Hedging objective Source of risk Curve behavior Hedging instrument Sensitivity measurement

Li and Luo (2011)

Interest rate risk (Spot) yield curve

Longevity risk (Spot) mortality curve Not necessarily shift in parallel Zero-coupon bonds Standardized q-forwards Key rate duration Key q-duration

KQD(P(q), j) ⫽

P(q ˜ ) ⫺ P(q) , ␦( j)

(8)

where P(q) is the price of the security on the basis of the mortality curve q. At this point we can see that Li and Luo’s framework is largely analogous to Ho’s (1992) framework for hedging interest rate–sensitive contracts. The analogy between these two frameworks is summarized in Table 2. Finally, we assign weights to the four q-forwards in the hedge portfolio. To make V and H have a similar sensitivity to q, we need a notional amount of KQD(V(q), j) KQD(Fj(q), j) on the q-forward linked to the jth key mortality rate, where Fj is the present value of the payoff from the q-forward with a notional amount of $1. It is often difficult to analytically calculate the key q-duration, KQD(V(q), j), for the pension liability. For practical purposes, we may estimate KQD(V(q), j) as follows: 1. Take E(q) as q 2. Assuming ␦( j) is 10 basis points, calculate ˜ q 3. Compute KQD(V(q), j) with equation (8). The formula for KRD(Fj(q), j) depends on whether population basis risk exists. For now let us assume that q-forwards linked to population (1), that is, the population associated with the pension plan, are available and used in the hedge portfolio. Then, as an example, F1(q) can be written as F1(q) ⫽ 100(1 ⫹ r)⫺5(q f ⫺ q(70,2011,1)), where q f is the corresponding forward mortality rate, and r is the interest rate at which the cash flows are discounted.15 It is obvious that F1(q) is unaffected by changes in other key rates, and that KQD(F1(q),1) is simply ⫺100(1 ⫹ r)⫺5. Now suppose that q-forwards linked to population (1) are unavailable and that q-forwards linked to population (2) are used instead. In this case KQD(F1(q),1) will be ⫺100(1 ⫹ r)⫺5 multiplied by the adjustment factor ⭸q(70,2011,2) , ⭸q(70,2011,1) which can be calculated from the mortality model we use. For instance, if the augmented common factor model is used, the derivative can be calculated as follows: ⭸q(x,t,i) q(x,t,i)(1 ⫹ 0.5m(x,t, j))2A(x,t,i) ⫽ , ⭸q(x,t, j) q(x,t, j)(1 ⫹ 0.5m(x,t,i))2A(x,t, j) 15

t ⫽ 2007, 2008, . . . ,

(9)

Recall that the net payoff to the hedger at maturity is 100 times the difference between the fixed (forward) and realized mortality rates.

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Table 3 Key q-Durations and Notional Amounts Needed

KQD(V(q), j ) KQD(Fj (q), j ) Notional of the jth q-forward

jⴝ1

jⴝ2

jⴝ3

jⴝ4

⫺62.2737 ⫺134.1680 0.4641

⫺42.9323 ⫺99.7704 0.4303

⫺27.1303 ⫺84.1244 0.3225

⫺15.0107 ⫺71.8477 0.2089

where A(x,t,i) ⫽ B(x)c ⫹ b(x,i)␾1(i)t⫺2006(␾0(i) ⫹ (␾1(i) ⫺ 1)k(2006,i)). The above relation is proved in the Appendix. In Table 3 we show the key q-durations for the pension plan and the four q-forwards in the hedge portfolio. Also shown in this table is the required notional amount of each q-forward.

4.3 Empirical Results We simulate 5,000 mortality scenarios on the basis of the augmented common factor model. Assuming r ⫽ 2%, we derive from the scenarios empirical distributions of (1) X, (2) X* when there is no basis risk,16 and (3) X* when basis risk exists. The resulting distributions are displayed in Figure 11. We observe from Figure 11 that the longevity hedge significantly reduces the dispersion of the unexpected cash flows, even if basis risk exists. To quantify the hedge’s effectiveness, we calculate the value-at-risk at a 95% confidence level. Without a longevity hedge, the VaR is 0.5787. When basis risk is absent, the longevity hedge can reduce the VaR by 73% to 0.1544. Even if we resort to using qforwards linked to U.S. mortality, the VaR can still be reduced to 0.2558, 56% lower than that in the Figure 11 Simulated Distributions of X and X*, with and without Basis Risk

16

This is calculated by using q-forwards linked to Canadian mortality in the hedge portfolio.

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unhedged position. The difference of 0.1014 (0.2558 ⫺ 0.1544) can be regarded as a measure of basis risk involved in this hedge. We also calculate R, the amount of risk reduction. We obtain R ⫽ 94.26% in the absence of basis risk. The value of R is lowered to 81.61% when basis risk is present, but it still indicates that the longevity hedge is reasonably effective. In deriving the distributions we assumed that the q-forwards are ‘‘free,’’ that is, the forward mortality rate is identical to (not lower than) the expected mortality rate. This is because prices of q-forwards are not available in the public domain. However, if the reader has access to pricing information, he or she can easily incorporate it into the stochastic simulations. The use of real forward mortality rates will make the distributions of X* shift rightward, but it will make no change to the values of R we obtained. As a sensitivity test, we recalculate the hedge effectiveness provided by portfolios with different numbers of q-forwards. We found that R will be dropped to 68.86% if only three q-forwards are used, and that R will be increased to 98.82% if five q-forwards are used. With information about pricing and the associated transaction costs, we would be able to conclude if the use of an additional q-forward is worthwhile.

5. SMALL SAMPLE RISK The calculations in the previous section assume that there is no small sample risk (or sampling risk), that is, the risk that the realized mortality experience is different from the true mortality rate. Small sample risk is diversifiable, so it does not matter much for a large population, such as one with more than 100,000 lives. However, for smaller populations, small sample risk may be significant, and hence, an estimate of its impact on the hedge’s effectiveness is needed. To take small sample risk into account, we treat the cohort of pensioners as a random survivorship group. Let l(x) be the number of pensioners who will survive to age x. Here, l(x) is still random even if the mortality curve, q ⫽ {q(65,2006,1), q(66,2007,1), . . .}, for the population associated with the annuitants is completely known. We model small sample risk with the following binomial death process: l(x) ⬃ Binomial(l(x ⫺ 1),1 ⫺ q(x,t,1)), t ⫽ 1941 ⫹ x,

x ⫽ 65, 66, . . .

The death process is incorporated into the simulations procedure as follows: 1. 2. 3. 4.

Simulate a mortality curve, q using the augmented common factor model For each simulated q, simulate the number of survivors l(x), x ⫽ 65, 66, . . . Calculate cash flows on the basis of the simulated l(x) Repeat steps 1–3 to derive empirical distributions of unexpected cash flows.

Note that the hedge portfolio needs no change. The impact of sampling risk, on top of population basis risk, is demonstrated in the simulated distributions shown in Figure 12. To better understand how the size of the cohort plays a role, we produce two simulated distributions, one with l(65) ⫽ 10,000 and the other with l(65) ⫽ 3,000. We observe that the impact of sampling risk is minimal if there are 10,000 pensioners initially, but it becomes material when the number is reduced to 3,000. In Table 4 we show the decrease in R when sampling risk is taken into account. These values reconfirm that sampling risk can be material if the cohort starts with only 3,000 individuals. However, even with a small sample size like 3,000, the hedge still yields R ⫽ 69.57%, indicating that a significant portion of risk is eliminated.

6. CONCLUDING REMARKS We have presented in this paper four models that can be used for predicting mortality of two populations. The models connect longevity improvements of two populations in different manners, giving

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Table 4 Values of R When Sampling Risk Considered l(65)

Longevity Risk Reduction (R)

⫹⬁ 10,000 3,000

81.61% 77.56 69.57

different predictions of mortality differentials. In the application to the female populations of Canada and the United States, the augmented common factor model stands out as being best in the goodnessof-fit tests and ex post evaluation. However, readers should note that we may reach another conclusion when we model mortality of other pairs of populations. A careful selection of model is always necessary. Using Li and Luo’s (2011) framework, we have created a longevity hedge for a hypothetical pension plan in Canada, for which a LifeMetrics Index is not available. Simulations based on the augmented common factor model indicate that the longevity hedge, calibrated carefully with key q-durations, is reasonably effective, even if population basis risk exists. We acknowledge that some smaller pension plans are subject to mortality sampling risk and have modeled it with a binomial death process. With a initial cohort size of 3,000, the hedge can still eliminate a meaningful portion (approximately 70%) of the pension plan’s longevity risk. The longevity hedge we used is static, by which we mean the hedge portfolio is kept unchanged over time. However, since the key q-durations for both the pension plan and the q-forwards may change over time, a hedge portfolio that is optimal at time 0 may no longer be optimal in the future. With changing q-durations, we may be able to achieve better hedge effectiveness by adjusting the hedge portfolio from time to time. The benefit from dynamic hedging has not been studied in this paper but certainly deserves investigation when sufficient information on the liquidity and transaction costs of q-forwards becomes available.

APPENDIX A PROOF

OF

EQUATION (9)

We begin by deriving (approximate) expressions for ⭸K(t)/⭸t and ⭸k(t, i)/⭸t for i ⫽ 1, 2. In the augmented common factor model, K(t) follows a random walk with drift, which means

ˆ ⫹ 1) ⫺ K(t) ˆ ⫽ ˆc, K(t

t ⫽ 2007, 2008, . . . ,

ˆ is the predicted value of K(t). Hence, where ˆ c is the estimated drift term in the random walk, and K(t) we may approximate ⭸K(t)/⭸t by c. On the other hand, k(t,i) follows an AR(1) process, which means ˆ ˆ 0(i) ⫹ ␾ˆ 0(i)k(t ˆ ⫺ 1,i), t ⫽ 2007, 2008, . . . , k(t,i) ⫽␾ ˆ ˆ 0(i) and ␾ˆ 1(i) are the estimates of ␾0(i) and ␾1(i), where k(t,i) is the predicted value of k(t,i), and ␾ respectively. By applying this relation recursively, we obtain, for t ⫽ 2007, 2008, . . . , ˆ ⫹ 1,i) ⫺ k(t,i) ˆ ˆ 1(i)t⫺2006(␾ˆ 0(i) ⫹ (␾ˆ 1(i) ⫺ 1)k(2006,i)). ˆ k(t ⫽␾ As a result, we may approximate ⭸k(t,i)/⭸t, t ⫽ 2007, 2008, . . . , as follows: ⭸k(t,i) ⫽ ␾1(i)t⫺2006(␾0(i) ⫹ (␾1(i) ⫺ 1)k(2006,i)). ⭸t Next, we estimate ⭸m(x,t,i)/⭸m(x,t, j). Differentiating m(x,t,i) with respect to t, we have

MEASURING BASIS RISK

IN

LONGEVITY HEDGES

199

⭸m(x,t,i) ⫽ m(x,t,i) ⭸t

冉

B(x)

冊

⭸K(t) ⭸k(t,i) ⫹ b(x,1) . ⭸t) ⭸t

Dividing ⭸m(x,t,i)/⭸t by ⭸m(x,t, j)/⭸t, we obtain, for t ⫽ 2007, 2008, . . . , ⭸m(x,t,i) A(x,t,i) ⫽ , ⭸m(x,t, j) A(x,t, j) where A(x,t,i) ⫽ B(x)c ⫹ b(x,i)␾1(i)t⫺2006(␾0(i) ⫹ (␾1(i) ⫺ 1)k(2006,i)). We then relate m(x,t,i) to q(x,t,i). Assuming uniform distribution of deaths for fractional ages, we have q(x,t,i) ⫽ m(x,t,i)/(1 ⫹ 0.5m(x,t,i)), which implies ⭸q(x,t,i) 1 ⫽ . ⭸m(x,t,i) (1 ⫹ 0.5m(x,t,i))2 Finally, since ⭸q(x,t,i) ⭸q(x,t,i) ⭸m(x,t, j) ⭸m(x,t,i) ⫽ , ⭸q(x,t, j) ⭸m(x,t,i) ⭸q(x,t, j) ⭸m(x,t, j) we get ⭸q(x,t,i) q(x,t,i)(1 ⫹ 0.5m(x,t, j))2A(x,t,i) ⫽ , ⭸q(x,t, j) q(x,t, j)(1 ⫹ 0.5m(x,t,i))2A(x,t, j)

t ⫽ 2007, 2008, . . . .

7. ACKNOWLEDGMENTS Financial support from the Natural Science and Engineering Research Council of Canada is acknowledged. The authors also thank Andrew Cairns and Guy Coughlan for their comments on an earlier draft of this paper. REFERENCES BROUHNS, N., M. DENUIT, AND J. K. VERMUNT. 2002. A Poisson Log-Bilinear Regression Approach to the Construction of Projected Lifetables. Insurance: Mathematics and Economics 31: 373–393. CAIRNS, A. J. G., D. BLAKE, AND K. DOWD. 2006. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance 73: 687–718. CAIRNS, A. J. G., D. BLAKE, AND K. DOWD. 2008. Modelling and Management of Mortality Risk: A Review. Scandinavian Actuarial Journal 108: 79–113. CAIRNS, A. J. G., D. BLAKE, K. DOWD, G. D. COUGHLAN, D. EPSTEIN, A. ONG, AND I. BALEVICH. 2009. A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States. North American Actuarial Journal 13(1): 1–35. CARTER, L. R., AND R. D. LEE. 1992. Forecasting Demographic Components: Modeling and Forecasting US Sex Differentials in Mortality. International Journal of Forecasting 8: 393–411. CONTINUOUS MORTALITY INVESTIGATION BUREAU. 1999. Standard Tables of Mortality Based on the 1991–94 Experiences. CMI Report no. 17. London: Institute of Actuaries and Faculty of Actuaries. CONTINUOUS MORTALITY INVESTIGATION BUREAU. 2002. An Interim Basis for Adjusting the ‘‘92’’ Series Mortality Projections for Cohort Effects. CMI Working Paper no. 1. London: Institute of Actuaries and Faculty of Actuaries. COUGHLAN, G. 2009. Longevity Risk Transfer: Indices and Capital Market Solutions. In The Handbook of Insurance Linked Securities, ed. P. M. Barrieu and L. Albertini, pp. 261–282. London: Wiley. COUGHLAN, G., D. EPSTEIN, A. ONG, A. SINHA, J. HEVIA-PORTOCARRERO, E. GINGRICH, M. KHALAF-ALLAH, AND P. JOSEPH. 2007. LifeMetrics: A Toolkit for Measuring and Managing Longevity and Mortality Risks. http://www.jpmorgan.com/pages/jpmorgan/investbk/ solutions/lifemetrics/library. COUGHLAN, G., D. EPSTEIN, A. SINHA, AND P. HONIG. 2007b. q-Forwards: Derivatives for Transferring Longevity and Mortality Risk. http://www.jpmorgan.com/pages/jpmorgan/investbk/solutions/lifemetrics/library. CZADO, C., A. DELWARDE, AND M. DENUIT. 2005. Bayesian Poisson Log-Bilinear Mortality Projections. Insurance: Mathematics and Economics 36: 260–284.

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DICKEY, D., AND W. FULLER. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74: 427–431. DOWD, K., A. J. G. CAIRNS, D. BLAKE, G. D. COUGHLAN, D. EPSTEIN, AND M. KHALAF-ALLAH. 2010. Backtesting Stochastic Mortality Models: An Ex-Post Evaluation of Multi-Period-Ahead Density Forecasts. North American Actuarial Journal 14(3): 281–298. ENGLE, R., AND C. GRANGER. 1987. Cointegration and Error Correction: Representation, Estimation and Testing. Econometrica 55: 251–276. HO, T. S. Y. 1992. Key Rate Durations: Measurers of Interest Rate Risks. Journal of Fixed Income 2: 29–44. HUMAN MORTALITY DATABASE. 2009. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). www.mortality.org or www.humanmortality.de. LEE, R., AND L. CARTER. 1992. Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association 87: 659–671. LEE, R. D., AND F. NAULT. 1993. Modeling and Forecasting Provincial Mortality in Canada. Paper presented at the World Congress of IUSSP, Montreal, Canada. LI, J. S.-H., AND A. LUO. 2011. Key q-Duration: A Framework for Hedging Longevity Risk. Working Paper, University of Waterloo. LI, S. H., M. R. HARDY, AND K. S. TAN. 2007. Report on Mortality Improvement Scales for Canadian Insured Lives. http://www.soa.org/ research/individuallife/cia-mortality-rpt.aspx. LI, J. S.-H., M. R. HARDY, AND K. S. TAN. 2009. Uncertainty in Mortality Forecasting: An Extension to the Classical Lee-Carter Approach. ASTIN Bulletin 39: 137–164. LI, J. S.-H., AND A. C. Y. NG. 2010. Canonical Valuation of Mortality-Linked Securities. Journal of Risk and Insurance. In press. LI, N., AND R. LEE. 2005. Coherent Mortality Forecasts for a Group of Population: An Extension of the Lee-Carter Method. Demography 42: 575–594. LIN, Y., AND S. H. COX. 2005. Securitization of Mortality Risks in Life Annuities. Journal of Risk and Insurance 72: 227–252. LIN, Y., AND S. H. COX. 2008. Securitization of Catastrophe Mortality Risks. Insurance: Mathematics and Economics 42: 628–637. PHILLIPS, P. C. B. 1991. Optimal Inference in Cointegrated Systems. Econometrica 59: 283–306. SAID, E., AND D. A. DICKEY. 1984. Testing for Unit Roots in Autoregressive Moving Average Models of Unknown Order. Biometrika 71: 599–607. SCHWARZ, G. 1978. Estimating the Dimension of a Model. Annals of Statistics 6: 461–464. SOCIETY OF ACTUARIES GROUP ANNUITY VALUATION TABLE TASK FORCE. 1995. 1994 Group Annuity Mortality Table and 1994 Group Annuity Reserving Table. Transactions of the Society of Actuaries 47: 865–915. TULJAPURKAR, S., N. LI, AND C. BOE. 2000. A Universal Pattern of Mortality Decline in the G7 Countries. Nature 405: 789–792. UNITED NATIONS. 1998. World Populations Prospects: The 1996 Revision. New York: Population Division, United Nations. WHITE, K. M. 2002. Longevity Advances in High-income Countries, 1955–96. Population and Development Review 28: 59–76. WILSON, C. 2001. On the Scale of Global Demographic Convergence 1950–2000. Population and Development Review 24: 593–600. WILLS, S., AND M. SHERRIS. 2008. Integrating Financial and Demographic Longevity Risk Models: An Australian Model for Financial Applications. UNSW Australian School of Business, Research Paper No. 2008ACTL05. WILMOTH, J. R. 1993. Computational Methods for Fitting and Extrapolating the Lee-Carter Model of Mortality Change. Technical report. Department of Demography, University of California, Berkeley.

Discussions on this paper can be submitted until October 1, 2011. The authors reserve the right to reply to any discussion. Please see the Submission Guidelines for Authors on the inside back cover for instructions on the submission of discussions.

Applied Financial Economics, 2012, 22, 1553–1569

Measuring operational risk in financial institutions Se´verine Plunusa,b,*, Georges Hu¨bnera,b,c and Jean-Philippe Petersd a

HEC-Management School, University of Lie`ge, Rue Louvrex, 14, B-4000, Lie`ge, Belgium b Gambit Financial Solutions, Lie´ge, Belgium c Faculty of Economics and Business Administration, Maastricht University, Maastricht, The Netherlands d Deloitte Luxembourg, Advisory and Consulting Group (Risk Management Unit), Luxembourg

The scarcity of internal loss databases tends to hinder the use of the advanced approaches for operational risk measurement (Advanced Measurement Approaches (AMA)) in financial institutions. As there is a greater variety in credit risk modelling, this article explores the applicability of a modified version of CreditRisk+ to operational loss data. Our adapted model, OpRisk+, works out very satisfying Values-at-Risk (VaR) at 95% level as compared with estimates drawn from sophisticated AMA models. OpRisk+ proves to be especially worthy in the case of small samples, where more complex methods cannot be applied. OpRisk+ could therefore be used to fit the body of the distribution of operational losses up to the 95%-percentile, while Extreme Value Theory (EVT), external databases or scenario analysis should be used beyond this quantile. Keywords: operational risk; Basel II; modelling; CreditRisk+ JEL Classification: G28; G21

I. Introduction Over the past decades, financial institutions have experienced several large operational loss events leading to big banking failures. Memorable examples include the Barings bank losing 1.4 billion USD from rogue trading in his branch in Singapore leading to the failure of the whole institution1; Allied Irish Banks losing 750 MM USD in rogue trading,2 or Prudential Insurance incurring 2 billion USD settlement in class action lawsuit,3 to name a few.

These events, as well as developments such as the growth of e-commerce, changes in banks’ risk management or the use of more highly automated technology, have led regulators and the banking industry to recognize the importance of operational risk in shaping the risk profiles of financial institutions. Reflecting this recognition, regulatory frameworks such as the New Capital Accord of the Basel Committee on Banking Supervision (‘Basel II’) have introduced explicit capital requirements for

*Corresponding author. E-mail: [email protected] 1 See Stonham (1996); Ross (1997); Sheaffer et al. (1998). 2 See Dunne and Helliar (2002). 3 See Walker et al. (2001). Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online ß 2012 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/09603107.2012.667546

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1554 operational risk. Similar to credit risk, Basel II does not impose a ‘one-size-fits-all’ approach to capital adequacy and proposes three distinct options for the calculation of the capital charge for operational risk: the basic indicator approach, the standardized approach and the Advanced Measurement Approaches (AMA). The use of these approaches of increasing risk sensitivity is determined according to the risk management systems of the banks. The first two methods are a function of gross income, while the advanced methods are based on internal loss data, external loss data, scenario analysis, business environment and internal control factors. In 2001, the Basel Committee was encouraging two specific AMA methods: (i) the Loss Distribution Approach (LDA) and (ii) an Internal Measurement Approach (IMA) developing a linear relationship between unexpected loss and expected loss to extrapolate credit-risk’s Internal Rating Based (IRB) approach to operational risk. While the Basel Committee dropped formal mention of the IMA in favour of Value-at-Risk (VaR) approaches in the final version of the Accord, it is still legitimate to be inspired by modelling approaches for credit risk in order to model the distribution of operational loss data. Indeed, both risk measurement frameworks have similar features, such as their focus on a 1-year measurement horizon or their use of an aggregate loss distribution skewed towards zero with a long right-tail. This article explores the possibility of adapting one of the current proposed industry credit-risk models to perform much of the functionality of an actuarial LDA model (see Crouhy et al., 2000; Gordy, 2000) for a comparative analysis of the main credit risk models). We identify CreditRisk+, the model developed by Credit Suisse, as an actuarial-based model, whose characteristics can be adapted to fit the LDA. The LDA is explicitly mentioned in the Basel II Accord as eligible among the AMAs to estimate risk capital, and has unambiguously emerged as the standard industry practice.4 After some adjustment, we construct a distribution of operational losses through an adapted CreditRisk+model, that we name as ‘OpRisk+’.5 As this model calibrates the whole distribution, not only can we retrieve the quantiles of the operational loss distribution, but also an estimate of its expectation, needed for the computation of the economic capital. 4

S. Plunus et al. Our research is aimed at answering the following questions: (i) How would the adaptation of CreditRisk+ model perform compared to sophisticated models such as the approach developed by Chapelle, Crama, Hu¨bner and Peters (2008) (henceforth CCHP) or Moscadelli (2004) among others? (ii) Does OpRisk+ provide a reasonable assessment of the body of the distribution of the operational losses? (iii) Are the VaR computed with OpRisk+ more conservative than the lower bound of Alexander (2003), an extended IMA approach? We address the questions with an experiment based on generated databases using three different Pareto distributions, proven to be appropriate to model operational loss data by Moscadelli (2004) and de Fontnouvelle and Rosengren (2004). The knowledge of the true distribution of losses is necessary to assess the quality of the different fitting methods. Had a real data set been used instead of controlled numerical simulations as proposed by McNeil and Saladin (1997), we would not be able to benchmark the observed results against the true loss distribution and to evaluate the performance of OpRisk+ for different loss generating processes and sample sizes. We assess the influence of the number of losses recorded in the database on the quality of the estimation. Indeed, Carrillo-Mene´ndez and Sua´rez (2012) have shown the difficulty of selecting the correct model from the data when only small samples are available. We also test our new adapted IRB model against Alexander’s existing improvement to the basic IMA formula. Alexander’s VaR on operational loss data (OpVaR) is effectively a quantile value from a normal distribution table which allows identification of the unexpected loss if one knows the mean and variance of the loss severity distribution and the mean of the frequency distribution. Our main findings are twofold. First, we note that the precision of OpRisk+ is not satisfactory to estimate the very far end of the loss distribution, such as the VaR6 at the 99.9% confidence level (VaR99.9). Yet, our model works out very satisfying quantile estimates, especially for thin-tailed Paretodistributions, up to a 95% confidence level for the computation of the VaR. Second, the simplicity of our model makes it applicable to ‘problematic’ business lines, that is, with very few occurrences of events, or with limited history of data. Procedures that rely on Extreme Value Theory (EVT), by

See Sahay et al. (2007) or Degen et al. (2007). We named our model OpRisk+ to keep its source in mind, that is, the CreditRisk+ model developed by Credit Suisse First Boston. Our model is not a new one but an adaptation of their model to make it useful in our specific situation: that is small samples of operational loss data. 6 The VaR is the amount that losses will likely not exceed, within a predefined confidence level and over a given time-period. 5

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Measuring operational risk in financial institutions contrast, are very data-consuming, and yield very poor results when used with small databases. Moreover, as argued by Malevergne et al. (2006), when there is a lack of data, nonparametric methods are useful to assess risk at probability level 95% but fail at high probability level such as 99% or larger. These findings make the OpRisk+ approach clearly not an effective substitute, but indeed a very useful complement to approaches that specifically target the extreme tail of the loss distribution. In particular, the body of the loss distribution can be safely assessed with our method, while external data or scenario analysis, as specifically mentioned in the Accord, can be used to estimate the tail. In the spirit of the multi-dimensional approach of operational risk management including more qualitative type of analyses, this model could also represent for a bank a very good cross-check method for scenario-based approaches. We can also position this article’s contribution in the other direction. With the reinforcement of the supervisory burden in the Basel III Accords, regulators could also be in a position to challenge banks using internal approaches that might seems too aggressive on small samples. Moreover, being able to simultaneously rely on the body and the tail of the distribution is crucial for the operational risk capital estimation, because one needs the full distribution of losses in order to capture the expected loss that enters the regulatory capital estimate. Next section describes the adjustment needed in order to apply CreditRisk+ model to operational loss data and presents two alternative methods to calibrate a VaR on operational loss data (OpVaR). We then describes our database, presents our results and compares them to the other approaches’ results. Finally, we discuss the application of our models to real loss data.

II. Alternative Approaches for the Measurement of Operational Risk This section presents three alternative ways to calibrate a VaR on operational loss data. The first one represents an adaptation of the CreditRisk+ framework, while the second one proposes an adaptation of the LDA in the context of operational losses with the use of EVT. Finally, we introduce an IMA approach developed by Alexander (2003). 7

OpRisk+: application of CreditRisk+ to operational loss data CreditRisk+ developed by Credit Suisse First Boston is an actuarial model derived from insurance loss models. It models the default risk of a bond portfolio through the Poisson distribution. Its basic building block is simply the probability of default of a counterparty. In this model, no assumptions are made about the causes of default: an obligor is either in default with a probability PA, or not in default with a probability 1  PA. Although operational losses do not depend on a particular counterparty, this characteristic already simplifies the adaptation of our model, as we do not need to make assumptions on the causes of the loss. CreditRisk+ determines the distribution of default losses in three steps: the determination of the frequency of defaults, approximated by a standard Poisson distribution, the determination of the severity of the losses, and the determination of the distribution of default losses. The determination of the frequency of events leading to operational losses can be modelled through the Poisson distribution as for the probability of default in CreditRisk+ Pð N ¼ n Þ ¼

n e n!

for n ¼ 0, 1, 2, . . .

ð1Þ

where  is the average number of defaults per period, and N is a stochastic variable with mean , p and SD . CreditRisk+ computes the parameter  by adding the probability of default of each obligor, supplied, for instance, by rating agencies. However, operational losses do not depend on a particular obligor. Therefore, instead of being defined as a sum of probabilities of default depending on the characteristics of a counterpart,  can be interpreted as the average number of loss events of one type occurring in a specific business line during one period. CreditRisk+ adds the assumption that the mean default rate itself is stochastic in order to take into account the fat right tail of the distribution of defaults. Nevertheless, the Poisson distribution being one of the most popular in operational risk frequency estimation, according to Cruz (2002)7 and Basel Committee on Banking Supervision (2009), we keep on assuming that the number of operational loss events follows a Poisson distribution with a fixed mean .

Cruz (2002) argues that this is due to its simplicity and to the fact that it fits most of the databases very well.

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Table 1. Allocating losses to bands

Table 2. Exposure, number of events and expected loss

Loss amount (A)

Loss in L (B)

Round-off loss j (C)

Band j (D)

1500 2508 3639 1000 1835 2446 7260

1.5 2.51 3.64 1.00 1.84 2.45 7.26

2.00 3.00 4.00 1.00 2.00 3.00 8.00

2 3 4 1 2 3 8

Notes: Illustration of the first three steps of the OpRisk+ approach: (1) Choose a unit amount of loss L. (1000 in the example); (2) Divide the losses of the available database (column A) by L (column B) and round up these numbers (column C); (3) Allocate the losses of different sizes to their band (column D).

In order to perform its calculations, CreditRisk+ proposes to express the exposure (here, the losses) in a unit amount of exposure L.8 The key step is then to round up each exposure size to the nearest whole number, in order to reduce the number of possible values and to distribute them into different bands. Each band is characterized by an average exposure, j, and an expected loss, "j, equal to the sum of the expected losses of all the obligors belonging to the band. Table 1 shows an example of this procedure. CreditRisk+ posits that "j ¼ j j

ð2Þ

where "j is the expected loss in band j, j is the common exposure in band j, and j is the expected number of defaults in band j. As the operational losses do not depend on a particular transaction, we slightly modify the definition of these variables. The aim is to calculate the expected aggregate loss. We will therefore keep the definition of "j unchanged. However, as noted earlier, j is not an aggregate expected number of defaults anymore but simply the (observed9) average number of operational loss events of size j occurring in 1 year. Consequently, in order to satisfy Equation 2, j must be defined as the average loss amount per event for band j. Table 2 illustrates the reprocessing of the data. Each band is viewed as a portfolio of exposures by itself. Because some defaults lead to larger losses than others through the variation in exposure amounts, the loss given default involves a second element of randomness, which is mathematically described through its probability generating function. 8 9

j

j

"j

1 2 3 4 5 6 7 .8 ..

9 121 78 27 17 15 8 4. ..

9 242 234 108 85 90 56 32 .. .

Notes: Illustration of step 5 of the OpRisk+ approach: ‘Compute the expected loss per band, "j, equal to the expected number of losses per band j, multiplied by the average loss amount per band, j, equal to j.’

Thus, let G(z) be the probability generating function for losses expressed in multiples of the unit L of exposure 1 X Gj ðzÞ ¼ Pðloss ¼ nLÞzn n¼0

¼

1 X n¼0

PðndefaultsÞznj

ð3Þ

As the number of defaults follows a Poisson distribution, this is equal to 1 ej n X j j nj Gj ðzÞ ¼ z ¼ ej þj z ð4Þ n! n¼0

As far as operational losses are concerned, we can no more consider a band as a portfolio but simply as a category of loss size. This also simplifies the model, as we do not distinguish exposure and expected loss anymore. For credit losses, one first sorts exposures, and then calculates the expected loss, by multiplying the exposures by their probability of default. As far as operational losses are concerned, the loss amounts are directly sorted by size. Consequently, the second element of randomness is not necessary anymore. This has no consequences on the following results except simplifying the model. Whereas CreditRisk+ assumes the exposures in the portfolio to be independent, OpRisk+ assumes the independence of the different loss amounts. Thanks to this assumption, the probability generating function for losses of one type for a specific business line is given by the product of the probability generating function for each band Pm Pm m Y j   þ j¼1 j zj j¼1 j GðzÞ ¼ ej þj z ¼ e ð5Þ j¼1

CreditRisk+’s authors argue that the exact amount of each loss cannot be critical in the determination of the global risk. The purpose of the model is to be applied to real loss data.

Measuring operational risk in financial institutions Finally, the loss distribution of the entire portfolio is given by  1 d n GðzÞ Pðloss of nLÞ ¼ for n ¼ 1, 2, . . . ð6Þ n! dzn z¼0

Note that this equation allows only computing the probability of losses of size 0, L, 2 L and so on. This probability of loss of nL will further be denoted An. Then, under the simplified assumption of fixed default rates, Credit Suisse has developed the following recursive equation (the Appendix presents the derivation of this equation): X "j Anj ð7Þ An ¼ n j:j n Pm " j Pm    j¼1 j j¼1 j ¼ e . where A0 ¼ Gð0Þ ¼ e ¼ e The calculation depends only on two sets of parameters: j and "j, derived from j, the number of events of each range, j, observed. With operational data, A0 is derived directly from A0 ¼ em. To illustrate this recurrence, suppose the database contains 20 losses, three (resp. 2) of which having a rounded-off size of 1 L (resp. 2 L) A0 ¼ e20 ¼ 2:06  109 A1 ¼

X "j A1j ¼ "1 A0 ¼ 3  2:06109 1 j: 1 j

¼ 6:18109 A2 ¼

X "j 1 A2j ¼ ð"1 A1 þ "2 A0 Þ 2 2 j: 2 j

 1 ¼ 3  6:18  109 þ 2  2:06  109 2 ¼ 1:13  108 Therefore, the probability of having a loss of size resp. 0, 1 L and 2 L is resp. 2.06  109, 6.18  109 and 1.13  108, and so on. From there, one can re-construct the distribution of the loss of size nL.

The loss distribution approach adapted to operational risk Among the AMAs developed over the recent years to model operational risk, the most common one is the 10

1557 LDA (), which is derived from actuarial techniques (see Frachot et al., 2001; Cruz, 2004; ChavezDemoulin et al., 2006; Peters et al., 2011, for an introduction). By means of convolution, this technique derives the Aggregate Loss Distribution (ALD) through the combination of the frequency distribution of loss events and the severity distribution of a loss given event.10 The operational VaR is then simply the 99.9th percentile of the ALD. As an analytical solution is very difficult to compute with this type of convolution, Monte Carlo simulations are usually used to do the job. Using the CCHP procedure with a Poisson distribution with a parameter  equal to the number of observed losses during the whole period to model the frequency,11 we generate a large number M of Poisson () random variables (say, 100 000). These M values represent the number of events for each of the M simulated periods. For each period, generate the required number of severity random variables (that is, if the simulated number of events for period m is x, then simulate  severity losses) and add them to get the aggregate loss for the period. The obtained vector represents M simulated periods and OpVaRs are then readily obtained (e.g. the OpVaR at 99.99% confidence interval is the 10th lowest value of the M sorted aggregate losses). Examples of empirical studies using this technique for operational risk include Moscadelli (2004) on loss data collected from the Quantitative Impact Study (QIS) of the Basel Committee, de Fontnouvelle and Rosengren (2004) on loss data from the 2002 Risk Loss Data Collection Exercise initiated by the Risk Management Group of the Basel Committee or CCHP (2008) with loss data coming from a large European bank. In the latter case, mixing two distributions fits more adequately the empirical severity distribution than a single distribution. Therefore, the authors divide the sample into two parts: a first one with losses below a selected threshold, considered as the ‘normal’ losses, and a second one, including the ‘large’ losses. To model the ‘normal’ losses, CCHP compare several classic continuous distributions such as gamma, lognormal or Pareto. In our example, we will use the lognormal distribution. To take extreme and very rare losses into account (i.e. the ‘large’ losses), CCHP apply the EVT on their results.12 The advantage of EVT is that it provides a

More precisely the ALD is obtained through the n-fold convolution of the severity distribution with itself, n being a random variable following the frequency density function. 11 While frequency could also be modelled with other discrete distributions such as the Negative Binomial for instance, many authors use the Poisson assumption (see, for instance, de Fontnouvelle et al., 2003). 12 This solution has been advocated by many other authors; see for instance King (2001), Cruz (2004), Moscadelli, M. (2004), de Fontnouvelle and Rosengren (2004) or Chavez-Demoulin et al. (2006).

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tool to estimate rare and not-yet-recorded events for a given database,13 hence providing an attractive solution for loss databases with limited collection history that are used to reach a very high confidence levels like the one required by Basel II (i.e. 99.9%).

Alexander’s IMA The basic formula of the IMA included in the AMAs of Basel II is UL ¼ EL

ð8Þ

where UL ¼ unexpected loss, determining the operational risk requirement,14 is a multiplier, and EL is the expected loss. Gamma factors are not easy to evaluate as no indication of their possible range has been given by the Basel Committee. Therefore, Alexander (2003) suggests that instead of writing the unexpected loss as a multiple ( ) of expected loss, one writes the unexpected loss as a multiple () of the loss SD (). Using the definition of the expected loss, she gets the expression for  VaR99:9  EL ¼ 

VaR99:9  L  ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2L þ L2

ð10Þ

where  is the average number of losses. For  L > 0, this formula produces slightly lower  than with no severity uncertainty, but it is still bounded below by the value 3.1. For a comparison purpose, we will use the following value for the needed OpVaRs, derived from Equation 10 in which we replace  by a value corresponding to the selected level of confidence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   OpVaR ¼   2L þ L2 þ L ð11Þ

III. An Experiment on Simulated Losses Data

ð9Þ

The advantage of this parameter is that it can be easily calibrated. The basic IMA formula is based on the binomial loss frequency distribution, with no variability in loss severity. For very high-frequency risks, Alexander notes that the normal distribution could be used as an approximation of the binomial loss distribution, providing for  a lower bound equal to 3.1 (as can be found from standard normal tables when the number of losses goes to infinity). She also suggests that the Poisson distribution should be preferred to the binomial as the number of transactions is generally difficult to quantify. Alexander (2003) shows that , as a function of the parameter  of the Poisson distribution, must be in a fairly narrow range: from about 3.2 for mediumto-high frequency risks (20 to 100 loss events per year) to about 3.9 for low frequency risks (one loss event every 1 or 2 years) and only above 4 for very rare events that may happen only once every 5 years or so. Table 3 illustrates the wide range for the gammas by opposition to the narrow range of the phi’s values. 13

Then, assuming the loss severity to be random, i.e. with mean L and SD  L, and independent of the loss frequency, Alexander writes the  parameter as

OpRisk+ makes the traditional statistical tests impossible, as it uses no parametric form but a purely numerical procedure. Therefore, as proposed by McNeil and Saladin (1997), in order to perform tests of the calibrating performance of OpRisk+ on any distribution of loss severity, we simulate databases to obtain an exhaustive picture of the capabilities of the approach. Moscadelli (2004) and de Fontnouvelle and Rosengren (2004) having shown that loss data for most business lines and event types may be well modelled by a Pareto-type distribution, we simulated our data on the basis of three different kinds of Pareto distributions: a heavy-tail, a mediumtail and a thin-tail Pareto distribution. A Pareto distribution is a right-skewed distribution parameterized by two quantities: a minimum possible value or location parameter, xm, and a tail index or shape parameter, . Therefore, if X is a random variable with a Pareto distribution, the probability that X is greater than some number x is given by  k x PrðX 4 xÞ ¼ ð12Þ xm for all x  xm., and for xm and k ¼ 1/ > 0.

See Embrechts et al. (1997), for a comprehensive overview of EVT. The unexpected loss is defined as the difference between the VaR at the 99.9% confidence level (VaR99.9) and the expected loss. 14

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Measuring operational risk in financial institutions Table 3. Gamma and phi values (no loss severity variability) 

100

50

40

30

20

10

VaR99.9 

131.81 3.18 0.32

72.75 3.22 0.46

60.45 3.23 0.51

47.81 3.25 0.59

34.71 3.29 0.74

20.66 3.37 1.07

17.63 3.41 1.21

14.45 3.45 1.41



5

4

8

6

3

2

1

0.9

0.8

0.7

12.77 3.48 1.55

10.96 3.48 1.74

9.13 3.54 2.04

7.11 3.62 2.56

4.87 3.87 3.87

4.55 3.85 4.06

4.23 3.84 4.29

3.91 3.84 4.59



0.6

0.5

0.4

0.3

0.2

0.1

0.05

0.01

VaR99.9 

3.58 3.85 4.97

3.26 3.90 5.51

2.91 3.97 6.27

2.49 4.00 7.30

2.07 4.19 9.36

1.42 4.17 13.21

1.07 4.54 20.31

0.90 8.94 89.40

VaR99.9 

Source: Alexander (2003, p. 151). Notes: Illustration of the wide range for the gammas by opposition to the narrow range of the phi’s values in the computation of the unexpected loss (UL¼. VaR99.9 – EL) determining the operational risk requirement. The basic formula of the IMA of Basel II is UL ¼ EL, where is a multiplier, and EL is the expected loss. As Gamma factors are not easy to evaluate, Alexander (2003) suggests to write unexpected loss as a multiple () of the loss SD ().

The parameters of our distributions are Pareto(100;0.3), Pareto(100;0.5) and Pareto(100;0.7); the larger the value of the tail index, the fatter the tail of the distribution. The choice of these functions has been found to be reasonable with a sample of real data obtained from a large European institution. We run three simulations: one for the thin-tailed, one for the medium-tailed and one for the fat-tailed Pareto severity distribution cases. For each of these cases, we simulate two sets of 1000 years of 20 and 50 operational losses respectively and two sets of 100 series of 200 and 300 operational losses respectively.15 Table 4 gives the characteristics of each of the 12 databases (each thus comprising 1000 or 100 simulated loss distributions) constructed in order to implement OpRisk+. For each series of operational losses we compute the expected loss, that is, the mean loss multiplied by the number of losses, as well as the SD, median, maximum and minimum of these expected losses. These results clearly show that data generated with a thin-tailed Pareto-distribution exhibit characteristics that make the samples quite reliable. The mean loss is very close to its theoretical level even for 20 draws. Furthermore, we observe a SD of aggregate loss that is very limited, from less than 10% of the average for N ¼ 20 to less than 3% for N ¼ 200. The median loss is also close to the theoretical value. For a tail index of 0.5 (medium-tailed), the mean loss

still stays close to the theoretical value but the SD increases. Thus, we can start to question the stability of the loss estimate. When the tail index increases, the mean aggregate loss becomes systematically lower than the theoretical mean, and this effect aggravates when one takes a lower number of simulations (100 drawings) with a larger sample. The SD and range become extremely large, thereby weakening inference based on a given set of loss observations. This highlights the difficulty of modelling operational risk losses (which often exhibit this type of tail behaviour) using classical distribution fitting methods when only a limited number of loss data points are available.

Application of OpRisk+ To apply OpRisk+ to these data, the first step consists of computing A0 ¼ e-, where  is the average number of loss events. For instance, for N ¼ 200, this gives the following value: A0 ¼ e200 ¼ 1:38  1087 . Then, in order to assess the loss distribution of the entire population of operational risk events, we use the recursive equation (7) to compute A1, A2 etc. Once the different probabilities An for the different sizes of losses are computed, we can plot the aggregate loss distribution as illustrated in Fig. 1.16

15 Only 100 years of data were simulated for high-frequency databases as the computation becomes too heavy for a too large number of data. However, we tested our model with 200 years of data for the sample of 200 events characterized by a Pareto (100; 0.7), and did not obtain significantly different OpVaRs. Detailed results are available upon request. 16 Note that this figure represents the distribution built from 1 year of data (200 losses), whereas Table 4 displays the average mean of the 100 years of 200 losses.

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Table 4. Characteristics of the 12 databases created for testing the different models Poisson parameter 

20

50

200

300

Panel A: Thin-tailed-Pareto distribution parameter ¼ 0.3) Theoretical mean 2857 7143 28 571 Mean 2845 7134 28 381 SD 287 472 847 Median 2796 7078 28 172 Maximum 4683 9026 30 766 Minimum 2268 6071 26 713

(shape

Panel B: Medium-tailed-Pareto distribution parameter ¼ 0.5) Theoretical mean 4000 10 000 40 000 Mean 3924 9913 39 871 SD 1093 1827 3585 Median 3676 9594 39 777 Maximum 15 680 29 029 54 242 Minimum 2567 7097 33 428

(shape

42 857 42 886 1118 42 763 45 582 40 383

60 000 59 431 5504 57 947 91 182 52 436

Panel C: Fat-tailed-Pareto distribution (shape parameter ¼ 0.7) Theoretical mean 6667 16 667 66 667 100 000 Mean 6264 16 165 61 711 93 724 SD 5940 13 018 13 899 24 514 Median 5180 13 721 57 713 87 646 Maximum 157 134 265 621 137 699 248 526 Minimum 2646 8304 45 315 69 991 Number of simulated years

1000

1000

100

100

Notes: For three Pareto severity distribution (thin-tailed, medium-tailed and fat-tailed), we simulate two sets of 1000 years of 20 and 50 operational losses respectively and two sets of 100 series of 200 and 300 operational losses respectively. For each of the 6600 simulated years (3  2  1100), the aggregate loss distribution is computed with the algorithm described in the Section ‘The loss distribution approach adapted to operational risk’.

With this information, we can compute the different Operational Values-at-Risk (OpVaRs). This is done by calculating the cumulated probabilities for each amount of loss. The loss for which the cumulated probability is equal to p% gives us the OpVaR at percentile p. We repeat the procedure for each year of losses and report the average values of the different yearly OpVaRs in Tables 5 and 6. Even though this procedure is likely to underestimate the true quantiles (see the Section ‘Comparison with OpRisk+ taking an average value of loss for each band’), we view this setup as more realistic than merely computing a single OpVaR on the whole number of years. Indeed, the operational risk manager is likely to be confronted with a few years of limited data, which is consistent with our simulation procedure. Table 5 compares the OpVaRs obtained using OpRisk+ with the simulated data for the

small databases. The first column represents the average observed quantiles of the aggregate distribution when simulating 100 000 years with a Poisson () distribution for the frequency and a Pareto(100, ) for the severity. The table also gives the minimum, maximum and SD of the 100(0) OpVaRs produced by OpRisk+. Panel A of Table 5 shows that OpRisk+ achieves very satisfactory OpVaRs for the Pareto-distribution with thin tail. The mean OpVaRs obtained for both the samples of 20 and 50 observations stays within a 3% distance from the true value. Even at the level of 99.9% required by Basel II, the OpRisk+ values remain within a very narrow range, while the Root Mean Square Error (RMSE) of the estimates is kept within 13% of the true value. The results obtained with the OpRisk+ procedure with medium and fat tails tend to deteriorate, which is actually not surprising as the adaptation of the credit risk model strictly uses observed data and does necessarily underestimate the fatness of the tails. However, we still have very good estimation for OpVaR95. It mismatches the true 95% quantile by 2% to 7% for the medium and fat tailed Paretodistribution, while the RMSE tends – naturally – to increase very fast. The bad news is that the procedure alone is not sufficient to provide the OpVaR99.9 required by Basel II. It severely underestimates the true quantile, even though this true value is included in the range of the observed values of the loss estimates, mainly because the support of the distribution generated by the OpRisk+ method is finite and thus truncates the true loss distribution. This issue had been pointed out by Mignola and Ugoccioni (2006) who propose to reduce the sources of uncertainty in modelling the operational risk losses, by lowering the percentile at which the risk measure is calculated and finding some other mechanism to reach the 99.9% percentile. Further reasons for this systematic underestimation can be found in the setup of the simulations. The procedure averages the individual yearly OpVaRs, each of them being computed using a very small number of losses. This modelling choice mimics a realistic situation as closely as possible. There is thus a small likelihood of observing extreme losses over a particular year, and the averaging process tends to lead to the dominance of too small OpVaR estimates for the extreme quantiles. Table 6 displays the results of the simulations when a large sample size is used. Table 6, Panel A already delivers some rather surprising results. The OpRisk+ procedure seems to overestimate the true operational risk exposure for all confidence levels. This effect aggravates for a high number of losses in the database. This phenomenon

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Fig. 1. Aggregate loss distribution derived from the application of OpRisk+ for a series of 200 loss events characterized by a Pareto(100;0.3) Table 5. VaR generated by OpRisk+ for small databases, with 20 and 50 loss events Panel A: Thin-tailed-Pareto distribution (shape parameter ¼ 0.3) N ¼ 20

N ¼ 50 OpRisk+ (L ¼ 10, bands ¼ 9)

Target OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

3770 4073 4712 5596

Mean 3880 4173 4744 5410

Bias

OpRisk+ (L ¼ 10, bands ¼ 13)

RMSE

3% 3% 1% 3%

13% 13% 13% 13%

Target 8573 9030 9942 11 141

Mean 8882 9334 10 209 11 250

Bias

RMSE

3% 3% 3% 1%

8% 9% 9% 10%

Panel B: Medium-tailed-Pareto distribution (shape parameter ¼ 0.5) N ¼ 20

N ¼ 50 OpRisk+(L ¼ 10, bands ¼ 11)

Target OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

5579 6364 8966 18 567

Mean 5672 6247 7329 8626

Bias

OpRisk+(L ¼ 10, bands ¼ 19)

RMSE

Target

Mean

Bias

RMSE

40% 46% 48% 60%

12 630 13 862 18 051 33 554

12 855 13 734 15 410 17 338

6% 7% 20% 52%

29% 32% 36% 55%

2% 2% 18% 54%

Panel C: Fat-tailed-Pareto distribution (shape parameter ¼ 0.7) N ¼ 20

N ¼ 50 OpRisk+ (L ¼ 50, bands ¼ 7)

OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

OpRisk+ (L ¼ 50, bands ¼ 13)

Target

Mean

Bias

RMSE

Target

Mean

Bias

9700 12 640 27 261 114 563

11 410 12 931 15 583 18 726

18% 3% 43% 84%

107% 99% 72% 85%

22 495 28 103 55 994 220 650

23 992 27 089 32 020 38 761

7% 3% 43% 83%

RMSE 116% 134% 99% 88%

Notes: The OpVaRs are calculated separately for each year of data, and we report their average (Mean), the average value of the spread between the ‘true’ value, Target, and the OpVaRs, as percents of the latest (Bias), and the RMSE as percents of the ‘true’ OpVaRs (RMSE). The ‘true’ value or target is approximated through a Monte Carlo simulation of 100 000 years of data, characterized by a frequency equal to a random variable following a Poisson (N) and a severity characterized by the selected Pareto-distribution. The unit amount chosen for the OpRisk+ implementation and the average number of corresponding bands is reported in parentheses.

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Table 6. OpVaRs generated by OpRisk+ for databases with 200 and 300 loss events Panel A: Thin-tailed-Pareto distribution (shape parameter ¼ 0.3) N ¼ 200

N ¼ 300 OpRisk+ (L ¼ 20, bands ¼ 13)

Target OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

31 448 32 309 33 995 36 063

Mean 33 853 34 728 36 397 38 310

Bias

OpRisk+(L ¼ 50, bands ¼ 8)

RMSE

7% 7% 7% 6%

Target

8% 8% 7% 7%

46 355 47 403 49 420 51 750

Mean 56 470 57 683 59 992 62 628

Bias 22% 22% 21% 21%

RMSE 22% 22% 22% 21%

Panel B: Medium-tailed-Pareto distribution (shape parameter ¼ 0.5) N ¼ 200

N ¼ 300 OpRisk+ (L ¼ 50, bands ¼ 14)

Target OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

45 757 48 259 55 919 83 292

Mean 51 836 53 816 57 668 62 237

OpRisk+(L ¼ 50, bands ¼ 11)

Bias

RMSE

Target

13% 12% 3% 25%

18% 18% 16% 29%

67 104 70 264 79 718 113 560

Mean 75 723 78161 82 817 88 309

Bias

RMSE

13% 11% 4% 22%

19% 20% 19% 27%

Panel C: Fat-tailed-Pareto distribution (shape parameter ¼ 0.7) N ¼ 200

N ¼ 300 OpRisk+ (L ¼ 50, bands ¼ 21)

OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

OpRisk+(L ¼ 50, bands ¼ 17)

Target

Mean

Bias

RMSE

Target

Mean

Bias

RMSE

82 381 96 971 166 962 543 597

82 539 88 248 98 972 111 875

0% 9% 41% 79%

30% 32% 47% 80%

120 654 139 470 234 442 733 862

119 943 127 037 140 665 156 642

1% 9% 40% 79%

29% 32% 47% 79%

Notes: The OpVaRs are calculated separately for each year of data, and we report their average (Mean), the average value of the spread between the ‘true’ value and the OpVaRs, as percents of the latter (Bias), and RMSE as percents of the ‘true’ OpVaRs (RMSE). The ‘true’ value is approximated through a Monte Carlo simulation of 100 000 years of data, characterized by a frequency equal to a random variable following a Poisson (N) and a severity characterized by the selected Paretodistribution. The unit amount chosen for the OpRisk+ implementation and the average number of corresponding bands is reported in parentheses.

may be due to an intervalling effect, where losses belonging to a given band are assigned the value of the band’s upper bound. Given that extreme losses are likely to occur in the lower part of the band, as the distribution is characterized by a thin tail Paretodistribution, taking the upper bound limit value for aggregation seems to deteriorate the estimation, making it too conservative. Nevertheless, the bias is almost constant in relative terms, indicating that its seriousness does not aggravate as the estimation gets far in the tail of the distribution. Sub-section ‘Comparison with OpRisk+ taking an average value of loss for each band’ investigates further this issue.

This intervalling phenomenon explains the behaviour of the estimation for larger values of the tail index. In Panel B, the adapted credit risk model still overestimates the distribution of losses up to a confidence level of 99%, while in Panel C, the underestimation starts earlier, around the 95% percentile of the distribution. In both cases, the process does not capture the distribution at the extreme end of the tail (99.9%), similar to what we observed for smaller sample sizes. Nevertheless, from Panels B and C altogether, the performance of OpRisk+ still stays honourable when the confidence level of 95% is adopted. The RMSE of the estimates also remains within 20% (with the tail

Measuring operational risk in financial institutions index of 0.5) and 32% of the mean (with a tail index of 0.7), which is fairly large but mostly driven by large outliers as witnessed in the last column of each panel. A correct mean estimate of the OpVaR95 would apply to a tail index between 0.5 and 0.7, which corresponds to a distribution with a fairly large tail index. Only when the tail of the Pareto-distribution is actually thin, one observes that the intervalling effect induces a large discrepancy between the theoretical and observed values. Let us mention that the good empirical application of OpRisk+ does not depend on the number of observed losses as it only affects the first term of the recurrence, namely A0.

Comparison with the CCHP and the Alexander’s approaches These results, if their economic and statistical significance have to be assessed, have to be compared with a method that aims at specifically addressing the issue of operational losses in the AMAs setup. We choose the CCHP approach, which is by definition more sensitive to extreme events than OpRisk+, but has the drawback of requiring a large number of events to properly derive the severity distributions of ‘normal’ and ‘large’ losses. For low frequency database, the optimization processes used by this type of approaches (e.g. Maximum Likelihood Estimation (MLE)) might not converge to stable parameters estimates. The graphs from Fig. 2 display the OpVaRs (with confidence levels of 90%, 95%, 99% and 99.9%) generated from three different kind of approaches, that is the sophisticated CCHP approach, OpRisk+ and the simpler Alexander (2003) approach (see the Section ‘Alexander’s IMA’) for each of the three tail index values (0.3, 0.5 and 0.7) and for each of the four sample size (20, 50, 200 and 300 loss events). From the graphs in Fig. 2, we can see that for most databases, OpRisk+ is working out a capital requirement higher than the Alexander’s IMA, but smaller than the CCHP approach. This last result could be expected as CCHP is more sensitive to extreme events. In next sub-section, we will discuss the fact that the database with 300 observations shows higher OpVaRs for OpRisk+ than CCHP. However, we can already conclude that our model is more risk sensitive than a simple IMA approach. Considering the thin-tailed Pareto-distribution in Panel A, we can observe that OpRisk+ produces the 17

1563 best estimations for the small database. Indeed, those are very close to the theoretical OpVaRs for all confidence level. However, for the large database, it is producing too cautious (large) OpVaRs. The comparison with other methods sheds new light on the results obtained with Panel A of Table 6: OpRisk+ overestimates the true VaR, but the CCHP model, especially dedicated to the measurement of operational risk, does frequently worse. Actually, Alexander’s (2003) approach, also using observed data but not suffering from an intervalling effect, works out very satisfactory results when the SD of loss is a good proxy of the variability of the distribution. For the medium and fat-tailed Pareto-distributions, neither of the models is sensitive enough for OpVaRs of 99% and more. This could raise some questions on the feasibility or appropriateness of a requirement of a 99.9% VaR by the Basel Accord, where it appears that even an LDA model is far from being able to estimating economic capital with such a high level of confidence. Nevertheless, as far as the small databases are concerned, it is interesting to note that OpRisk+ is producing the best estimations for OpVaR95. While none of these approach seems good enough for the level of confidence required by Basel II, we would first recommend OpRisk+ or Alexander’s for low frequency databases, as none of these needs the pre-determination of the shape of the distribution. Then, although Alexander’s approach is simpler and provides as good OpVaRs as our model for the thintail Pareto distribution, this method has the drawback of deteriorating much faster than OpRisk+ for larger tails. Unfortunately, risk managers usually do not know the type of distribution they are dealing with, and in this case, we would recommend the OpRisk+ method that seems a bit more complicated but yields more consistent results.

Comparison with OpRisk+ taking an average value of loss for each band As shown above, taking the upper bound limit value for aggregation as described in the CreditRisk+ model tends to overestimate the true operational risk exposure for all confidence levels; especially with larger databases. A solution could be to take the average value of losses for each band.17 Table 7 displays the results of the simulations when a relatively large sample size is used.

That is, every loss between 15 000 and 25 000 would be in band 20, instead of every loss between 10 000 and 20 000 being in band 20.

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Fig. 2. Comparison of CCHP, OpRisk+ and Alexander’s IMA approach Notes: On the basis of N simulated losses, characterized by a thin, medium or fat-tailed Pareto distribution, we computed OpVaR with level of confidence at 90, 95, 99 and 99.9 percents using three different approaches. The ‘Simulated’ value corresponds to the true value to estimate.

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Fig. 2. Continued. Table 7. Comparison of the average of the yearly OpVaRs computed with OpRisk+ using resp. an upper bound limit value (rounded up) and an average value (rounded) for the allocations into bands (see step 2 of the OpRisk+ procedure described in the Section ‘OpRisk+: application of CreditRisk+ to operational loss data’), for large databases N ¼ 200

N ¼ 300 OpRisk+

Target

Roundup

OpRisk+ Bias

Panel A: Thin-tailed-Pareto distribution 31 448 33 853 OpVaR90 OpVaR95 32 309 34 728 33 995 36 397 OpVaR99 36 063 38 310 OpVaR99.9

Round

Bias

Target

Roundup

Bias

Round

Bias

(shape parameter ¼ 0.3) 8% 30 576 3% 7% 31 404 3% 7% 32 991 3% 6% 34 813 3%

46 355 47 403 49 420 51 750

56 470 57 683 59 992 62 628

22% 22% 21% 21%

43 558 44 563 46 486 48 687

6% 6% 6% 6%

Panel B: Medium-tailed-Pareto distribution (shape parameter ¼ 0.5) 45 757 51 836 13% 44 338 3% OpVaR90 48 259 53 816 12% 46 222 4% OpVaR95 OpVaR99 55 919 57 668 3% 49 885 11% 83 292 62 237 25% 54 257 35% OpVaR99.9

67 104 70 264 79 718 113 560

75 723 78 161 82 817 88 309

13% 11% 4% 22%

64 523 66 849 71 296 76 544

4% 5% 11% 33%

Panel C: Fat-tailed-Pareto distribution (shape parameter ¼ 82 381 82 539 0% 75 696 OpVaR90 96 971 88 248 9% 81 375 OpVaR95 OpVaR99 166 962 98 972 41% 91 991 543 597 111 875 79% 104 699 OpVaR99.9

120 654 139 470 234 442 733 862

119 943 127 037 140 665 156 642

1% 9% 76% 79%

112 596 120 850 135 481 152 904

7% 13% 42% 79%

0.7) 8% 16% 45% 81%

Notes: The average value of the spread between the ‘true’ value and the mean of the yearly OpVaRs, as percents of the latter, is reported under the ‘Bias’ column. The ‘true’ value is approximated through a Monte Carlo simulation of 100 000 years of data, characterized by a frequency equal to a random variable following a Poisson (N) and a severity characterized by the selected Pareto distribution.

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Table 8. Comparison of the average of the yearly OpVaRs computed using OpRisk+ with an upper bound limit value (round up) and an average value (rounded) for the allocations into bands (see step 2 of the OpRisk+ procedure described in the Section ‘OpRisk+: application of CreditRisk+ to operational loss data’), for small databases N ¼ 20

N ¼ 50 OpRisk+

Target

Roundup

OpRisk+ Bias

Round

Bias

Target

Roundup

Bias

Round

Bias

Panel A: Thin-tailed-Pareto distribution (shape parameter ¼ 0.3) 3770 3880 3% 3535 6% OpVaR90 4073 4173 2% 3815 6% OpVaR95 4712 4744 1% 4363 7% OpVaR99 5596 5410 3% 5010 10% OpVaR99.9

8573 9030 9942 11 141

8882 9334 10 209 11 250

4% 3% 3% 1%

8074 8501 9332 10 311

6% 6% 6% 7%

Panel B: Medium-tailed-Pareto distribution (shape OpVaR90 5579 5672 2% 6364 6247 2% OpVaR95 8966 7329 18% OpVaR99 18 567 8626 54% OpVaR99.9

parameter ¼ 0.5) 5332 4% 5901 7% 6945 23% 7904 57%

12 630 13 862 18 051 33 554

12 855 13 734 15 410 17 338

2% 1% 15% 48%

11 323 12 152 13 668 14 377

10% 12% 24% 57%

Panel C: Fat-tailed-Pareto distribution (shape parameter ¼ 0.7) 9700 11 410 18% 9413 3% OpVaR90 OpVaR95 12 640 12 931 2% 10 914 14% 27 261 15 583 43% 13 353 51% OpVaR99 114 563 18 726 84% 16 290 86% OpVaR99.9

22 495 28 103 55 994 220 650

23 992 27 089 32 020 38 761

7% 4% 43% 82%

25 235 28 537 33 837 40 024

12% 2% 40% 82%

Notes: The average value of the spread between the ‘true’ value and the mean of the yearly OpVaRs, as percents of the latter, is reported under the ‘Bias’ column. The ‘true’ value is approximated through a Monte Carlo simulation of 100 000 years of data, characterized by a frequency equal to a random variable following a Poisson (N) and a severity characterized by the selected Pareto distribution.

Panel A of Table 7 shows that OpRisk+ achieves very good results for the Pareto-distribution characterized by a thin tail when using an average value for each band (‘round’ column). The OpVaR values obtained for the sample of 200 observations is very close to the theoretical value, whereas it stays within a 6% range from the ‘true’ value with a 300 observations sample, including at the Basel II level of 99.9%. When the loss Pareto-distributions are mediumtailed, the results obtained with the OpRisk+ procedure with the databases are very good for quantiles up to 95% but deteriorate for more sensitive OpVaRs. OpRisk+ is still totally unable to capture the tailedness of the distribution of aggregate losses for very high confidence interval, such as the Basel II requirement. Table 8 compares the two methods when applied to small databases of 20 and 50 observations. In such cases, OpRisk+ provides better results with the ‘round up’ solution than with the ‘round-off’ one. This bias could be due to the fact that with the second method we tend to loosen the ‘EVT’ aspect of the model. Small databases tend indeed to lack extreme losses and taking the upper bound limit value for the

aggregation makes the resulting distribution’s tail fatter.

IV. An Application to Real Loss Data As an illustration of the application of our model, we have applied the three models on operational loss data provided by a large European bank. Given that we only had a collection of 1 year of data, we could not apply the CCHP model on low frequency data. We applied the three models on the Asset management/Execution, process and delivery management cell, characterized by 169 losses for a total loss amount of 815 000 Euros and on the ‘Agency Services/Execution, process and delivery management’ cell, characterized by 18 losses for a total loss amount of 72 000 Euros. Table 9 presents the OpVaRs computed with the different models for different confidence levels. As we can see, OpRisk+ worked out, as expected, OpVaRs systematically higher than the lower bound of Alexander for high and low frequency datasets. Regarding the comparison with CCHP approach, we found close VaRs for all the percentiles, except the one required by Basel II (99.9% level of confidence),

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Table 9. Comparison of the OpVaRs (in thousands) computed on real loss data of a large European bank, using OpRisk+, CCHP and the lower bound of Alexander Asset management (169 losses)

OpVaR90 OpVaR95 OpVaR99 OpVaR99.9

Trading and sales (18 losses)

OpRisk+

CCHP

Alexander

OpRisk+

CCHP

Alexander

1085 1140 1249 1378

1041 1111 1246 1434

1081 1128 1218 1319

135 154 192 238

 – – –

133 147 174 203

Notes: The loss data were collected from the Asset management/Execution, process and delivery management and Trading and Sales/Execution, process and delivery management cells of the Basel Matrix.

where the OpVaR remains lower than the CCHP VaRs.

V. Conclusions This article introduces a structural operational risk model, named OpRisk+, that has been inspired from the well-known credit risk model, CreditRisk+, which has characteristics transposable to the operational risk modelling. In a simulation setup, we work out aggregate loss distributions and OpVaR for various confidence levels, including the one required by Basel II. The performance of our model is assessed by comparing our results to theoretical OpVaRs, to an OpVaR issued from a simpler approach, that is, the IMA approach of Alexander (2003), and to a more sophisticated approach proposed by CCHP (2008), or ‘CCHP’ approach which uses a mixture of two distributions to model the body and the tail of the severity distribution separately. The results show that OpRisk+ produces OpVaRs closer to theoretical ones than the approach of Alexander (2003), but that it is not receptive enough to extreme events. On the other hand, our goal is not to produce a complete compliant AMA model to compute regulatory capital requirements, but rather to propose a first solution to the lack of low frequency operational risk models. Besides, whereas the CCHP approach has better sensitivity to very extreme losses, the simplicity of OpRisk+ gives the model the advantage of requiring no large database in order to be implemented. Specifically, we view the value-added of the OpRisk+ procedure as twofold. First, it produces average estimates of operational risk exposures that are very satisfactory at the 95% level, which makes it a very useful complement to approaches that specifically target the extreme tail of the loss distribution. Indeed, even though the performance of OpRisk+ is clearly not sufficient for the measurement

of unexpected operational losses as defined by the Basel II Accord (the VaR should be measured with a 99.9% confidence level), it could be thought of as a sound basis for the measurement of the body of losses; another more appropriate method must relay OpRisk+ for the measurement of the far end of the distribution. Moreover, it appeared to us, that the 99.9% level of confidence required by Basel II might be quite utopian when we observe that even an LDA approach with 300 losses do not even get close to the theoretical level when the distribution is characterized with a Pareto(100;0.7). Second, despite the fact that we cannot conclude that OpRisk+ is an adequate model to quantify the economic capital associated to the bank’s operational risk, its applicability to approximate the loss distribution with small databases is proven. Even for such a small database as one comprising 20 observations, the estimation could make it attractive as a complement to more sophisticated approaches requiring large number of data per period. The fit is almost perfect when the Pareto-distribution has a thin tail, and the OpVaR95 is the closest among the three specifications tested when the tail gets fatter. Of course, this approach is still subject to refinements, and could be improved in many ways. Indeed, internal data rarely includes very extreme events (banks suffering those losses probably would no more be there to tell us), whereas the last percentiles are very sensitive to the presence of those events. The problem would therefore be to determine which weight to place on the internal data and on the external ones. From our study, we could imagine that fitting a distribution calibrated with external data, EVT or relying on scenario analysis beyond the 95% percentile would justify the use of OpRisk+ preferably to other models. This advantage can prove to be crucial for business lines or event types where very few internal observations are available, and thus where more data intensive approaches such as the CCHP would be powerless.

1568 References Alexander, C. (2003) Statistical models of operational loss, in Operational Risk: Regulation, Analysis and Management, Ft Prentice Hall Financial Times, Upper Saddle River, NJ, pp. 129–70. Basel Committee on Banking Supervision (2009) Observed range of practice in key elements of advanced measurement approaches, BIS Report. Carrillo-Mene´ndez, S. and Sua´rez, A. (2012) Robust quantification of the exposure to operational risk: bringing economic sense to economic capital, Computers and Operations Research, 39, 792–804. Chapelle, A., Crama, Y., Hu¨bner, G. and Peters, J. (CCHP) (2008) Practical methods for measuring and managing operational risk in the financial sector: a clinical study, Journal of Banking and Finance, 32, 1049–61. Chavez-Demoulin, V., Embrechts, P. and Neslehova, J. (2006) Quantitative models for operational risk: extremes, dependence and aggregation, Journal of Banking and Finance, 30, 2635–58. Credit Suisse (1997) CreditRisk+: a credit risk management framework, Credit Suisse Financial Products, Appendix A4, p. 36. Crouhy, M. G., Galai, D. and Mark, R. (2000) A comparative analysis of current credit risk models, Journal of Banking and Finance, 24, 59–117. Cruz, M. G. (2002) Modeling, Measuring and Hedging Operational Risk, Wiley Finance, New York. Cruz, M. (2004) Operational Risk Modelling and Analysis: Theory and Practice, Risk Waters Group, London. de Fontnouvelle, P., Dejesus-Rueff, V., Jordan, J. and Rosengren, E. (2003) Capital and risk: new evidence on implications of large operational losses, Working Paper No. 03-5, Federal Reserve Bank of Boston. de Fontnouvelle, P. and Rosengren, E. (2004) Implications of alternative operational risk modeling techniques, Federal Reserve Bank of Boston. Degen, M., Embrechts, P. and Lambrigger, D. (2007) The quantitative modelling of operational risk: between gand-h and EVT, Astin Bulletin, 37, 265–91. Dunne, T. and Helliar, C. (2002) The Ludwig report: implications for corporate governance, Corporate Governance, 2, 26–31. Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin.

Appendix – CreditRisk+: The Distribution of Default Losses – Calculation Procedure CreditRisk+ mathematically describes the random effect of the severity distribution through its probability generating function G(Z)18: X1 Pðaggregated loss ¼ n  LÞzn G ð zÞ ¼ n¼0

18

Source : Credit Suisse (1997).

S. Plunus et al. Frachot, A., Georges, P. and Roncalli, T. (2001) Loss distribution approach for operational risk, Groupe de Recherche Ope´rationnelle, Cre´dit Lyonnais. Gordy, M. (2000) A comparative anatomy of credit risk models, Journal of Banking and Finance, 24, 119–49. King, J. (2001) Operational Risk: Measurement and Modelling, Wiley, New York. Malevergne, Y., Pisarenko, V. and Sornette, D. (2006) On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns, Applied Financial Economics, 16, 271–89. McNeil, A. J. and Saladin, T. (1997) The peak over thresholds method for estimating high quantiles of loss distributions, ETH Preprint. Mignola, G. and Ugoccioni, R. (2006) Sources of uncertainty in modeling operational risk losses, Journal of Operational Risk, 1, 33–50. Moscadelli, M. (2004) The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee, No. 517, Banca d’Italia. Peters, G. W., Shevchenko, P. V., Young, M. and Yip, W. (2011) Analytic loss distributional approach models for operational risk from the stable doubly stochastic compound processes and implications for capital allocation, Insurance: Mathematics and Economics, 49, 565–79. Ross, J. (1997) Rogue trader: how I brought down Barings Bank and shook the financial world by Nick Leeson, Academy of Management Review, 22, 1006–10. Sahay, A., Wan, Z. and Keller, B. (2007) Operational risk capital: asymptotics in the case of heavy-tailed severity, Journal of Operational Risk, 2, 61–72. Sheaffer, Z., Richardson, B. and Rosenblatt, Z. (1998) Early-warning-signals management: a lesson from the Barings crisis, Journal of Contingencies and Crisis Management, 6, 1–22. Stonham, P. (1996) Whatever happened at Barings? Part two: unauthorised trading and the failure of controls, European Management Journal, 14, 269–78. Walker, P. L., Shenkir, W. G. and Hunn, C. S. (2001) Developing risk skills: an investigation of business risks and controls at Prudential Insurance Company of America, Issues in Accounting Education, 16, 291–313.

Comparing this definition with the Taylor series expansion for G(z), the probability of a loss of n  L, An, is given by  1 dn GðzÞ ¼ An Pðloss of nLÞ ¼ n! dzn z¼0 In CreditRisk+, G(Z) is given in closed form by GðzÞ ¼

m Y j¼1

j

ej þj z ¼ e



Pm

j¼1

Pm

j þ

j¼1

j zj

1569

Measuring operational risk in financial institutions Therefore, using Leibniz formula we have  ! m  1 dn GðzÞ 1 dn1 dX j  GðzÞ: j z  ¼  n n1  n! dz  n! dz dz j¼1 z¼0 z¼0 ! n1 n  1 dnk1 1X ¼ GðzÞ dznk1 n! k¼0 k ! m  dkþ1 X  j zj   kþ1  dz j¼1

and by definition   dnk1  G ð z Þ   dznk1

! m  dkþ1 X j   z  j  dzkþ1 j¼1 z¼0

j ðk þ 1Þ! if k ¼ j  1 for some j ¼ 0 otherwise

¼ ðn  k  1Þ!Ank1

Therefore An ¼

z¼0

However

z¼0

¼

X

kn1 k¼j 1for some j m X j j

j:j n

n

 1 n1 ðk þ 1Þ!ðn  k  1Þ!j Ank1 n! k

An

Using the relation "j ¼ vj : j , the following recursive equation is obtained: X "j Anj An ¼ n j:j n

Journal of Risk Research, 2014 http://dx.doi.org/10.1080/13669877.2014.910678

Misclassifications in financial risk tolerance Caterina Lucarellia*, Pierpaolo Ubertib and Gianni Brighettic a

Faculty of Economics, Department of Management, Università Politecnica delle Marche, Ancona, Italy; bDepartment of Economics, Università degli Studi di Genova, Genova, Italy; c Department of Psychology, Università degli Studi di Bologna, Bologna, Italy (Received 14 April 2012; final version received 11 March 2014) This paper analyses the empirical risk tolerance of individuals and the role of physiological measures of risk perception. By using a test that mimics the financial decision process in a laboratory setting (N = 445), we obtained an ex-post empirical measure of individual risk tolerance. Predictive classification models allow us to evaluate the accuracy of two alternative risk-tolerance forecasting methods: a self-report questionnaire and a psycho-physiological experiment. We find that accuracy of self-assessments is low and that misclassifications resulting from questionnaires vary from 36 to 65%: individuals asked to self-evaluate their risk tolerance reveal a high probability of failing their judgement, i.e. they behave as risk takers, even if, before the task, they define themselves as risk averse (and vice versa). Conversely, when the risk-tolerance forecast is obtained from individuals’ physiological arousal, observed via their somatic activation before risky choices, the rate of misclassification is considerably lower (~17%). Emotions are confirmed to influence the financial risk-taking process, enhancing the accuracy of the individual risk-tolerance forecasting activity. Self-report questionnaires, conversely, could lead to inadequate risk-tolerance assessments, with consequent unsuitable investment decisions. Bridging these results from the individual to the institutional level, our findings should enhance cautiousness, among regulators and financial institutions, on the (ab)use of risk tolerance questionnaires as tools for classifying individuals’ behaviour under risk. Keywords: risk perception; decision-making; emotion recognition; classification methods; Iowa Gambling Task; skin conductance response Subject JEL Classification Codes: D81, C91, C38

1. Introduction Risk propensity affects judgement and decision-making in many different domains (e.g. health, career, recreation, safety, finance), which renders it complex to define and to measure (Nicholson et al. 2005). Finance is a particular domain in which risk tolerance ordinarily guides behaviours and decisions, and it represents a natural realm for testing evaluation methods, at the agent level. This paper aims at measuring and predicting tolerance for financial risk, by comparing two forecasts: the first, resulting from self-assessment of individual risk propensity/aversion; the second, using physiological measures of risk perception. We administer a self-report *Corresponding author. Email: [email protected] © 2014 Taylor & Francis

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financial risk tolerance questionnaire, and we measure physiological arousal recorded before risky choices via change in individual skin conductivity. We compare these two forecasts with a standardized ex-post risk-tolerance measure, shown during an experimental task. Predictive classification models (confusion matrices and revisions of Receiver Operating Characteristic [ROC] curves) indicate the accuracy of the two predictions. The theoretical framework of this paper is the ‘risk-as-feelings’ hypothesis of Loewenstein et al. (2001), which supports the relevance of emotions when people engage in risky decision-making: from a range of clinical and physiological studies, they provide evidence that emotional reactions to risky situations often diverge from their cognitive assessments. Measures and forecasts of personal risk tolerance here are empirically obtained from an in-person experimental task (N = 445) based on the Iowa Gambling Task (IGT) and the simultaneous collection of skin conductance response (SCR). Before the psychophysiological experiment, individuals complete a financial risk tolerance questionnaire. We use a self-report tool similar to those commonly employed to evaluate ex-ante individual risk tolerance, at both operational and institutional levels. Therefore, the assessment of its forecasting accuracy seems to be particularly intriguing. The IGT, conducted jointly with the SCR measurement, is an experiment generally used in psychological and clinical studies; it mimics a risk-taking decision process in a laboratory setting (Bechara and Damasio 2002; Bechara et al. 2005). This task has been also exploited to experimentally validate the Somatic Marker Hypothesis (henceforth, SMH), according to which Antonio Damasio postulated that somato-visceral signals from the body ordinarily guide individuals’ decision-making and risk engagement processes (Damasio 1994; Bechara et al. 1997). We demonstrate that the IGT allows one to distinguish, in a mean-variance framework, behaviours of financial risk aversion from those related to risk propensity. The task returns an immediate empirical (ex-post) quantification of risk aversion/propensity and allows us to avoid any arbitrary theoretical statement on agents’ utility function.1 During the task, we measured physiological arousal as a change in electrodermal activity, specifically, the anticipatory somatic response of individuals to the IGT risk-taking activity; it corresponds to the SCR calculated 5 s before each risky choice. Even if it is a multifaceted phenomenon, SCR is considered a valuable tool, in judgement and decision-making research, for studying psychological processes related to sympathetic arousal and affective processes (Figner and Murphy 2011). It is clear that both the PDFRT assessment tool (self-report questionnaire) and the anticipatory SCR work as forecasts because they provide individual risktolerance measurements technically recorded ‘before’ the IGT risky choices made by individuals. 2. Emotions, decisions under risk and investment behaviour Weber, Blais, and Betz (2002, 222) refer to risk attitude as ‘a person’s standing on the continuum from risk aversion to risk seeking’, and they contend that the degree of risk taking is highly domain specific. As said, this paper explores tolerance of financial risk. Psychology and neuroscience widened the understanding of judgement and behaviour under risk, including affect and emotional processes (among others,

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Grossberg and Gutowski 1987; Zuckerman 1994; Loewenstein 2000; Peters and Slovic 2000; Olsen 2001; Loewenstein and Lerner 2003). This approach constitutes a clear development of traditional economic literature that correlates individual heterogeneity in financial risk tolerance with socio-demographic features (e.g. Gooding 1975; McInish 1982; Riley and Chow 1992) and that traditionally attributes decision-making to cognitive processes, from the Expected Utility Theory of Von Neumann and Morgenstern (1944) to the Modern Portfolio Theory of Markowitz (1952). Our conceptual background also overcomes behavioural theories, such as the Prospect Theory of Kahneman and Tversky (1979) or the Behavioural Portfolio Theory (Shefrin and Statman 2000), because they remain strongly cognitive and consequentialist (Loewenstein et al. 2001). The inclusion of emotions in decision-making gives rise to a vast literature that explores at least three areas of interest. A first issue is whether affect improves, or worsens, financial performance. Some studies observe patients with neurological disease that impairs their emotional response to risk taking and offer contradictory findings: some report that individuals with emotional dysfunction tend to perform poorly (Bechara et al. 1997), and others report the opposite (Shiv et al. 2005). The negative, or positive, role of emotions in investment decision-making has also been explored with traders. Lo and Repin (2002), for example, record the results of psycho-physiological measures of emotional responses (skin conductance, blood volume pulse, heart rate, electromyographical signals, respiration and body temperature) on a pilot sample of traders during live trading sessions, and find that trading experience influences physiological responses to financial choices. In a subsequent study, Lo, Repin, and Steenbarger (2005), from an online survey, observe that one component of successful trading may be a reduced level of emotional reactivity. A second stream of literature investigates whether personal traits, or specific emotions, influence behaviours under risk. Among others, Nicholson et al. (2005) indicate that risk propensity is strongly rooted in personality, influenced by extraversion, openness, neuroticism, agreeableness and conscientiousness. Lerner and Keltner (2001), on the other hand, find that fear and anger differently affect risk preferences: fearful people tend to show risk-averse behaviours, whereas angry people, similar to happy people, tend to make risk-seeking choices. The third area of interest is a natural consequence of the so-called emotion revolution (Weber and Johnson 2009) of the last 10 years of behavioural studies, and it addresses issues of the localization and measurement of emotions. On the one hand, neurophysiology research seeks to understand the neural mechanisms that govern brain activations under risky and ambiguous situations, by exploiting the potential of functional magnetic resonance imaging (fMRI; among others, see Levy et al. 2010). On the other hand, research on judgement and decision-making intensifies the emphasis on measures of emotional arousal, like SCR, that could, conversely, be based on cheap and non-intrusive experimental tools, suitable for large-scale experiments (Figner and Murphy 2011). Even if measures of physiological arousal are ‘neutral’ gauges because they cannot deal with a valence-based approach to affect (i.e. to distinguish good or bad emotions, like pleasure, joy, pain or panic), they offer an objective and reliable estimate of the ‘intensity’ of individual emotional activation. This renders physiological measures particularly appropriate for studying people’s reactions in conditions of risk, in various domains. Skin conductance, or galvanic skin response, for example, has been largely used to assess autonomic

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response in subjects suffering from various diseases, for example, in those with permutation alleles (55–200 CGG repeats) of the fragile X (Hessl et al. 2007), or those with obsessive-compulsive disorders (Starcke et al. 2009 ) or in Alexithymic patients (Nandrino et al. 2012). SCR is applied in different stimulus conditions and related to risk perception. Kinnear et al. (2013) show the relationship between SCR and driver’s risk: SCR levels double in experienced drivers when they drive fast and hazardously, and triple in novice drivers under the same conditions. More broadly, various psycho-physiological gauges have been tested within augmented reality systems for the training of actors operating in contexts of great danger, such as natural disasters or war operations (St. John et al. 2007; Callam 2010). This paper is situated within the third stream of studies on emotions recognition and measurement: we test the predictive accuracy of traditional self-report tools against a physiological measure of risk perception, by preserving the accuracy of the analysis, with the psycho-physiological experiment, on a large and qualified sample of individuals. The paper contributes to the existing literature because it supports evidence of different reliabilities of alternative methods for predicting financial risk tolerance. 3. Methods Our empirical analysis involves mainly bank customers, but we also include financial professionals (N = 645). The width of the sample is relevant, considering the use of physiological measures (e.g. in Lo and Repin 2002, N = 10; in Bechara and Damasio 2002, N = 46 substance-dependent individuals, N = 10 with lesions of the ventromedial prefrontal cortex, and N = 49 controls). In order to recruit people seriously committed to the task, we offered participants personal psychological profiling, as feedback, instead of a monetary reward. First, we administered the financial risk tolerance questionnaire. We use the 13item risk scale developed by Grable and Lytton (1999, 2003) because it is a psychometrically designed scale used to self-assess financial risk tolerance with acceptable reliability levels and is not under copyright (Davey and Resnik 2008). It is a representative example of Psychometrically Derived Financial Risk Tolerance questionnaires (hereinafter, PDFRT assessment tools). Its scores return our first forecast of individual financial risk propensity/aversion. Next, participants engage in the psycho-physiological task based on the IGT and the simultaneous measurement of SCR. 3.1. The IGT and the ex-post empirical risk aversion/propensity Although originally intended to explain decision-making deficits in people with specific frontal lobe damage, the IGT mimics real-life decisions under conditions of reward, punishment and, overall, uncertainty and risk. The IGT requires individuals to make a sequence of 100 risky choices (from card decks A, B, C and D), reaping rewards and punishments in terms of artificial money, with the goal of maximizing profits (and minimizing losses) in terms of overall performance at the end of the task. Pre-existing financial knowledge or skills are not relevant because of the anonymity of risky assets, and the sequence of monetary outcomes is based on a ‘cognitive impenetrability’ (Dunn, Dalgleish, and Lawrence 2006, 245).

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The decks’ pay-off and risk/return combination are a reasonable simplification of investment alternatives. Some decks (C and D) are set to give low returns, but at lower risk (losses are neither frequent nor severe); other decks (A and B) offer higher returns but at higher risk (losses are more frequent and more severe). The four risk return profiles are shown in Table 1. It is evident that decks C and D satisfy the positive risk return relationship generally assumed in finance, whereas decks A and B do not. This is just the peculiar feature that renders IGT appropriate for this paper’s goal, as well as for those who used it in the previous literature, in other words, to focus on human behaviours under risk. The IGT and the risk return combination of its decks can be used in finance to classify the risk aversion and risk propensity of agents. The preference structure is easily defined starting from the assumption that an investor prefers odd moments of returns distribution (expected return, skewness, etc.) while avoiding even moments (variance, kurtosis, etc.). Intuitively, higher mean values imply higher probabilities of positive returns; conversely, higher values of variance, which is a dispersion measure, imply increasing risk (Scott and Horvath 1980). Consequently, we introduce a definition of stochastic dominance in the E-V framework, where E-V stands for Expected return-Variance. DEFINITION 1. Return rA E-V dominates rB (rA > EV rB) if lðAÞ  lðBÞ and rðAÞ  rðBÞ. The E-V dominance is a generalization of the classical second-order stochastic dominance. Under suitable regularity conditions on the individual utility function U, the E-V dominance can be characterized as U′ > 0, U″ < 0. However, it is always possible to find a couple rA and rB, and a utility function U*, such that rA > EV rB but U*(rB) > U*(rA); that is, there is no general compatibility between expected utility theory and stochastic dominance (see e.g. Brockett and Kahane 1992). This incompatibility does not affect our analysis, for two reasons. First, it is always possible to restrict the set of Von Nuemann–Morgenstern expected utility functions in order to obtain a correspondence with the E-V dominance rule. Second, our theoretical foundations (‘risk-as-feelings’ hypothesis) render inappropriate the use of cognitive utility functions (Bechara and Damasio 2005). In accordance with our preference structure based on stochastic dominance, the IGT returns an immediate empirical (ex-post) quantification of risk aversion/propensity: agents are considered to show risk aversion when choosing from decks C and D with a frequency higher than 0.5 at the end of the task, and a risk propensity when choosing from decks A and B with a frequency higher than 0.5 at the end of the task. This statement appears coherent with the original experimental design and with Bechara and Damasio’s (2002, 1677) qualification of decks: A and B are defined ‘in the long run disadvantageous’ and C and D ‘advantageous’, because, at the end of Table 1. Moments of the pay-off distribution of the four possible choices.

Expected pay-off Standard deviation

A

B

C

D

−28.233 136.613

−31.933 384.083

26.447 26.864

28.449 70.168

Note: Table 1 shows the features of monetary outcomes related to each deck. Even if participants during the IGT visualized symbols of banknotes (gained or lost), the currency was not specified.

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the task, agents who prefer decks C or D gain; the others lose. When individuals prefer decks C or D it means they are aware, either consciously or unconsciously, of the strict dominance of these choices, compared to the others. Their behaviour is defined as risk averse because they take risks, but only if this risk is properly rewarded. On the contrary, decks A and B induce one to face a higher risk that is not adequately remunerated. We assume that individuals who prefer such decks make this choice because they are just willing to take risks, and this reveals their risk propensity. Table 2 offers values of the ex-post empirical risk aversion/propensity shown by individuals, also according to specific sub-groups by gender, education and profession. One might presume that the preference for ‘disadvantageous decks’ is unlikely in ‘normal’ agents, without any specific neurological deficit, because Table 1 clearly shows that decks A and B involve higher risks with negative expected outcomes. Nevertheless, Table 2 indicates that a large number of individuals (N = 264) prefer decks A and B, thus showing empirical risk propensity, compared with those (N = 181) preferring decks C and D, thus showing empirical risk aversion. 3.2. The SCR measurement SCR is a cheap, unobtrusive and reliable measure, one that works as a proxy for neural and brain activation (Figner and Murphy 2011) because of the network synchronization between central and peripheral systems. SCR is measured as the voltage drop between two electrodes placed on the skin of the participant (Figner and Murphy 2011). Changes in SCR occur when the eccrine sweat glands, which are innervated by the sympathetic autonomic nervous system fibres, receive a signal from a certain part of the brain. In our experiment, recording of SCR begins at least 10 min before the beginning of the IGT, and continues throughout. We use the Biopac MP150 system (Biopac

Table 2. Sample and relevant sub-groups by empirical risk aversion/propensity. Total

Empirical risk aversion (frequency of choices C or D > 0.5)

Overall sample 445 Sub-sample 1: gender Males 348 Females 97 Sub-sample 2: education Not degree 199 Degree 246 Sub-sample 3: financial profession Financial 301 customers Traders and 144 asset managers

Empirical risk propensity (frequency of choices A or B > 0.5)

181

264

141 40

207 57

67 114

132 132

106

195

75

69

Note: Table 2 shows some relevant socio-demographic features of our sample and offers a synthetic portrait of how individuals behaved during the IGT. The majority preferred the ‘disadvantageous’ decks (frequency of choices A and B higher than 50%). Only for the sub-sample of traders and asset managers did the majority of individuals prefer the ‘advantageous’ decks.

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Systems, CA, USA). Filtering rate is set at 1 Hz. The task duration is about 30 min for each participant. For the majority of healthy subjects, the task consists of two phases: an early phase (first 20 choices), where subjects learn to make choices, but without having any explicit knowledge of the contingencies guiding their decisions (decision under ambiguity), and a later phase (last 80 choices), where risks associated with each deck become explicit (decision under risk). Coherent with our paper’s purpose, computations refer to this last set of choices. Somatic reactions to rewards (gains at the IGT) and punishments (losses) are generated after each card selection so that individuals begin to trigger anticipatory reactions (anticipatory SCR) that will guide their forthcoming choices. The somatovisceral signals from the body are affective reactions measured ‘before’ the choice. Coherent with Damasio’s SMH, these anticipatory SCRs should ordinarily guide the individual’s decision-making, according to his or her personal risk tolerance. The anticipatory SCR is technically ‘before’ each risky choice and works as our second risk-tolerance forecast. 4. Predictive classification models In a binary risk-tolerance classification model, the two classes can be represented by 1 and 0, that is, by the presence or absence of risk aversion. A classification model provides an assignment procedure for classifying an agent according to one of the two pre-defined categories. Binarization of behaviours clearly simplifies reality and neglects the nuances of individual risk aversion. Nevertheless, the partition of riskseekers from risk-averse investors is necessary to answer our research question. In general, any assignment procedure is imprecise: errors occur, meaning that sometimes an agent is assigned to an incorrect class. Incorrect classifications persist independent of the kind of information processed to classify. Although misclassifications can be minimized, in a predictive framework they are always present. The information used to classify an agent is a vector of descriptive variables, characteristics or features. Information depends on the typology of the classifying variable: for a nominal variable, values are categories; for a binary variable, values are nominal variables corresponding to two categories (e.g. the presence or absence of something); for an ordinal variable, values are categories that are ordered in some way (e.g. absent, mild, moderate or high-risk aversion); for a discrete numerical variable, values are a finite set of possible numbers; finally, for a continuous numerical variable, values belong to either a finite or an infinite range (e.g. an investment performance). Suppose that t is the value of the threshold T in a particular classification rule, so that an individual is assigned to population P if its classification score s exceeds t and to population N otherwise. This is a simple example of a binary classification rule: a threshold t has to be chosen in order to assign an individual to one of the two categories. In order to assess the efficacy of this classifier, one must compute the probability of making an incorrect classification. This probability indicates the rate at which individuals requiring classification, in the future, will be misallocated. Let us define four probabilities and the associated rates for the classifier: (1) The probability that an individual from P is correctly classified, that is, the true positive rate tp ¼ pðs [ tjPÞ.

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(2) The probability that an individual from N is misclassified, that is, the false positive rate fp ¼ pðs [ tjN Þ. (3) The probability that an individual from N is correctly classified, that is, the true negative rate tn ¼ pðs  tjN Þ. (4) The probability that an individual from P is misclassified, that is, the false negative rate fn ¼ pðs  tjPÞ. The presence of misclassifications justifies the need to evaluate the performance of classification procedures, which can be assessed in two different ways. First, we can compare classification procedures through ‘confusion matrices’.2 Second, we can evaluate the performance of models by changing the parameters of the classification procedure and observing the classification’s performance as the threshold’s values change. This second approach is the ROC curve. ROC curves were originally used in the Signal Detection Theory (Green and Swets 1966; Egan 1975) to detect the presence of a particular signal, missing as few genuine occurrences as possible, while simultaneously raising as few false alarms as possible. ROC curves are used to compare classifiers and to visualize their performance.3 This technique is widely employed in medical decision-making, and increasingly in machine learning (Bradley 1997) and data mining research (Breiman et al. 1984). It is interesting that Signal Detection Theory has been recently transferred from ‘psychophysics’ to socio-psychological studies, to address issues of marginal trust and risk perception (Eiser and White 2006), in line with our methodological approach. The ROC curve would be worthwhile in our analysis because for both the PDFRT assessment tool and the anticipatory SCR we cannot state an exogenous and objective rule to separate risk-seeking from risk-averse individuals. Hence, the flexibility of a threshold is required. Nevertheless, given that we need to compare the accuracy of two different forecasts, we require homogenous performance evaluations. The following sections demonstrate that whereas the SCR venue offers a binomial definition of risk aversion/propensity, the PDFRT tool offers ordinal scores, that is, risk-tolerance scores resulting from the questionnaire. Consequently, the ROC curve is not applicable because results from the two classifications are not directly comparable. In order to exploit the utility of a threshold, as well as to obtain comparable performance evaluations, we use an instrument that mimics ROC curve information content and we preserve its interpretation. We draw the graph of the function f ðtÞ ¼ fpðtÞ þ fnðtÞ

(1)

where t is the threshold. This function simply describes how the total number of misclassifications changes with t. Note that in our case, the probability of a true or false classification is not required. In fact, we directly use the frequencies, absolute or relative, of correct/incorrect classifications resulting from comparisons of the two forecasts (PDFRT questionnaire and anticipatory SCR) with the IGT empirical ex-post quantification of risk aversion/propensity. 4.1. Predictive classifications from the PDFRT assessment tool The PDRT assessment tool is a questionnaire that provides a score associated with each agent. This score has been normalized to fall within a [0, 1] scale. The

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information obtained is ‘ordinal’, in the sense that we can compare every pair of agents based on their score and find the one that is more risk averse than the other. In order to obtain an absolute classification, we calculate a threshold t in some optimal way and define risk aversion/propensity by comparing the score of the questionnaire and threshold t. Note that the choice of t is not trivial. If threshold t is too close to the maximum or minimum possible score, the result of the classification will not make sense. In fact, in those two extreme situations almost all the agents are considered to be risk seekers or risk avoiders. The procedure of binarization in this case is trivial and does not need any particular explanation. It directly follows from the ordinal information provided by the PDFRT scores. 4.2. Predictive classifications from SCR The anticipatory SCR is a real number associated with a single choice of each agent. A natural choice for a classification problem based on such a measure is to select threshold t in some optimal way in order to define significant individual risk activation. The importance of considering individual risk activation relates to the fact that SCR is a physiological measure of (uncontrollable) activation that varies among agents. Whereas Damasio’s model considers only reactions of individuals to disadvantageous decks, in our experiments we uncovered a significant activation before advantageous decks as well. Keeping in mind the pay-off distribution of IGT (Table 1), we assume that agents who show stronger activation to disadvantageous decks than to advantageous ones reveal a ‘somatic reaction to risk’ (they ‘feel the risk’). Therefore, we define them as risk-averse individuals. Agents who show the opposite behaviour disclose a ‘somatic reaction to reward’, and we therefore define them as risk seekers. For this reason, we consider SCR, separately, on the ‘disadvantageous decks’ (A and B) and on the ‘advantageous’ ones (C and D). Coherent with this division, the average activation shown by each
we compute  agent before choices A or B SCR , on the one hand, and before choices C or A or B
 D SCRC or D , on the other. In addition, we calculate the standard deviation of the SCR before the choice of ‘disadvantageous decks’, rA or B , and of ‘advantageous’ ones, rC or D . Then, we compare mean values of SCR with their respective standard errors. An agent is classified as risk averse on the basis of SCR if   0 62 SCRA or B trA or B ; SCRA or B þ trA or B ; where t is the threshold. Varying t, we change the length of the confidence interval around the value of SCRA or B . In other words, t can be interpreted as a parameter that describes the level of significance for SCRA or B . For notational convenience, we rename the condition in the previous equation SCRA or B . In the same way, an agent is classified as a risk seeker on the basis of SCR if   0 62 SCRC or D trC or D ; SCRC or D þ trC or D : The choice of the classification procedure based on SCR is supported for two reasons: first, it is for the most part coherent with existing literature (Bechara et al. 1997, 2005); second, it seems to perform the best among the classification procedures we tested. Thus, our classes of risk aversion/propensity from SCR are, respectively:

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(1) true risk-averse investors (tp): when SCRA or B or SCRA or B [ SCRC or D , and frequency of choices (C or D) > 0.5. These individuals somatically activate themselves before disadvantageous decks and prefer advantageous ones (they win). This means that they ‘react to risk’ and avoid it (i.e. they refuse to take risk if not adequately rewarded). (2) false risk seekers (fn): when SCRC or D or SCRC or D [ SCRA or B , and frequency of choices (C or D) > 0.5. These individuals somatically activate themselves before advantageous decks and prefer advantageous ones (they win). They ‘react to reward’ but avoid riskier decks (i.e. even if they are attracted by earnings, they take risks only if rewarded). (3) false risk-averse investors (fp): when SCRA or B or SCRA or B [ SCRC or D , and frequency of choices (C or D) < 0.5. These individuals somatically activate themselves before disadvantageous decks and prefer disadvantageous ones (they lose). They ‘react to risk’ (they ‘feel the risk’) but end up taking on a risk that is not proportionally rewarded. (4) true risk seekers (tn): when SCRC or D or SCRC or D [ SCRA or B , and frequency of choices (C or D) < 0.5. These individuals somatically activate themselves before advantageous decks and prefer disadvantageous ones (they lose). In other words, they ‘react to reward’, and their temptation for earnings induces them to take risks that are not proportionally rewarded. 5. Results and discussion 5.1. Forecasting accuracy of the PDFRT assessment tool Function (1) allows us to control for misclassifications by changing the value of t, and provides information comparable among classifications. In Figure 1, we represent the percentage of misclassifications obtained using the PDFRT assessment tool as a function of threshold t, that is, fp + fn = f(t), where function f indicates the classification procedure. Figure 1 shows that the procedure achieves the best classification for values of the threshold close to 0. Intuitively, if the optimal threshold – in other words, the one that minimizes the misclassifications – is close to 0 or 1, all the subjects are, respectively, classified as being either risk averse or risk seeking. This implies that classification is independent of the questionnaire scores, and that its information is not useful for classifying agents. This empirical evidence indicates that the selfreport questionnaire seems to be inadequate to unambiguously distinguish between risk seekers and risk averters. Conversely, the next section shows a different finding for the anticipatory SCR. Figure 1 clearly indicates that the best classification corresponds to an optimal threshold value t* = 0. Remember that the individual score obtained from the PDFRT questionnaire falls within a [0, 1] scale. The interpretation of our result is immediate: setting the threshold t equal to 0 implies that an agent is considered to be a risk seeker if the score from the questionnaire is strictly positive, and a risk averter if the score is non-positive. The optimal confusion matrix corresponding to the optimal value of t* is presented in Table 3. It is manifest that when the threshold is set to 0, for any normalized ‘score’ obtained from the risk tolerance questionnaire, all the agents will be classified as risk

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Figure 1. Misclassifications from the PDFRT assessment tool. Table 3. The optimal confusion matrix from the PDFRT assessment tool. Risk aversion

Risk propensity

tp = 0 fp = 0

fn = 159 tn = 286

Note: This table shows classification results when comparing risk aversion of individuals (N = 445) forecasted by scores of the financial risk tolerance questionnaire, with the risk effectively taken during the IGT experiment. This is an optimal confusion matrix because it returns individuals’ categorization when threshold minimizes misclassifications; this condition is reached when t is strictly positive, that is, when all the individuals show risk propensity. Nevertheless, this is true (tn, meaning they are true risk seekers) only for 286 of them. Misclassification rate is 159 445  36%.

seekers. It follows that the confusion matrix is singular, and it contains only tn and fn. The (optimal) classification corresponds to the case where all the agents are considered to be risk seekers, independent of the score from the questionnaire. That is the case of N = 286 agents. As a final consequence, the optimal classification results independently of the questionnaire’s information. The self-report questionnaire forecasting accuracy appears quite weak, with an (optimal) misclassification rate at ~36%.4 5.2. Forecasting accuracy of SCR Performance assessment of the SCR classification requires that function f(t) be transformed as: f ðtÞ ¼ fp þ fn þ 0:5 ni

(2)

where ni (not identifiable) stands for those agents who, during the experiment, revealed an average somatic activation not significantly different from zero. It

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follows that the classification procedure described in Section 4.2 is not clearly applicable and that their risk aversion on the basis of SCR results is dubious. Agents ni do not have a position in the confusion matrix because, on the one hand, they do not technically represent a classification error; on the other hand, based on their (low) somatic activation they could be classified as either risk averse or risk seeking. This is why in function (2) we weight the class ni by 0.5: while agents belonging to fp and fn are wrongly classified with a probability of 1, the agents that belong to ni may be well classified with a probability of 0.5. Our results for function (2) are shown in Figure 2. Figure 2 is of interest because it shows how the misclassifications vary with t. The optimal threshold t*, that is, the one that minimizes the misclassifications, is equal to 0.48. The shape of function f(t)2 shows that SCR provides useful information to describe risk propensity. In particular, comparing Figures 1 and 2, we note that the first one is non-decreasing with t while the second one is non-monotone. The optimal confusion matrix corresponding to the optimal value of t* = 0.48 is presented in Table 4. Note that the confusion matrix obtained from SCR does not contain the whole sample, N = 445, but only 390 individuals, because we exclude 55 ni agents. Moreover, it is manifest that the classification procedure based on SCR has a higher predictive accuracy than the PDFRT assessment tool. First, f(t) shows that there is no independence between SCR and risk aversion. Second, the optimal classification achieved with SCR clearly performs better than that achieved from the questionnaire, with a misclassification rate of ~0.17, which is lower than the ~36% rate obtained from the PDFRT assessment tool. Nevertheless, a consistent comparison requires that we exclude the same ni agents also from the performance evaluation of the questionnaires misclassifications. This way the PDFRT forecast returns a misclassification rate of ~65%.

Figure 2. Misclassifications from the SCR assessment tool.

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Table 4. The optimal confusion matrix from SCR. Risk aversion tp = 126 fp = 53

Risk propensity fn = 14 tn = 197

Note: This table shows classification results when comparing risk aversion of individuals (N = 390; 55 ni agents excluded) predicted by the anticipatory SCR, with the risk effectively taken during the IGT experiment. This is an optimal confusion matrix because it returns individuals’ categorization when threshold minimizes misclassifications; this condition is reached when t = 0.48. In this case, we have 126 true positive (tp) individuals, because they are predicted to be risk avoiders based on the SCR and because during the IGT they prefer ‘advantageous’ decks; we also have 197 true negative (tn) individuals, because they are predicted to be risk seekers based on the SCR and because during the IGT they prefer ‘disadvantageous’ decks. Therefore, regularly classified individuals are N = 323. Misclassification rate is 14þ53 390  17%.

6. Implications and conclusion The empirical evidence in this paper indicates that the predictive accuracy of a selfreport financial risk tolerance questionnaire is weak: individuals asked to self-assess their risk tolerance reveal a relevant probability of failing in their judgement. Misclassifications of risk aversion/propensity are manifest in 36–65% of our sample. This means that an extensive share of the interviewees define themselves as risk averse in response to the PDFRT questionnaire but show risk-inclined behaviour during the experimental task (and vice versa). We believe that this is not particularly due to a different qualification of risk tolerance (appetitive vs attitude vs capacity), but because a self-report assessment is basically influenced by socialization and social construct (Brighetti and Lucarelli 2013). The predictive accuracy of anticipatory SCR, on the other hand, is considerably better. Only 17% of the individuals are misclassified when we consider their affective reactions before taking risky choices. Emotions are confirmed to guide the (financial) risk-taking process. The implications of our findings are both theoretical and practical. As far as research is concerned, in our experiment we observe a relevant somatic activation before advantageous decks, probably due to an anticipatory reward effect. In partial contrast to Damasio’s model, we suppose that this is due to the cerebral default network, that is, the wandering mind network, following Mason et al. (2007). Future experimental tasks, supported by the complementary use of fMRI, could shed light on this neurological issue. Further implications correspond to operational and regulatory levels. In this paper, we demonstrate the limited reliability of a self-reported financial risk tolerance questionnaire: individuals, when asked to self-assess their financial risk tolerance, are highly likely to fail in their judgement/forecast. This raises concerns about the regulatory role assigned to financial questionnaires for guiding investors’ decision-making. For example, the Markets in Financial Instruments Directive 2004/39/ EC (MiFID) directs intermediaries to submit questionnaires in order to profile customers and assess the appropriateness and suitability of their financial services. Frequently, within such questionnaires, the way questions elicit risk tolerance does not control for either cognitive or behavioural biases, thus inducing flawed answers (Linciano and Soccorso 2012) and inadequate risk profiling.

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European regulators have recently indicated that they seek to limiting the (ab)use of self-report questionnaires. In July 2012, after an alarming increase in the number of complaints from investors and evidence of unsuitable investment advice in many European countries (Marinelli and Mazzoli 2012), the European Securities and Markets Authorities (ESMA) published the final guidelines on certain aspects of the MiFID suitability requirements. Here, the role assigned to questionnaires appears better specified: ‘Questionnaires should not be excessively relied nor used by investment firms to reverse the burden of proof. Live discussion and interaction between firm and clients is the best method for understanding clients’ needs’ (ESMA/2012/387, 14). Ultimately, this paper indicates that individuals are not perfectly aware of some aspects of their own decision-making under risk. They are mainly responsible for these inadequate risk perceptions, because they portray themselves as different from how they act. Therefore, as Han, Lerner, and Keltner (2007) also point out, we argue that our research could benefit investors (consumers) themselves, by increasing their consciousness of the emotional component of their decision-making under risk. In this sense, communication campaigns and risk communication guidelines (Loftedt 2010), extended to the financial domain, would complement the regulatory effort in making investors/customers take fair and suitable risky decisions. Acknowledgements This research was supported by a grant from the Italian Ministry of University and Research as ‘Research of National Interest’ – PRIN 2007 (September 2008–September 2010). We thank CONSOB (Italian Securities and Exchange Commission) for valuable comments and observations.

Notes 1. We deliberately avoid traditional measures of individual risk aversion because, as Bechara and Damasio point out (2005, 337), economic models of expected utility state that ‘people established their values of wealth on the basis of the pain and pleasure that it would give them’, but these models practically neglect the role of emotion in human decisions, and are absolutely ‘inconsistent with their foundations’. 2. The confusion matrix is a mathematical instrument that allows one to describe a classification model showing correct and incorrect classifications: the ‘closer’ the confusion matrix to a diagonal matrix, the better the performance of the associated classification model. 3. The ROC curve is a graph showing a true positive rate on the vertical axis and a false positive rate on the horizontal axis, as the classification threshold t varies. 4. We are confident that results do not depend on the kind of PDFRT questionnaire used.

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Quantitative Finance, 2013 Vol. 13, No. 12, 1977–1989\, http://dx.doi.org/10.1080/14697688.2011.592854

Modeling of commercial real estate credit risks YONG KIM* Model Validation, KeyCorp, 4910 Tiedeman Road, Brooklyn, OH 44144, USA (Received 16 September 2009; revised 11 October 2010; in final form 27 May 2011) Modeling the probability of default for commercial real estate mortgages is more complicated than that for non-commercial real estate loans. This is because borrowers will default only if both the net operating income and the property value fall below the threshold levels. To make modeling more complicated, the property value at the time of default will determine the lossgiven default. In this paper, I derive closed-form solutions for the probability of default and the expected loss of commercial real estate mortgages in a Merton framework. The model is in its essence still a single risk factor model, although there is a sector risk factor that influences both the net operating income and the property value. I obtain analytically the economic capital for the corporate-wide commercial real estate portfolio, with granularity adjustments for name concentration and sector concentration. Keywords: Commercial real estate; Credit risk; Probability of default; Expected loss; Bivariate normal distribution; Concentration risk; Granularity adjustment; Economic capital JEL Classification: C1, C16, G1, G2, G13, G21

1. Introduction Commercial real estate (CRE) mortgages are major asset holdings for commercial banks, life insurance companies and thrift institutions. The slumping market for commercial real estate is the next big item that threatens further to drag down regional banks and other smaller financial institutions in the current financial crisis. Despite the prominence of CRE mortgages, modeling and analysing credit risk of CRE mortgages has been lagging behind those of non-CRE commercial loans. One of the main reasons for this is because most CRE mortgages are privately originated and placed. There is some secondary trading in whole loans, but the transactions are seldom publicly reported. Hence, a broad database is unlikely to be available in the near future. Another reason is because

credit risk of CRE mortgages is theoretically more difficult to model than that of non-CRE loans in a Merton-type model. For non-CRE commercial loans, the distance to default for the asset value is a standardized default barrier. For CRE mortgages, there are two distances to default,y one for the net operating income and the other for the property value under the assumption that borrowers will default only if the net operating income falls below the debt service amount and the property value below the mortgage balance.z To further complicate modeling, the property value at the time of default will determine the level of loss, unlike in the case of non-CRE commercial loans where the loss-given default (LGD) is usually given. Goldberg and Capone (2002) considered the ‘double trigger’ in their default model for multifamily mortgages similar to this model.

*Current address: Model Validation, BBVA Compass, 15 South 20th Street, Birmingham, AL 35232, USA. Email: yong.kim@ bbvacompass.com yIn terms of building a quantitative model for CRE credit risk, one can build a model based only on a distance to default for the property value by equating the present value of the net operating income stream to the property value as in Ciochetti et al. (2003). There are two reasons why we believe two underlying stochastic processes describe the credit risk of CRE mortgages better than a single stochastic process. First, CRE markets are not as efficient as stock markets so that changes in commercial property net operating incomes (NOI) are not quickly translated into changes in the property values. I assume in this paper that the regional economy will drive both cash flows and the property value in addition to some influence of cash flows on the value. Second, the financial industry and regulators seem to believe that NOI influences more the probability of default and the property value influences more the loss. See Real Estate/Portfolio Strategist (2001) and Board of Governors of the Federal Reserve System (2003) White Paper for this second point. zOne way to resolve the difficulty of two factors is to formulate the distance to default as a weighted average of the distance to default for NOI and that for the property value as shown in Real Estate/Portfolio Strategist (2001). ß 2011 Taylor & Francis

1978 2

Y. Kim

Their model, however, did not provide closed-form solutions for the probability of default and the expected loss. Goldberg and Capone argued that the observed low default frequencies for multifamily mortgages may be due to the ‘double trigger’ default feature built into CRE mortgages. One of the main contributions of this paper is that closed-form solutions for the probability of default and the expected loss of CRE mortgages are derived in a Merton framework. Both the probability of default and the expected loss are expressed in a bivariate normal distribution governed by two underlying stochastic processes. In particular, the expected loss is expressed as the expected value of a put option at the expiration date (i.e. at the time horizon, typically one year after). The economic capital for CRE mortgage portfolios is analytically calculated in the framework of a single systemic risk factor, not relying on Monte Carlo simulations. Most of the CRE loan portfolios are not well diversified, but more so than non-CRE loan portfolios. The number of sector CRE portfolios separated by combinations of geographic region and property type is limited. Consequently, the actual loss of the corporatewide CRE portfolio is likely to contain significant portions of idiosyncratic property-specific losses as well as sector-specific losses. The analytical calculation of the idiosyncratic risk measure in the portfolio is known as the granularity adjustment for name concentration, first proposed by Gordy (2003, 2004) and refined by Martin and Wilde (2002) using the work of Gourieroux et al. (2000).y Pykhtin (2004) recently extended the granularity adjustment formula to account for sector concentration in a multi-factor framework. Du¨llmann and Masschelein (2006) examined the impact of sector concentration on economic capital using Pykhtin’s model. Garcia Cespedes et al. (2006) took a different route by developing an adjustment to the single risk factor model in the form of a scaling factor similar to the Herfindahl index.z Most CRE credit risk models have yet to address these name/sector concentration risks in the portfolios. In this paper, I derive a granularity adjustment formula for name concentration in each portfolio, following Martin and Wilde (2002) and Pykhtin (2004). I then propose an approach to measure approximately the sector concentration risks in the corporate-wide CRE portfolio within a single systemic risk factor framework, similar to, but simpler than, that in Garcia Cespedes et al.

One reason for separating portfolios this way is because commercial real estate market research is generally done by property types and by regions. I will suppress subscripts for borrowers until the portfolio loss is considered. A representative borrower’s property value (At) and its net operating income (Ct) at time t are assumed to follow geometric Brownian motions (GBM) such that and

dAt ¼ At dt þ At dWt

ð1Þ

dCt ¼ Ct dt þ Ct dZt ,

ð2Þ

PrðAt  NÞ ¼ ðdA Þ,

ð3Þ

respectively, where dWt and dZt are Wiener processes drawn from normal distributions with mean zero and variance pdt. can be written as ffiffiffiffiffi The Wiener processes pffiffiffiffiffi dWt ’ w dt and dZt ’ z dt, where w and z are standard normal random variables. For the time being, w and z are treated as independent. Let N and D be the mortgage balance and the debt service amount of the CRE mortgage, respectively. CRE mortgages are assumed to be homogeneous to the extent that they share the same GBM parameters, ,  ,  and  , within the portfolio. Note that the signs of  and  can be negative during a down-market cycle. Let A0 and C0 be the initial property value and the property’s initial net operating income, respectively. The probability that the property value At falls below the mortgage balance N at time t is where dA is the distance to default for the property value such that dA ¼

lnðN=A0 Þ  ð   2 =2Þt pffiffi :  t

ð4Þ

The probability that the net operating income Ct falls below the debt service amount D at time t is PrðCt  DÞ ¼ ðdC Þ,

ð5Þ

where dC is the distance to default for the net operating income such that dC ¼

lnðD=C0 Þ  ð  2 =2Þt pffiffi :  t

ð6Þ

N/A0 in equation (4) is the loan-to-value (LTV) ratio, and D/C0 in equation (6) the inverse of the debt service coverage ratio at the individual loan level.

3. Probability of default and expected loss 2. Model structure Consider a portfolio of CRE mortgages with similar characteristics, i.e. the same property type in the same region.x

The property value is assumed to be positively related to the net operating income such that pffiffiffiffiffiffiffiffiffiffiffiffiffi w ¼ ’z þ 1  ’2 e, ð7Þ

yFor a more recent development of granularity adjustment for name concentration risk, see Gordy and Lu¨tkebohmert (2007). zFor comparisons among the many alternative approaches for adjustments of sector concentration risks, see a working paper by the Basel Committee on Bank Supervision (2006) and references therein. xBanks hold portfolios of CRE mortgages of many different types of properties in various regions. That means that there will be as many CRE portfolios to be considered as number of property types times number of regions that the banks engage in the CRE lending.

19793

Modeling of commercial real estate credit risks where ’ 4 0 is the correlation between At and Ct, e is a standard normal random variable, and e and z are independent of each other. The conditional probability of At  N given z is ! ! dA  ’z dA  ’z PrðAt  NjzÞ ¼ Pr e  pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð8Þ 1  ’2 1  ’2

I assume that the property owner will default if both the net operating income and the property value fall below the default barriers, i.e. At  N and Ct  D. The rationale behind this assumption is as follows. Consider the following two cases. (i) If the property value is under the water but the net operating income still exceeds the debt service amount, then the borrower will keep the excess cash flow without defaulting on the mortgage obligation. On the other hand, (ii) if the property value is above the mortgage balance but the net operating income does not cover the debt service amount, the borrower will either sell the property to repay the balance or will request the lender to restructure the mortgage, rather than default on the mortgage obligation. The probability of default (PD) is therefore PD ¼ PrðAt  N, Ct  DÞ ! Z dC dA  ’z  pffiffiffiffiffiffiffiffiffiffiffiffiffi ðzÞ dz ¼ 2 ðdA , dC , ’Þ: ¼ 1  ’2 1

ð9Þ

Equation (9) is the probability of default expressed as a cumulative bivariate normal distribution at two standardized default barriers with a correlation ’. Let L ¼ ðN  At Þþ denote the loss of the CRE mortgage upon default. The conditional loss given z is E½Ljz ¼ E½ðN  At Þþ jz ! dA  ’z ¼ N pffiffiffiffiffiffiffiffiffiffiffiffiffi  E½At jAt  N, z, 1  ’2

pffi ð1=2Þð1’2 Þ 2 tþ’ tz

¼ A t e

ð11Þ

and ð12Þ

The unconditional expected loss is 2

EðLÞ ¼ N2 ðdA , dC , ’Þ  A t eð1=2Þ t 2 ðdA pffiffi pffiffi  ð1  ’2 Þ t, dC  ’ t, ’Þ:

and

w ¼ wy y þ wx x þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2wy  2wx ,

ð15Þ

where x is a property-specific idiosyncratic risk factor for the net operating income, and  the property-specific idiosyncratic risk factor for the property value. zy 4 0 is the correlation between Ct and y, wy 4 0 the correlation between At and y, and wx 4 0 the correlation between At and x. x, y and  are mutually independent standard normal random variables. Assume further that the sector risk factor y is related to the economy-wide systemic risk factor m such that pffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ m þ 1  2 , ð16Þ

where is a sector-specific risk factor that cannot be explained by the economy-wide systemic movement. 4 0 is the correlation between y and m, and and m are mutually independent standard normal random variables. Note that wx is related to ’ such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ’ ¼ EðwzÞ ¼ E ðwy y þ wx x þ 1  2wy  2wx Þðzy y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i þ 1  2zy xÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð17Þ ¼ wy zy þ wx 1  2zy or

! pffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi dA  ’z 2  pffiffiffiffiffiffiffiffiffiffiffiffiffi   t 1  ’ 1  ’2

2 A t ¼ A0 eðð1=2Þ Þt :

The net operating income Ct and the property value At are assumed to be related to a sector risk factor y. The sector risk factor represents a regional economy that affects borrowers’ defaults in a systemic way. Then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ z ¼ zy y þ 1  2zy x

’  wy zy wx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2zy

ð10Þ

where

E½At jAt  N, z

4. Conditional probability of default, expected loss and variance of loss

ð13Þ

Expression (13) is the expected value of a put option at the expiration date that is governed by two underlying stochastic processes. Note that the loan loss allowance for this CRE mortgage is measured as E(L) in equation (13).

ð18Þ

Given two underlying stochastic processes (for At and Ct) and two systemic risk factors (one regional and the other economy-wide) in this model, it would be convenient if the conditional trivariate normal distribution on the systemic risk factor m could be expressed as a bivariate normal distribution in calculating the conditional probability of default, expected loss and variance on m. A standard trivariate normal distribution function can be defined as 3 ðb1 , b2 , b3 , RÞ

Z b1 Z b2 Z b3 1 T 1 p ffiffiffiffiffiffiffiffiffiffi eX R X=2 dx3 dx2 dx1 3=2 detR 1 1 1 ð2Þ Z b1 1 2 ex1 =2 Fðx1 Þ dx1 , ð19Þ ¼ pffiffiffiffiffiffi 2 1 where R ¼ ðij Þ is a correlation matrix, and 0 1 ¼

32  31 21 C Bb2  21 x b3  31 x FðxÞ ¼ 2 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, 1  221 1  231 1  221 1  231

ð20Þ

1980 4

Y. Kim

where 2 ðh, k, rÞ denotes a standard bivariate normal distribution function. Lemma 1 is from Genz (2004).y Lemma 1: The trivariate normal distribution can be expressed as 3 ðb1 , b2 , b3 , RÞ ¼ 3 ðb1 , b2 , b3 , R0 Þ 0 Z1 1 ef3 ð21 tÞ=2 B þ @21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu^ 3 ðtÞÞ 2 0 1  221 t2 1 f ð tÞ=2

e 2 31 C þ31 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu^ 2 ðtÞÞA dt, 1  231 t2

ð21Þ

ð22Þ

f2 ðrÞ ¼

b21 þ b23  2rb1 b3 , 1  r2

ð23Þ

f3 ðrÞ ¼

b21 þ b22  2rb1 b2 , 1  r2

ð24Þ

2

1 0 R0 ¼ 4 0 1 0 32

3

0 32 5, 1

Lemma 3: Given equations (4) and (6) for standardized threshold levels for Ct and At, and the relationships among z, w and y expressed in equations (14), (15) and (16), respectively, the conditional probability of default and the conditional expectation of loss on m can be expressed as 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B dA  wy m dC  zy m wy 1  C PDðmÞ ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1  ðwy Þ2 1  ðzy Þ2 1  ðwy Þ2 ð28Þ

and

where 3 ðb1 , b2 , b3 , R0 Þ ¼ ðb1 Þ2 ðb2 , b3 , 23 Þ,

Lemma 3 shows the conditional probability of default and the conditional expectation of loss on the systemic risk factor m.

ð25Þ

b2 ð1  231 t2 Þ  b1 tð21  31 32 Þ  b3 ð32  31 21 t2 Þ ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u^ 2 ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  231 t2 1  231 t2  221 t2  232 þ 2t2 31 21 32

ð26Þ

E½Ljm

0

1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B dA  wy m dC  zy m wy 1  C ¼ N2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1  ðwy Þ2 1  ðzy Þ2 1  ðwy Þ2  E½At jAt  N, m,

for E½At jAt  N, m

2 2 2 ¼ A t eð1=2Þð1wy Þ tþwy m 0

ð30Þ

respectively, where a1 ¼

ð27Þ

Lemma 2 further simplifies Lemma 1 as b1 ! 1. Lemma 2:

and

1  ðwy Þ2  2wx pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  t 1  ðwy Þ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zy wy ð1  2 Þ þ wx 1  2zy pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼  t: 1  ðzy Þ2

ð31Þ

ð32Þ

Proof: See appendices A and B. 0

3 ðb1 , b2 , b3 , R Þ ¼ 2 ðb2 , b3 , 23 Þ for b1 ! 1: Proof:

pffi t

1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dC  zy m wy 1  C B dA  wy m 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1  ðwy Þ2 1  ðzy Þ2 1  ðwy Þ2

and b3 ð1  221 t2 Þ  b1 tð31  21 32 Þ  b2 ð32  31 21 t2 Þ ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u^ 3 ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  221 t2 1  221 t2  231 t2  232 þ 2t2 31 21 32

ð29Þ

As b1 ! 1, f2 ðrÞ ! 1 and f3 ðrÞ ! 1. Hence, ef3 ð21 tÞ=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 and 1  221 t2

ef2 ð31 tÞ=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0: 1  231 t2

Consequently, the integral term in equation (21) approaches zero. As a result, equation (21) will become equation (22). Further, as b1 ! 1, ðb1 Þ ! 1. œ

Note that a typical non-CRE commercial loan’s expected loss is much simpler than expression (29). This is primarily because of the assumption of constant lossgiven default (LGD) for non-CRE loans.z The granularity adjustment for the degree of portfolio diversification will be considered in the next section. The calculation of the granularity adjustment, however, requires the conditional variance and its derivative with respect to m, and the conditional expectation of loss, and its first and second derivatives. Lemma 4 shows the

yFor more information on this subject, refer to Genz (2004) and references therein. zExceptions to this constant LGD assumption in the literature are Frye (2000) and Pykhtin (2003), who consider the correlation between default and recovery rates.

1981 5

Modeling of commercial real estate credit risks

such that Lki ¼ ðNki  Aki ðtÞÞþ , where Nki is the notional balance and Aki(t) is the property value at the time of default. For an individual mortgage weight, let Pk Lemma 4: The conditional variance of loss can be wki ¼ Nki =Nk for Nk ¼ nj¼1 Nkj , and for the P sector written as portfolio weight, let wk ¼ Nk =Np for Np ¼ K k¼1 Nk . varðLjmÞ Let Lk denote an individual portfolio loss, expressed as Pn k an average loss of mortgages, i.e. L w L ¼ 2 ki ki . k i¼1 ¼ N P1 ðmÞð1  P1 ðmÞÞ p ffi The expected loss of the kth portfolio is EðL Þ ¼ k 2 2 2 Pn k  2NA t eð1=2Þð1wy Þ tþwy m t P2 ðmÞð1  P1 ðmÞÞ w EðL Þ, where the expression for E(L ) is in ki ki i¼1 ki pffi 2 2 2 2 2 2 þ A 2t eð1wy Þ tþ2wy m t ðeð1wy Þ t P3 ðmÞ  P2 ðmÞ2 Þ, equation (13). The conditional expectation of the kth portfolio loss is ð33Þ nk X where L^ k ðmÞ ¼ E fEðLk jm, Þg ¼ wki EðLki jmÞ 1 0 i¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffi nk X B dA  wy m dC  zy m wy 1  2 C P1 ðmÞ ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, ¼ ð38Þ wki L^ ki ðmÞ, 2 2 2 i¼1 1  ðwy Þ 1  ðzy Þ 1  ðwy Þ conditional variance, and lemma 5 the derivative of a bivariate normal distribution.y

ð34Þ

P2 ðmÞ

0

1

pffiffiffiffiffiffiffiffiffiffiffiffiffi dC  zy m wy 1  2 C B dA  wy m ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1  ðwy Þ2 1  ðzy Þ2 1  ðwy Þ2 ð35Þ

and P3 ðmÞ

0

dC  zy m B dA  wy m ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2a1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðwy Þ2 1  ðzy Þ2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi wy 1  2 C 2a2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, 1  ðwy Þ2

¼

nk X i¼1

nk X i¼1

w2ki varðLki jmÞ

w2ki Vki ðmÞ,

ð39Þ

for varðLki jmÞ in equation (33). For the corporate-wide CRE portfolio loss, the conditional expectation is K X k¼1

wk L^ k ðmÞ,

ð40Þ

and the conditional variance is

ð36Þ

See appendix C.

Lemma 5: Let h and k be functions of m in a bivariate normal distribution 2 ðh, k, Þ. The derivative with respect to m is ! @ h  k 2 ðh, k, Þ ¼ k0  pffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ @m 1  2 ! k  h 0 ð37Þ þ h ðhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2

Proof:

Vk ðmÞ ¼ varðLk jmÞ ¼

L^ p ðmÞ ¼

and a1 and a2 are in equations (31) and (32), respectively. Proof:

for EðLki jmÞ in equation (29). The conditional variance is

See appendix D.

5. Diversification measure and granularity adjustment for the portfolio Suppose that there are K sector portfolios. Let Lki denote the ith mortgage loss of the kth (sector) portfolio at time t

Vp ðmÞ ¼

K X k¼1

w2k Vk ðmÞ:

ð41Þ

Suppose that the bank engages in the CRE lending business in many and for various types of P regions 2 properties, i.e. K k¼1 wk ! 0, and each portfolio is well Pk 2 diversified, i.e. ni¼1 wki ! 0, 8k. Then, the conditional corporate-wide portfolio loss in equation (40) would be a good approximation of the actual portfolio loss upon realizing the systemic risk factor m as both the idiosyncratic risk factor and the sector risk factor are diversified away. That is, as nk ! 1, Lk ! L^ k ðmÞ and Vk ðmÞ ! 0, 8k, and as K ! 1, Lp ! L^ p ðmÞ and Vp ðmÞ ! 0. Vasicek (2002) called this the limiting loss distribution, while Gordy (2003) called this the asymptotic loss limit. The loss of the well-diversified kth portfolio will be within L^ k ðmq Þ with the probability Pr½Lk  L^ k ðmq Þ ¼ 1  q. The 1  q confidence level is determined by the corporation’s target debt rating. For example, for AAA (or Aaa) debt rating, q ¼ 0.001. The conditional portfolio loss on mq, L^ k ðmq Þ, is known as the unexpected loss (UL) as opposed to the expected loss (EL) in equation (13). The expected loss is covered by the loan loss allowance. The difference between the unexpected loss and the expected loss is covered by the economic capital (EC).

yThe second derivative is a matter of taking an additional derivative from the result of the first derivative, and thus is not in lemma 5.

1982 6

Y. Kim

Hence, the economic capital is determined by a target debt rating and the portfolio loss distribution. Most portfolios of CRE loans are typically not well diversified. The portfolio loss is likely to contain some portion of idiosyncratic property-specific losses, and thus the conditional variance for an individual portfolio in equation (39) would be some positive number. Further, it is very unlikely that the number of sector portfolios would approach anywhere close to infinity. Consequently, sector-specific losses are likely present in the portfolio loss as the conditional variance for the corporate-wide portfolio in equation (41) would be some positive number. Let tq ðLk Þ denote the qth percentile loss of the kth portfolio at the 1  q confidence level, i.e. Pr½Lk  tq ðLk Þ ¼ 1  q. The loss of a well-diversified portfolio is an unlikely event that mq ¼ 1 ðqÞ will be close to tq ðEðLk jmÞÞ. The difference tq ðLk Þ  tq ðEðLk jmÞÞ is the expression for the exact measure of the undiversified idiosyncratic risk portion in the portfolio, which can be approximated through Taylor expansion. A granularity adjustment (GA) formula conditional on m for the kth portfolio loss is ! 1 1 d Vk ðmÞðmÞ GAk ðmÞ ¼ nk 2ðmÞ dm L^ 0k ðmÞ ! " # 1 1 L^ 00k ðmÞ 0 V ðmÞ  m þ Vk ðmÞ : ¼ nk 2L^ 0 ðmÞ k L^ 0 ðmÞ k

the mapping function is not unique. Third, since many sector factors in CRE markets tend to move closely together, the correlation coefficient matrix for sector factors in CRE markets is not likely to be positive symmetric, in which case a Monte Carlo simulation is likely to encounter difficulties in generating sector factors from a Cholesky decomposition of the correlation coefficient matrix.y P 2 Let H ¼ K k¼1 wk denote the Herfindahl index for the corporate-wide CRE portfolio, where wk is the weight for the kth sector portfolio. This is one way to quantify the sector concentration risk. Let SAk be the kth sectorspecific concentration risk adjustment. It approximates the loss due to the kth sector-specific risk present in the corporate-wide portfolio. Fromffi the volatility measure of pffiffiffiffiffiffiffiffiffiffiffiffi the kth portfolio loss qV k ðmÞ, first isolate the sectorffiffiffiffiffiffiffiffiffiffiffiffiffi

specific risk factor as 1  2k , and then account for the sector-specific risk on the property value by multiplying k,wy as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ð43Þ SAk ðmÞ ¼ k,wy ð1  2k Þw2k Vk ðmÞ: The dollar amount of the economic capital for the kth portfolio is

ECk ¼ Nk ½L^ k ðmq Þ  EðLk Þ þ GAk ðmq Þ þ SAk ðmq Þ, ð44Þ where mq ¼ 1 ðqÞ:

k

ð42Þ

Expression (42) is the adjustment for name concentration risks in a portfolio of CRE mortgages. However, for the credit risk of the corporate-wide CRE portfolio, the single systemic factor assumption (which is the one used so far) seems difficult to discern. With a limited number of portfolios separated based on the combination of geographic regions and property types, sector-specific concentration risks are not going to be diversified away completely. In this paper, undiversified sector-specific concentration risks in the corporate-wide CRE portfolio are quantified by using the Herfindahl index, an approach similar to Garcia Cespedes et al. (2006). I choose not to follow Pykhtin’s (2004) analytical GA formula in a multifactor framework for the following reasons. First, this model is in essence still a single risk factor model as the systemic sector risk factor y is assumed to be related to the economy-wide systemic risk factor m. Second, in Pykhtin’s model, the correlations between the sector risk factor and the systemic risk factor are determined by a mapping function whose parameter values are optimally chosen. However, the set of optimum parameter values in

ð45Þ

Since I have already adjusted for name and sector concentration risks, the economic capital for the corporate-wide CRE portfolio is the sum of the economic capital for individual portfolios, ECp ¼

K X k¼1

ECk :

ð46Þ

6. Numerical example Suppose that the kth sector CRE market condition is described as { ¼ 0,  ¼ 0:1,  ¼ 0,  ¼ 0:1, and ’ ¼ 0:5}.z There are a number of studies that estimate parameter values of the markets, and examine the performance of commercial real estate using the National Council of Real Estate Investment Fiduciaries (NCREIF) data.x Both mean returns for the property value and the net operating income are set to zero. The purpose of this example is to demonstrate how formulas would yield numbers. Hence, the parameters here are not necessarily representative of the current

yDu¨llmann and Masschelein (2006) who examined sector concentration risks across industries were able to obtain a positive-definite correlation coefficient matrix. zCiochetti et al. (2003) reported, among other estimations, that volatilities of the property value for different types vary between 7 and 13%, and for different regions between 11 and 23% over the period 1978–2002. Hence, the assumption of the property value volatility being 10% seems reasonable. Later, the volatility at 20% will be considered, which may be close to the current market environment. xSee Ciochetti et al. (2003) and references therein for empirical studies in this area and about NCREIF data. In NCREIF data, the property value is the appraisal value and the cash flow is the operating income for four property types in eight divisions.

1983 7

Modeling of commercial real estate credit risks Table 1. Probability of default and expected loss for five mortgages.

1 2 3 4 5

LTV

DSCR

PD

EL

EL/PD

Rating

0.95 0.90 0.85 0.80 0.75

1.10 1.15 1.20 1.25 1.30

0.1124 0.0404 0.0110 0.0022 0.0003

0.0490 0.0171 0.0045 0.0009 0.0001

0.4358 0.4238 0.4118 0.3996 0.3871

B3 B2 Ba3 Baa3 Aa3

A higher loan-to-value ratio (LTV) and a lower debt-service-coverage ratio (DSCR) indicate a higher probability of default (PD) and expected loss (EL) of CRE mortgages. PD and EL were calculated based on { ¼ 0,  ¼ 0:1,  ¼ 0,  ¼ 0:1, and ’ ¼ 0:5}. EL was calculated based on 1 dollar of notional balance and a 40% discount of the property value at the time of default.

market conditions. The probabilities of default (PD) and the expected losses (EL) of five mortgages with various terms are calculated in table 1. The last column of table 1 is for mortgages’ ratings based on PD in Moody’s rating convention. The expected loss is calculated based on 1 dollar of notional balance. It turns out, however, that the EL of CRE mortgages in this example is much lower than the conventional EL. This may be due to the fact that an important cost may have been ignored in the modeling of credit risks in CRE markets. Upon default, lenders most likely have to pay the costs incurred during the sales of the foreclosed properties in addition to the steep drop in the property value.y To capture these costs, it is assumed that there would be a 40% discount of the property value at the time of default.z This assumption brings the expected loss to a more realistic level.x EL/PD in the sixth column is approximately the lossgiven default (LGD). The sixth column shows that the loss severity increases with the PD. This result is consistent with that of Frye (2000) and Pykhtin (2003). Frye and Pykhtin, however, assume that the PD and the LGD are positively correlated,ô which is not assumed in this paper. In this model, when the PD is high, the property value would already be close to the mortgage balance. By the time the obligors default on the mortgages, the property values with a high PD are likely to fall below the balance much deeper than those with a low PD.

This positive relation between the PD and the loss severity does not require the estimation of the loss severity over at least one real estate market cycle for the following reason. During a down-turn real estate market period, many mortgages in CRE portfolios would have a higher loanto-value ratio (LTV) and a lower debt-service-coverage ratio (DSCR). Consequently, the portfolios would realize a much higher loss per mortgage than during a normal real estate market period. It seems reasonable to examine whether the discount size of property values would increase as the market moves to the down turn.? In table 2, the PD for the mortgage with LTV ¼ 0.85 and DSCR ¼ 1.20 is examined by changing one variable at a time for the worse from the CRE market parameters { ¼ 0,  ¼ 0:1,  ¼ 0,  ¼ 0:1, and ’ ¼ 0:5}. The PD increases accordingly, and changes in parameter values for the net operating income (NOI) affect the PD more severely than those for the property value, as expected. Table 3 shows the composition of the kth sector CRE portfolio with correlations { ¼ 0:5, wy ¼ 0:5, zy ¼ 0:4, and wx ¼ 0:3}k. Note that the portfolio is more heavily overloaded with high-LTV and low-DSCR mortgages to reflect the current unfavorable CRE market conditions. Table 3 assumes that this is an AA (or Aa) rating corporation whose confidence interval level is 1  q ¼ 99:5%. The economy-wide systemic risk factor mq ¼ 2:5758 that can occur with ð2:5758Þ ¼ 0:005

yUpon the default of CRE loans, lenders may need to outsource the property maintenance to firms outside of the organization. The foreclosure costs incurred to lenders may be the discounted portion of the property value, the usual transaction costs for the sales, and the excess maintenance costs until the sales of the properties are finalized. zThere was some awareness in the popular press on how costly the foreclosure processes of residential properties were. There are well-known studies on the loss severities of non-CRE industrial loans and corporate bonds. For example, Keenan et al. (2000) and Altman et al. (2005), to name just a few. However, besides the Board of Governors of the Federal Reserve System (2003) White Paper, there are few studies on the loss severities of CRE loans, and no studies for the foreclosure costs of commercial properties that I am aware of. xThe Board of Governors of the Federal Reserve System (2003) White Paper (and thus, the supervisory group) recommended 0.35 for LGD during normal periods and 0.55 during turbulent market periods for advanced IRB banks. These are for all ratings, and for both income producing real estate properties (IPRE) and high correlation properties (HVCRE). Hence, the assumption of a 40% discount in the property value at the time of default in table 2 may not be that far off. ôBoth the Basel committee and the supervisory group noted this positive correlation between PD and loss severity in CP3 (2003) and in The Board of Governors of the Federal Reserve System (2003) White Paper and ANPR (2003), and recommended that loss severities should be estimated at least over one credit cycle. ?Such empirical studies may reveal the degree of collateral deterioration during the down market environment. That is, as property owners struggle to keep up with payments, it is likely that they may neglect the maintenance of the properties. Furthermore, the effect of a down market may be disproportionately more severe in some regions and for certain types of property. For example, during a severe recession period, apartment buildings may not encounter a sharp increase in vacancy rates while office buildings may. Hence, certain types of properties in some regions may experience a downward spiral much faster due to acute competition for a limited number of tenants than, say, apartment buildings in other regions, and consequently experience further decline in property values. The empirical question is whether one can quantify the degree of deterioration and property value decline. kNote that wx is calculated from equation (18).

1984 8

Y. Kim Table 2. Comparative statistics of the probability of default. Property value Expected 0.00 0.00 0.00 –0.05 0.00 0.00

NOI Volatility

Expected

Volatility

Correlation

PD

0.10 0.15 0.10 0.10 0.10 0.10

0.00 0.00 0.00 0.00 –0.05 0.00

0.10 0.10 0.15 0.10 0.10 0.10

0.50 0.50 0.50 0.50 0.50 0.60

0.0110 0.0204 0.0252 0.0192 0.0218 0.0141

Mortgages are characterized by LTV ¼ 0.85 and DSCR ¼ 1.20. The PD in the first row was calculated based on the CRE market parameters { ¼ 0,  ¼ 0:1,  ¼ 0,  ¼ 0:1, and ’ ¼ 0:5}. By changing the market parameters one variable at a time for the worse, the PD increased, as expected.

Table 3. The kth sector portfolio.

1 2 3 4 5

LTV

DSCR

No. of loans

PDi ðmq Þ

L^ i ðmq Þ

ELi

EC1 i

L^ 0i ðmq Þ

L^ 00i ðmq Þ

Vi ðmq Þ

V0i ðmq Þ

0.95 0.90 0.85 0.80 0.75

1.10 1.15 1.20 1.25 1.30

40 20 15 15 10

0.2640 0.1205 0.0421 0.0106 0.0018

0.1200 0.0534 0.0182 0.0045 0.0008

0.0490 0.0171 0.0045 0.0009 0.0001

0.0710 0.0363 0.0137 0.0036 0.0006

–0.0377 –0.0227 –0.0101 –0.0031 –0.0006

0.0062 0.0068 0.0045 0.0019 0.0005

0.0406 0.0210 0.0076 0.0019 0.0003

–0.0090 –0.0080 –0.0041 –0.0013 –0.0003

For given correlations ¼ 0:5, wy ¼ 0:5, zy ¼ 0:4, and wx ¼ 0:3, the EL and the conditional loss were calculated based on the assumption that there would be a 40% discount of the property value at the time of default. For the sector portfolio, UL (L^ k ðmq Þ) was 0.0622 and EL 0.0238, thus the economic capital for the asymptotic case (EC1 k ) was 0.0383. The granularity adjustment for the name concentration risk in the sector CRE portfolio was calculated to be 0.0222.

will determine the conditional probability of default PDi ðmq Þ and the unexpected loss L^ i ðmq Þ. The economic capital for an asymptotic portfolio, EC1 i , is the difference between the unexpected loss and the expected loss for each mortgage i. For the kth sector CRE portfolio, the unexpected loss (L^ k ðmq Þ) was calculated at 0.0622, the expected loss (ELk) at 0.0238, and thus the economic capital rate for the asymptotic case at 0.0383. Without accounting for the concentration risk adjustments, the actual loss upon the default of CRE mortgages could be worse than that in this example, specifically due to a much worse actual portfolio composition and/or higher loss severity of the mortgages, i.e. a higher discount for the property value at the time of default in this model. Table 3 also shows the first and second derivatives of the conditional losses, L^ 0i ðmq Þ and L^ 00i ðmq Þ, respectively, and the conditional variances Vi ðmq Þ and the first derivatives V0i ðmq Þ of individual mortgages of the kth sector CRE portfolio. Using these, the kth sector CRE portfolio’s granularity adjustment for the name concentration risk in the sector CRE portfolio was calculated to be 0.0222. Without considering sector concentration risks in the corporate-wide CRE portfolio, the economic capital for this portfolio now reaches 0.0605. Suppose that there were 10 sector CRE portfolios in the organization. Table 4 shows these sector portfolios with

differing betas (the effects of the economy-wide systemic risk on each sector) between 0.3 and 0.7.y For each beta, consider two volatility levels, one for low volatilities at  ¼  ¼ 0:1 and the other for high volatilities at  ¼  ¼ 0:2. CRE portfolios with low betas are, in general, for income-producing real estate properties (IPRE), and those with high betas are for high correlation properties (HVCRE).z With these two categories, there seems to be a positive relation between the volatility and the beta (which is a correlation between the common systemic factor and the sector factor). For non-CRE credits, asset volatilities tend to be negatively related to asset correlations. Whether the volatilities are positively related to the betas for CRE credits is ultimately an empirical issue. For simplicity, it is assumed in this paper that all 10 submarkets share the same other characteristics,  ¼ 0,  ¼ 0, ’ ¼ 0:5, wy ¼ 0:5, and zy ¼ 0:4, and that the size is the same for all sector portfolios. Note that the economic capital for the asymptotic (well-diversified) portfolio should increase with the beta as the asymptotic portfolio loss should increase with the common systemic credit risk. However, table 4 shows that the granularity adjustment for the name concentration risk decreases with the beta. The rationale for this somewhat counterintuitive result is as follows. As the economy-wide systemic credit risk increases, the idiosyncratic credit

yTable 3 of the Board of Governors of the Federal Reserve System (2003) White Paper shows correlations for CRE credits to be between 16.6 and 29.5%, depending on data sets and methodologies used. In this paper, the effect of beta on CRE portfolios is wy . Hence, the equivalent correlations are between 15 and 35%, which is close to the above range in the White Paper. zIn the Board of Governors of the Federal Reserve System (2003) White Paper, certain acquisition, development, and construction loans (ADC) receive high correlation treatments (HVCRE), and most other CRE loans receive low correlation treatments (IPRE). However, the US banking agencies have the responsibility of determining which types of CRE lending should receive HVCRE.

19859

Modeling of commercial real estate credit risks

Table 4. EC for the asymptotic loss, the granularity adjustment for the name concentration, the sector adjustment for the sector concentration, and EC after adjusting for both name and sector concentrations.

1 2 3 4 5 6 7 8 9 10

Beta

Volatility

UL

EL

UL–EL

GA

SA

EC

0.30 0.40 0.50 0.60 0.70 0.30 0.40 0.50 0.60 0.70

0.10 0.10 0.10 0.10 0.10 0.20 0.20 0.20 0.20 0.20

0.0445 0.0529 0.0622 0.0723 0.0832 0.1216 0.1408 0.1615 0.1835 0.2068

0.0238 0.0238 0.0238 0.0238 0.0238 0.0708 0.0708 0.0708 0.0708 0.0708

0.0206 0.0290 0.0383 0.0484 0.0594 0.0508 0.0700 0.0907 0.1127 0.1360

0.0366 0.0276 0.0222 0.0184 0.0157 0.0359 0.0268 0.0213 0.0175 0.0147

0.0036 0.0037 0.0038 0.0037 0.0034 0.0054 0.0054 0.0053 0.0050 0.0046

0.0609 0.0604 0.0643 0.0705 0.0785 0.0920 0.1023 0.1172 0.1352 0.1552

The same assumptions as in table 3 were maintained except for beta and the volatility. The kth sector portfolio in table 3 is the third portfolio in this table. All portfolios were assumed to have the same size. The beta measures the effect of the economy-wide systemic risk on each sector. For the corporate-wide CRE portfolio, EC was 0.0937, and GA and SA were 0.0237 and 0.0044, respectively.

risk due to the name concentration deceases in relative terms. On the other hand, the effect of the beta on the sector-specific risk adjustment is weaky and not monotonic. For those sectors with high betas, an increase in the systemic credit risk factor would decrease the sectorspecific risk adjustment due to the decreased sectorspecific risk. Note that the economic capital increases, in general, with the beta after adjusting for both name and sector concentration risks in table 4. For the corporatewide CRE portfolio, the granularity adjustment (GA) and the sector-specific risk adjustment (SA) are 0.0237 and 0.0044, respectively. The economic capital after adjusting for these concentration risks is 0.0937. An analytical, closed-form solution model is obviously much more convenient than a Monte Carlo simulation model. Because of the immediate availability, transparency and tractability, it can perform various stress tests and be used as a risk management tool. For example, the risk manager can set the limit on the size of the sector portfolio based on the sensitivity test combined with the size. 7. Conclusion I have shown analytical solutions for the probability of default and the expected loss for CRE mortgages in a Merton-type model under the assumption that default occurs if both the net operating income and the property value fall below the threshold levels. Both the probability of default and the expected loss are expressed in a bivariate normal distribution, and the expected loss is expressed as the expected put option value at the expiration date. The results are easy to interpret. The closedform solutions in this model are useful in the following way. (1) In the absence of public ratings on CRE mortgage obligors, the transparency and tractability of the formulas are important for internal ratings. (2) The model can easily perform various stress tests, and be used as a risk management tool. For

example, the risk manager can set the limit on the size of the sector portfolio based on the sensitivity test combined with the size. (3) Given the lack of data, a theoretical model of credit risk is crucial in collecting further data and in designing empirical studies. For example, the loss severity is positively related to the probability of default for CRE credits in this model without an explicit assumption. Unlike for non-CRE credits, the estimation of loss severity may not be required over at least one (real-estate) market cycle. The result is useful given the lack of historical data for CRE credits. Most CRE loan portfolios are typically not that granular, i.e. the size of loans is large and the number of loans is fewer in a portfolio. Further, the number of sector portfolios separated by the combination of geographic regions and property types is limited. Hence, it is important to adjust for the name concentration risk present in individual CRE portfolios as well as to adjust for the sector concentration risk in the corporate-wide portfolio in calculating economic capital for CRE portfolios. However, most CRE credit risk models have yet to address these name/sector concentration risks in the portfolios. An analytical formula for the granularity adjustment for individual CRE portfolios and the quantification of the sector risks for the corporate-wide CRE portfolio in this paper are not only easy to implement, but are also useful for risk management in terms of transparency, tractability, and not having to rely on Monte Carlo simulations.

Acknowledgements This paper was written while I was with KeyCorp. The views expressed here are mine, not those of KeyCorp or BBVA Compass. I would like to thank Zhihong Li for discussions, and two anonymous referees for comments and suggestions. All errors are mine.

yAlthough a table with varying sector correlations similar to table 4 is not presented in this paper, the sector adjustment is much more sensitive with the sector correlation than with beta.

1986 10

Y. Kim

References Altman, E., Brooks, B., Resti, A. and Sironi, A., The link between default and recovery rates: theory, empirical evidence, and implications. J. Bus., 2005, 78(6), 2203–2227. ANPR, Risk-based capital guidelines: implementation of new Basel Capital Accord. Federal Register/Vol. 68, No. 149/ Proposed Rules, 2003. Basel Committee on Bank Supervision, The new Basel Capital Accord. Bank for International Settlements, Consultative Document, 2003. Basel Committee on Bank Supervision, Studies on credit risk concentration. Working paper, Bank for International Settlements, 2006. Board of Governors of the Federal Reserve System, Loss characteristics of commercial real estate loan portfolios. White Paper, 2003. Ciochetti, B., Fisher, J. and Gao, B., A structured model approach to estimating return and volatility for commercial real estate. Working Paper, University of North Carolina, 2003. Du¨llmann, K. and Masschelein, N., Sector concentration in loan portfolios and economic capital. Working Paper, National Bank of Belgium, 2006. Frye, J., Collateral damage. Risk, 2000, April, 91–94. Garcia Cespedes, J., Herrero, J., Kreinin, A. and Rosen, D., A simple multi-factor ‘factor adjustment’ for the treatment of diversification in credit capital rules. J. Credit Risk, 2006, Fall, 57–85. Genz, A., Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Statist. Comput., 2004, 14, 151–160. Goldberg, L. and Capone, C., A dynamic double-trigger model of multifamily mortgage default. Real Estate Econ., 2002, 30(1), 85–113. Gordy, M., A risk-factor model foundation for rating-based bank capital rules. J. Financ. Intermed., 2003, July, 199–232. Gordy, M., Granularity adjustment in portfolio credit risk measurement. In Risk Measures for the 21st Century, edited by G. Szego¨, 2004 (Wiley: New York). Gordy, M. and Lu¨tkebohmert, E., Granularity adjustment for Basel II. Working Paper, Deutsche Bundesbank, 2007. Gourieroux, C., Laurent, J-P. and Scaillet, O., Sensitivity analysis of value at risk. J. Empir. Financ., 2000, 7, 225–245. Keenan, S., Hamilton, D. and Carty, L., Historical defaults and recoveries for corporate bonds. In Credit Derivatives and Credit Linked Notes, 2nd ed., edited by S. Das, 2000 (Wiley: New York). chapter 10. Martin, R. and Wilde, T., Unsystematic credit risk. Risk, 2002, November, 123–128. Pykhtin, M., Unexpected recovery risk. Risk, 2003, August, 74–78. Pykhtin, M., Multi-factor adjustment. Risk, 2004, March, 85–90. Real Estate/Portfolio Strategist, Modeling expected default frequency: a new perspective. 2001, 5(6), 1–7. Vasicek, O., A series of expansion for the bivariate normal integral. J. Comput. Financ., 1998, 1(4), 5–10. Vasicek, O., Loan portfolio value. Risk, 2002, December, 160–162.

Appendix A: Proof of lemma 3 In order to show that the conditional probability of default can be expressed as in equation (28), we need to calculate the correlation between z and after netting the

effect of the economy-wide systemic risk factor from equations (14) and (16). Rewrite equation (14) as pffiffiffiffiffiffiffiffiffiffiffiffiffi z  zy m zy 1  2 z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  ðzy Þ2 1  ðzy Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2zy þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 5 dC ðmÞ, 1  ðzy Þ2 0

where

dC  zy m dC ðmÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  ðzy Þ2

ðA:1Þ

ðA:2Þ

The correlation between z0 and in equation (A.1) is z jm

pffiffiffiffiffiffiffiffiffiffiffiffiffi zy 1  2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  ðzy Þ2

ðA:3Þ

From equations (A.1) to (A.3), equation (A.1) can be rewritten as x 5 dC ðm, Þ,

ðA:4Þ

where dC ðm, Þ ¼

dC ðmÞ  z jm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2z jm

ðA:5Þ

Similarly, the correlation between w and after netting the effect of the economy-wide systemic risk factor can be written as follows. From equations (15) and (16), pffiffiffiffiffiffiffiffiffiffiffiffiffi w  wy m wy 1  2 wx w ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 2 1  ðwy Þ 1  ðwy Þ2 1  ðwy Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2wy  2wx ðA:6Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   dA ðmÞ, 1  ðwy Þ2 0

where

dA  wy m dA ðmÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  ðwy Þ2

ðA:7Þ

From equation (A.6), the correlation between w0 and

can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffi wy 1  2 w jm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðA:8Þ 1  ðwy Þ2 and the correlation between w0 and x as wx wxjm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  ðwy Þ2

ðA:9Þ

1987 11

Modeling of commercial real estate credit risks From equations (A.6) to (A.9), equation (A.6) is rewritten as w0  w jm

wxjm w ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 1  2w jm 1  2w jm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2w jm  2wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ   dA ðm, Þ, 1  2w jm 00

Note that wzjm ¼ EðwzjmÞ ¼ E½ðw jm þ wxjm x þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1  2z jm xÞjm

pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wy 1  2 ¼ w jm z jm þ wxjm 1  2z jm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  ðwy Þ2

ðA:10Þ

where

dA ðmÞ  w jm

dA ðm, Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2w jm

ðA:17Þ

ðA:11Þ

The correlation between w00 and x is

Hence,

wxjm wxjm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2w jm 

dA ðm, Þ  wxjm x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  2wxjm

ðA:12Þ

BdC ðmÞ  z jm C ¼ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: 1  2z jm

ðA:13Þ

ðA:14Þ

Similarly, the conditional probability that At 5 N given x and fm, g is obtained from equations (A.11) and (A.13) as 1 0 dA ðm, Þ  wxjm xC B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Pðwjx, m, Þ ¼ Pr@  1  2wxjm

1 0 BdA ðm, Þ  wxjm xC ¼ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: 1  2wxjm

Hence, the conditional correlation in equation (A.16) satisfies the correlation expression of the bivariate normal distribution in equation (20), i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   þ  1  2z jm  w jm z jm w jm z jm wxjm wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : 1  2w jm 1  2z jm 1  2w jm

ðA:18Þ

The conditional probability that Ct 5 D given m and

can be obtained from equations (A.4) and (A.5), i.e. 0 1 dC ðmÞ  z jm C B Pðzjm, Þ ¼ Pr@x  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1  2z jm 0 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2w jm  2wxjm Þðz jm

ðA:15Þ

The conditional probability of default on fm, g can be calculated as

The probability of default conditional only on the economy-wide systemic risk factor m can be expressed as PDðmÞ ¼ PrðAt 5 N, Ct 5 DjmÞ Zb dA ðmÞ  w jm dC ðmÞ  z jm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ¼ 2 1 1  2w jm 1  2z jm  !  wxjm  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ d  :  2 1 w jm

b!1

ðA:19Þ

The solution to equation (A.19) involves the trivariate normal integral. Applying lemmas 1 and 2, the formula for the standard trivariate normal integral in terms of the bivariate normal integral becomes  PDðmÞ ¼ ðbÞ2 ðdA ðmÞ, dC ðmÞ, wzjm Þb!1 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi B dA  wy m dC  zy m wy 1  2 C ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: 1  2zy 2 1  ðwy Þ2 1  ðwy Þ2 ðA:20Þ

Appendix B: Proof of lemma 3 (continued)

PDðm, Þ ¼ PrðAt 5 N, Ct 5 Djm, Þ 0

1

dA ðm, Þ  wxjm x dC ðmÞ  z jm

C B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , x  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jm, A ¼ Pr@  2 2 1  wxjm

1  z jm 0 1 wxjm C BdA ðmÞ  w jm dC ðmÞ  z jm

¼ 2 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: 1  2w jm 1  2z jm 1  2w jm

ðA:16Þ

The conditional expectation of loss given x and fm, g is E½Ljx, m,  ¼ E½ðN  At Þþ jx, m, :

ðB:1Þ

Then, 0

1

BdA ðm, Þ  wxjm xC E½NjAt  N, x, m,  ¼ N  @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1  2wxjm

ðB:2Þ

1988 12

Y. Kim

and

and

E½At jAt  N, x, m,  ¼ A t e 0

ð1=2Þð12wy 2wx Þ 2 tþðwy ð mþ

pffiffiffiffiffiffiffiffi

1 2 Þþ

pffi wx xÞ t

Note that

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi C BdA ðm, Þ  wxjm x @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  2wy  2wx  tA: 2 1  wxjm

ðB:3Þ

Fð 0 Þ ¼ 2

ðB:4Þ

It is easy to see now that E½NjAt  N, m,  ¼ N2 ðdA ðm, Þ, dC ðm, Þ, wxjm Þ ðB:5Þ

where

2

¼ A t eð1=2Þð1wy Þ

1 2 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi   1   wx C pffiffi wy B a1 ¼ @w jm wy 1  2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A t 1  ðwy Þ2 ¼ and

2

tþwy ð mþ

pffiffiffiffiffiffiffiffi

pffi 1 2 Þ t

1  2wy  2wx pffiffi dA ðmÞ  w jm

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  t, 1  2w jm 1  2w jm 1  ðwy Þ2 ! pffiffi dC ðmÞ  z jm

wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  wx  t, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðB:6Þ 1  2z jm 1  2w jm  2

Finally, the conditional expectation of loss given m is

ðB:12Þ

0

and E½At jAt  N, m, 

ðB:11Þ

dA ðmÞ  a1  w jm 0 dC ðmÞ  a2  z jm 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , , 1  2w jm 1  2z jm !

wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1  2w jm

Integrating both sides of equation (B.1) over x, the conditional expectation of loss given fm, g becomes E½Ljm,  ¼ Ex fE½Ljx, m, g ¼ Ex fE½ðN  At Þþ jx, m, g:

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi b0 ¼ b  wy 1  2  t:

1  ðwy Þ2  2wx pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  t 1  ðwy Þ2

ðB:13Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ z jm wy 1  2 þ wx 1  2z jm  t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zy wy ð1  2 Þ þ wx 1  2zy pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  t: 1  ðzy Þ2

ðB:14Þ

Hence, equation (B.9) becomes

pffi 2 2 2 E½At jAt  N, m ¼ A t eð1=2Þð1wy Þ tþwy m t 2 ðdA ðmÞ ðB:15Þ  a1 , dC ðmÞ  a2 , wxjm Þ:

E½Ljm ¼ E fE½Ljm, g ¼ E fE½ðN  At Þþ jm, g: ðB:7Þ Given m, the conditional expectation of the notional balance in the event of default is E½NjAt  N, m 0

1

pffiffiffiffiffiffiffiffiffiffiffiffiffi B dA  wy m dC  zy m wy 1  2 C ¼ N2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, 1  ðwy Þ2 1  ðzy Þ2 1  ðwy Þ2

ðB:8Þ

and the conditional expectation of the property value can be expressed as E½At jAt  N, m  Zb   ¼ E½At jAt  N, m, ð Þ d   1 b!1  Z 0  b p ffi 2 2 2  ¼ A t eð1=2Þð1wy Þ tþwy m t Fð 0 Þð 0 Þ d 0   1

The conditional variance of loss can be written as varðLjmÞ ¼ EðL2 jmÞ  ½EðLjmÞ2 :

ðC:1Þ

From equation (29), it can be shown that ½EðLjmÞ2 ¼ N2 P1 ðmÞ2

pffi 2 2 2  2NP1 ðmÞA t eð1=2Þð1wy Þ tþwy m t P2 ðmÞ pffi 2 2 2 ðC:2Þ þ A 2t eð1wy Þ tþ2wy m t P2 ðmÞ2 :

Since EðL2 jmÞ ¼ EðN2  2NAt þ A2t jAt  N, mÞ, EðL2 jmÞ ¼ N2 PDðmÞ  2NEðAt jAt  N, mÞ þ EðA2t jAt  N, mÞ:

ðC:3Þ

For E½A2t jAt  N, m, we first require to calculate ,

b0 !1

ðB:9Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi

0 ¼  wy 1  2  t

Appendix C: Proof of lemma 4

ðB:10Þ

E½A2t jAt  N, x, m,  2 2 2 ¼ A 2t e2ð1wy wx Þ tþ2 0

wy mþ

pffiffiffiffiffiffiffiffi



pffi 1 2 þwx x  t

1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiC BdA ðm, Þ  wxjm x  @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1  2wy  2wx  tA: 1  2wxjm

ðC:4Þ

1989 13

Modeling of commercial real estate credit risks Then, E½A2t jAt  N, m,  2

¼ A 2t e2ð1wy Þ

2

tþ2wy ð mþ

Substituting equations (C.2) and (C.12) into equation (C.1), the conditional variance becomes the expression in equation (33).

pffiffiffiffiffiffiffiffi

pffi 1 2 Þ t

1  2wy  2wx pffiffi dA ðmÞ  w jm

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  t, 1  2w jm 1  2w jm 1  ðwy Þ2 ! pffiffi dC ðmÞ  z jm

wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2wx  t, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðC:5Þ 1  2z jm 1  2w jm

 2

Finally,

E½A2t jAt  N, m  Zb   2 ¼ E½At jAt  N, m, ð Þ d   1 b!1  Z b00  p ffi 2 2 2  2 2ð1 Þ tþ2 m t 00 00 00 wy wy ¼ A t e Fð Þð Þ d   1

where

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi

00 ¼  2wy 1  2  t

and

Note that Fð 00 Þ ¼ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi b00 ¼ b  2wy 1  2  t:

,

ðC:7Þ ðC:8Þ

ðC:9Þ

E½A2t jAt  N, m

Since P3 ðmÞ

ðC:10Þ

0

1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d   m d   m  1  wy C zy wy B A C ¼ 2 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2a1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2a2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, 2 2 2 1  ðwy Þ 1  ðzy Þ 1  ðwy Þ ðC:11Þ

equation (C.3) becomes pffi 2 2 2 EðL2 jmÞ ¼ N2 P1 ðmÞ  2NA t eð1=2Þð1wy Þ tþwy m t P2 ðmÞ pffi 2 2 2 þ A 2t e2ð1wy Þ tþ2wy m t P3 ðmÞ:

ðC:12Þ

ðD:1Þ

The first derivative of equation (D.1) is ! h  k 0 0 ½2 ðh, k, Þ ¼ k  pffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ 1  2 h0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2

ðC:6Þ

where a1 and a2 are in equations (31) and (32), respectively. Consequently, equation (C.6) becomes pffi 2 2 2 ¼ A 2t e2ð1wy Þ tþ2wy m t 2 ðdA ðmÞ  2a1 , dC ðmÞ  2a2 , wxjm Þ:

A bivariate normal distribution can be written as ! Zk h  y  pffiffiffiffiffiffiffiffiffiffiffiffiffi ð yÞ dy: 2 ðh, k, Þ ¼ 1  2 1

b00 !1

dA ðmÞ  2a1  w jm 00 dC ðmÞ  2a2  z jm 00 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , , 1  2w jm 1  2z jm !

wxjm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1  2w jm

Appendix D: Proof of lemma 5

! h  y  pffiffiffiffiffiffiffiffiffiffiffiffiffi ð yÞ dy: 1  2 1

Z

k

ðD:2Þ

Within the integration in equation (D.2) is ! h  y  pffiffiffiffiffiffiffiffiffiffiffiffi ðyÞ 1  2 2 !2  3 1 h  y 1 25 pffiffiffiffiffiffi exp½y2 =2 ¼ pffiffiffiffiffiffi exp4 pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1



1 h2 . 1 ðy  hÞ2  ðhÞ2 . 2 pffiffiffiffiffiffi exp  ¼ pffiffiffiffiffiffi exp  2 1  2 1  2 2 2

1 1 ðy  hÞ2 . ¼ pffiffiffiffiffiffi exp½h2 =2 pffiffiffiffiffiffi exp  2 1  2 2 2 ¼ ðhÞðzÞ,

ðD:3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 y ¼ hþ where pffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ ð y  hÞ= 1  p.ffiffiffiffiffiffiffiffiffiffiffiffiffiSince 2 5 k, 2. z 1  p z 5 ðk  hÞ= Note that 1   ffiffiffiffiffiffiffiffiffiffiffiffiffi dy ¼ dz 1  2 . Applying this to the second term in equation (D.2) results in ! Zk h0 h  y pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi ð yÞ dy 1  2 1  2 1 Z ðkhÞ=pffiffiffiffiffiffiffiffi 12 pffiffiffiffiffiffiffiffiffiffiffiffi h0 1 pffiffiffiffiffiffi expðz2 =2Þ 1  2 dz ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðhÞ 2 1  2 1 p ffiffiffiffiffiffiffiffi Z ðkhÞ= 12 1 0 pffiffiffiffiffiffi expðz2 =2Þ dz ¼ h ðhÞ 2 1 ! k  h ¼ h0 ðhÞ pffiffiffiffiffiffiffiffiffiffiffiffi : 1  2 ðD:4Þ

Journal of Sustainable Finance & Investment, 2013 Vol. 3, No. 1, 17– 37, http://dx.doi.org/10.1080/20430795.2012.738600

Modern portfolio theory and risk management: assumptions and unintended consequences Mehdi Beyhaghia and James P. Hawleyb∗ a

Schulich School of Business, York University, Toronto, ON M3J 1P3, Canada; bElfenworks Center for the Study of Fiduciary Capitalism, School of Economics and Business, Saint Mary’s College of California, Moraga, CA 94575, USA (Received 16 August 2012; final version received 25 September 2012) This article presents an overview of the assumptions and unintended consequences of the widespread adoption of modern portfolio theory (MPT) in the context of the growth of large institutional investors. We examine the many so-called risk management practices and financial products that have been built on MPT since its inception in the 1950s. We argue that the very success due to its initial insights had the unintended consequence, given its widespread adoption, of contributing to the undermining the foundation of the financial system in a variety of ways. This study has relevance for both the ongoing analyses of the recent financial crisis, as well as for various existing and proposed financial reforms. Keywords: benchmarking; contagion; efficient market hypothesis; financial crisis; financial reforms; investor herding; modern portfolio theory; risk management; systemic risk

1. Introduction Pensions (defined benefit-DB) and mutual funds (defined contribution-DC) are encountering and will continue to face mounting benefit obligations to their fund beneficiaries and retirement investors. While our specific focus is on that proportion of (US) mutual funds that are 401-k, IRA and other dedicated and tax incented retirement investments, a large but unknown proposition of nonretirement dedicated investment are nevertheless also for retirement purposes. The unprecedented amount of capital accumulated in pension and mutual funds, on the one hand, and a stock market that can be declining or illiquid over significant periods of time, on the other hand, have created pressures to expand their portfolios beyond the traditional investment securities of equity and fixed income into so-called alternative investment (e.g. hedge funds, private equity, commodities and foreign exchange). Partly, this is to offset a decade of volatility in equity markets and of closeto-zero growth (2001– 2011). Yet, more fundamentally it reflects the logical extension of risk management, built on modern portfolio theory (MPT).1 DB and DC funds are by far the largest players in equity (and other) markets. In 2009 (the most recent data available), U.S. pension funds (private trustees, private insured and state and local funds) held 39.9% of all institutionally owned assets (about $10.1 trillion). U.S. mutual funds as a whole (open and closed end, DC retirement funds and all other types of mutual fund investments) held 29.9% of all institutionally owned assets (about $7.2 trillion).2 DB and ∗

Corresponding author. Emails: [email protected]; [email protected]

# 2013 Taylor & Francis

18

M. Beyhaghi and J.P. Hawley

DC retirement funds continue to dominate equity markets. Along with insurance companies, foundations and banks, institutional ownership of equity was just over 50% in 2009, while those institutions owned 73% of the largest 1000 firms.3 Institutional investment in hedge funds has expanded rapidly in the last decade making both DC and DB plans significant suppliers of funds to hedge funds (and hedge funds of funds). Data gathered on the largest 200 DB plans indicate that these funds invested just over $70 billion in hedge funds, although there is very wide variation among individual DB funds.4 Retirement funds in the broadest sense are exposed to various types of risks as a result of an ever-increasing sophistication of investment strategies they utilize. The boom of derivative markets, the inception of futures on indices and currencies, as well as indexed options, and the development of collateralized debt and loan obligations among other innovations have changed the face of the investment industry.5 Yet, in spite of having more investment opportunities, retirement funds specifically and mutual funds more generally, as fiduciaries, are expected to establish a solid risk management system by which they can identify, assess and process different related risks, especially the new risks to which they are exposed to as a result of practicing innovative investment strategies. Risk management is far more important for both DB and DC retirement funds than other institutional investors because the fund participants and investors are in theory at least protected by the fiduciary obligations of fund managers managing ‘other people’s money’. Yet, as we discuss below, as various asset markets are ever more tightly linked, even indirectly all institutional investors can affect other market participants, so risk is generalized, as the 2007– 2009 financial crisis demonstrated.6 Even as the importance of risk management has increased in light of the 2007 – 2009 financial crisis, MPT has long since become embedded in law, legal practice and standards. In 1974, MPT became the basis for what was then called the ‘prudent man standard’ in the U.S. Employee Retirement Income Security Act. Subsequently, the prudent man/person standard was changed to the prudent investor standard, to better reflect the rise of large institutional investors and the logic of MPT itself. (The prudent investor standard encompasses the whole portfolio, whereas the prudent person standard was not clear on this.) MPT was incorporated in the 1992 revisions to the Third Restatement of Trusts as well as in the 1994 passage of the Uniform Prudent Investor Act. In the U.K. the Pensions Act of 1995 incorporate similar reforms on the basis of financial economics and MPT. The significance of the incorporation of MPT in these and other legal reforms, and, importantly, in practices in jurisdictions where there were not necessarily legal mandates, was the widespread adoption of similar and indeed often identical risk management theories and techniques. In the U.S. these developments resulted in the duty of care (also called the duty of prudence) becoming the benchmark standard for pension and other fiduciaries and the practices of other institutional investors. As we argue below, this lemming-like standard had unintended consequences, including increasing systemic risk due in part to herding behaviour, unintentionally magnified by legal mandates. Additionally, herding has increased as a perceived means of mitigating the legal liability of fiduciaries and pension fund trustees: if we are only doing our ‘fiduciary duty’ like everyone else, we are ‘prudent investors’. Following the financial crisis financial reforms in the US, the UK, other EU countries and at the EU level as a whole, the issue of systemic risk itself has been enshrined in law and various legal mandates. In particular, the US Dodd-Frank reforms established the Financial Stability Oversight Council whose tasks, among others, is to monitor systemic risk.7 This article examines the growth and underlying assumptions of MPT as it developed into contemporary financial risk management, focused in part on MPT and financial risk managements’ unintended contribution to systemic risk. The article also notes the gap in analysis of MPT’s contribution to systemic risk. In Section 2 we examine the effectiveness of current risk management practices focusing on four major areas of risk. In Section 3 we look at the evolution of MPT as the underpinning of

Journal of Sustainable Finance and Investment

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almost all contemporary approaches to risk management. In this section we look at some of the paradoxes of indexation and benchmarking and their relation to what are taken to be sophisticated financial products. In Section 4 we look at the efficient market hypothesis (EMH), noting its centrality to both MPT in its original form(s), as well as to much of what has been built on it. We note a number of important problems with EMH, including information asymmetry and bounded rationality; the joint hypothesis problem and random walk hypotheses. In Section 5, we look at some findings of behaviour finance including irrational behaviour and inconsistent preferences. Section 6 looks at herding and feedback loops in terms of systemic risk, while Section 7 continues a discussion on systemic risk calculations and some neglected risks. Section 8 concludes the article. 2. The effectiveness of current risk management practices Understanding the potential undesirable outcomes of investment decisions and timely reaction to changes in the economic environment requires adoption of an effective risk management system by financial institutions (FIs). Risk management goal is to achieve a probabilistic view of the known potential outcomes, a clear view of the limitations imposed on these by the assumptions in the models, and, some sense of what is not known, that is, what is uncertain. How effective a risk management system is depends on how effectively it performs the tasks of identifying, evaluating and dealing with risks, the correlations between various risk exposures and uncertainty. In the first task, FIs need to identify the main group to which each risk belongs. Most FIs categorize risks into four groups: market risks, credit risks, liquidity risks and operational risks. Market risks are risks of a market break-down which include a wide variety of macro-economic events that can negatively affect FIs’ portfolios, for example unexpected changes in equity prices, interest rates, foreign exchange rates and commodity prices. Credit risks are related to the likelihood that a specific counterparty fails to repay part or all of its short-term or a long-term obligation to the FI. Liquidity risks are associated with the possibility that the FI will be unable to sell an investment at any time without incurring a significant cost. Finally, operational risks, concern losses due to operational inadequacies within the organization resulting from failures in processes, people, technical systems and/or fraudulent activities. As we discuss below, most financial models are based on normal distributions or their transformations. Yet, if the actual distribution is nonnormal, the models will fail, as they have on a variety of occasions.8 Figure 1 compares the observed historical probability distributions of returns on two most liquid market indices with their relevant normal distribution functions (normal distribution functions with similar mean and standard deviation). FIs with the help of experience and developments in the financial economic theory have identified different types of risks in each group. However, the complexity of the new investment world and the diversity of financial products it contains have made it very difficult for institutional investors to accurately identify all risks. Yet, beyond what has prior to the 2007 financial crisis been designated ‘risk’ is the underlying (and sometime over-riding) issue that ‘risk’ is not essentially in the things that we know and think we can calculate probabilistically, but rather the real ‘risk’ as Bookstaber (2007) says is ‘the one you can’t see’. That is, it is uncertain, both the ‘known unknown’ and the ‘unknown unknown’. Almost one hundred years ago, the University of Chicago economist Frank Knight made this distinction between risk, which was calculable, and uncertainty, which was neither calculable nor necessarily known (Knight 1921; Bhide´ 2010, 87– 88).9 We will return to this ‘neglected risk’ below. The second task of a risk management system is to determine how accurately a risk management system evaluates each identified risk. The new sophisticated algorithms and software pension funds and others used to assess risks did not accurately measure the real amount of risk to which the funds are exposed because these algorithms and software are based on their

20

M. Beyhaghi and J.P. Hawley

Figure 1. Historical probability density functions for S&P 500 and Dow Jones Industrial indices. The top panel represents the observed probability density function of historical daily return on S&P 500, since its inception (3 January 1950) to 6 May 2011. Density functions are compared with a normal distribution function that has the same mean and standard deviation. The bottom chart represents the observed density functions for Dow Jones Industrial historical daily returns since its inception (1 October 1928) to 6 May 201. This figure shows that in the real world stock returns (represented by two most liquid stock indices) are not normally distributed. Non-normality can be observed by looking at the Kurtosis of the distributions. Kurtosis is the degree of peakedness of a distribution. Both indices have a pure leptokurtic distribution. That is they have a higher peak than the normal distribution and have heavier tails. Also J.P. Morgan Asset Management (available at www.jpmorgan.com/insight) provides other empirical evidence of non-normality of returns in financial markets.

assumptions and model specifications. Most importantly, the models used prior to the 2007 crisis were not adequately tested before being applied by funds because the historical data that might have been used to test these models belonged to a different market and economic situation than arose prior to and with the onset of the crisis. The financial crisis was the first real ‘laboratory’ to assess the accuracy and efficacy of these models. Many, if not most, turned out to be not useful at best or perverse at worst, exacerbating the crisis. Failure to measure market risk effectively (Khandani, Lo, and Merton 2009; Gennaioli, Shleifer, and Vishney 2012) and failure to consider investors’ irrationality in addition to the severity of market interrelatedness and contagion, the impact of herding and cascading as well as inaccurate assumptions about returns distributions and anomalies (Taleb 2007; among others) and/or market inefficiencies are some of the possible important shortcomings of the models that are used by institutional investors generally and more importantly retirement funds specifically. These and other issues are treated in detail below. The third task of a good risk management system is how it deals with each identified risk. In the 40 or so years of its existence financial economics has come a long way from the simple idea

Journal of Sustainable Finance and Investment

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that diversification is able to deal with idiosyncratic firm or sector specific risk to the complex multi-dimensional risk management models FIs are currently using. If model assumptions are correct and market conditions are accurately considered in a risk management system, then the solutions that the system provides are probably the most optimal ones.10 The system may suggest selling, keeping, taking or insuring against a certain risk. However, the effectiveness of each decision and its feasibility depends on how realistic the assumptions of the models used are, the collective action impacts of who, what institution or institutions are using them, and the degree to which they are widely adopted which can lead to herding. In such situations each institution is no longer a price taker, but rather a part of (an often unwitting) ‘super-portfolio’ that can move markets. By ‘super-portfolio’ is meant a large number, and sometimes a majority, of institutions move in one direction (buying or selling) more or less at the same time, their combined values creating a de facto giant portfolio, albeit it one not consciously coordinated. Or again, models cannot implement a decision to use hedging or to sell that may be impossible if the market is illiquid. These examples per se do not satisfy the models’ assumptions, creating failure of the model. Moreover, evidence from empirical work in behavioural finance suggests that there are significant problems with investors’ reasoning abilities.11 Therefore, even if a model identifies and assesses a certain risk, it does not mean that an effective action can be conducted to cope with that risk. A rough parallel to the Heisenberg principle in physical would suggest that the observation of the object changes its characteristics and actions. Bhide´ (2010, 105, 108 – 09) concludes: ‘Assumptions of universal omniscience about probability distributions and the absence if Knightian uncertainty is at the heart of many breakthrough models’. 3. The evolution of modern portfolio theory The history of modern portfolio management (also known as MPT) originates with the seminal academic work of Markowitz (1952, 1959). MPT introduced the concepts of the risk return trade-off, correlations in returns of different assets, portfolio selection and investment optimization. Using these concepts MPT is a prescriptive rather than a descriptive theory which provides solution for investors by showing what is the best combination of available assets in a portfolio in order to maximize the total expected return for a given amount of risk, or, alternatively, in order to minimize the portfolio risk, for a given level of expected return. The risk of an asset is measured as the variance of the return on that asset, where variance is a measure of how returns can deviate from their expected value. The portfolio’s return variance then is the sum over all assets of the square of the fraction held in a specific asset (weight) times the asset’s return variance. There are a number of critical underlying assumptions of MPT about the behaviour of individuals, which are typically and usually implicitly also made about institutional behaviour, whatever the problems with that linkage. Some of these assumptions are also taken by the EMH. First, that the investor is rational. The rational investor assumption (‘homo economicus’, that is utility maximizing and calculating) is the basis for the EMH, which itself is assumed by MPT. The EMH assumes, in turn, that information is symmetric (all actors have equal and timely access to all relevant information), that it is immediately available and ‘knowable’, which is understood, digested and thus becomes ‘knowledge’ and that such knowledge is immediately acted upon. Some have argued (for instance, Ross 1976; Chen, Roll, and Ross 1986; Chamberlain and Rothschild 1983) that even if a market is inefficient at a given moment, such inefficiencies will be made efficient through arbitrage, thereby reverting to a default efficiency. Critical to the EMH is the idea that returns on stock price are stochastic, following a normal bell curve distribution. Implicit in this formulation is the idea that there is finite variance and returns are independent of each other.12 However, most evidence suggests, especially with the rise of institutional investors, is that returns are not independent; distributions are not normal; and there is possible infinite variation.

22

M. Beyhaghi and J.P. Hawley

MPT’s second assumption is that investors are risk averse and make decisions based on the axioms of expected utility theorem. That is, a risk-averse investor is one who when presented with two different portfolios with the same expected utility/return prefers the portfolio with the lower overall risk over the one with the higher overall risk. Furthermore, MPT assumes that risk aversion (the risk-return trade-off) is linear or constant,13 while there is significant evidence that it is non-linear (Rabin and Thaler 2001). A third assumption of MPT is that of a monotonic investor always prefer a portfolio with a higher expected return over another portfolio with a lower expected return.14 A fourth assumption is that investors are price takers who cannot affect a security price, which is in critical instances contradicted by ‘super portfolio’ collective movements of large institutional investors. A fifth assumption is that the investor knows the expected return of each asset in his/her portfolio. The expected return is then the sum of all returns at different ‘states of the world’ weighting by the probability that each state happens. Therefore, to calculate the expected return of an asset one needs to know the distribution of the return of that asset, that is, its net present value. Thus, taking these five assumptions into account MPT concludes that the overall risk of a portfolio depends on the risk of each asset in the portfolio, the proportion of the portfolio in that asset and the correlations among different assets in the portfolio when considering the pattern of returns. Sharpe (1964) in an extension to the original work of Markowitz (1952, 1959) introduced the concepts of systematic and idiosyncratic risks, portfolio beta, diversification and the linear relationship between beta as a measure of portfolio/asset risk and the expected return of portfolio/asset. Through diversification, an investor can reduce the total risk of his portfolio by choosing different assets the returns to which are not completely correlated. This reduces idiosyncratic risk or asset-specific risk which can be, in theory, diversified away. Systemic risk, on the other hand, is common to all assets, and it cannot be diversified away. Consequently, risk-averse investors will prefer a well-diversified portfolio only exposed to systematic risk. Sharpe (1964) argues that since idiosyncratic risk can be diversified away it is the correlation of each asset with a single common underlying factor that matters. He also shows that the most optimized portfolio is the market portfolio which has the best return versus risk ratio because it is the best diversified of all possible portfolios. If investors select their portfolios based on risk and return, and if they have the same opinion about each asset in terms of expected return, risks and correlations, then, Sharpe shows that in equilibrium there is a linear relationship between the expected return of an asset and its sensitivity to the return on the market portfolio, or beta. The reason for the proportional relationship between the risk and the return of an asset is first of all the risk aversion of the investors who require higher return for taking higher risks, and secondly the equilibrium in which supply and demand forces adjust asset prices based on their risks. The assumption of a single common underlying factor reduces calculations for portfolio optimizations dramatically.15 The Capital Asset Pricing Model (CAPM), based on the works of Treynor (1962), Sharpe (1964) and Lintner (1965), is the first important economic model that explains the relationship between risk and expected return, as mentioned before. Based on this model, there is a linear relationship between expected return for a security and the market risk premium. In addition to all the assumptions about investors (rationality, risk aversion and monotonic preferences), CAPM assumes all investors are identical, that they maximize economic utility and that the shape of their utility function is assumed to be fixed. Moreover, it assumes they can lend and borrow a risk-free asset without any restriction as well as that correlations between different assets do not change over time. There are different extensions to the original CAPM, however. For instance, Black (1972) suggests a version of CAPM in which it is not assumed that investors

Journal of Sustainable Finance and Investment

23

can lend and borrow at a common risk-free rate. Nevertheless, the new framework of asset pricing not only changed the curriculum of business schools forever (Bookstaber 2007) but also it changed the ways institutional investors including mutual funds and investment banks form their portfolios. This is extremely important. Index funds are an outstanding innovation based on MPT. However, the low explanatory power of these single-factor models when working with real data (Fama and French 1992), motivated financial economists to propose new models to relate the variations in expected returns with common risk factors.16 The Arbitrage Pricing Theory (APT) suggested by Ross (1976) is based on the assumption that asset returns are adjusted in a way that no arbitrage opportunity would be left for investors to take advantage of. In the APT framework asset returns are sensitive to different economic factors and hence ‘factor loadings’ came to replace the beta of CAPM.17 3.1. Employing market indices and benchmarking MPT concludes that the market portfolio mentioned above provides the best risk and return combination. As a result two major trends started to appear in the investment world: first, widespread use of different market indices as performance benchmark, and, second creation of index funds, special types of mutual funds that aim to replicate the movement of some specific market index. The latter became popular very rapidly as index funds charge investors a much lower management fee since there is no active portfolio management. Based on the idea that the market provides best risk-return trade-off as over time it is not possible to beat the market, index fund managers only need to replicate the proportion of different securities in their portfolio the way an underlying market index represents. The popularity of index funds had an unpredicted consequence increase the demand for stocks generated simply by their inclusions in a market index in turn causing the price of these stocks to move to some degree independently from their fundamental values. This ‘indexation paradox’ has led to undermining the efficacy of the risk metrics essential to calculating risk-adjusted returns since, paradoxically; a market index is no longer representative of a market. Rather it fluctuates due to the demand of index funds and portfolio managers that attempt to replicate the index. Therefore, performance evaluation based on these indices (both alpha and beta) are subject to inefficiencies as these indices are not really an ‘index’ of the market. We elaborate on the herding behaviour implicit in this benchmarking paradox below. MPT also provides a standard to measure a portfolio’s relative benchmark performance.18 Therefore, MPT provides a framework for performance attribution. There are two major weakness of this framework. First, it relies significantly on past data which are mostly based on a factor model with a static structure. That is, underlying assumption is that returns (both for the market and the underlying security) follow a normal distribution which in turn is based on the assumption that sudden movements are unlikely and that the portfolio as well as the market will continue to maintain the same trend of risk taking and performance in the future. The second major weakness of using a market index as a benchmark is the extent to which a market index really represents the ‘market’ as MPT uses it.19 Therefore, performance measurements based on alpha or beta that is estimated according to a market index are inaccurate. 3.2.

Development of sophisticated financial products

MPT provided a platform for academics and financial engineers to price other sophisticated asset classes aside from equity such as derivatives whose price is derived from an underlying asset. The most famous model stemming from MPT is the Black-Scholes-Merton option pricing formula

24

M. Beyhaghi and J.P. Hawley

(BSM), a model widely used in the financial services industry. It is based on the assumption that there are no arbitrage opportunities in the market, that is if the future stream of cash flow created by a security can be replicated by the future stream of cash flows created by another or another group of securities in the market, then the current price of the former and the latter must be equal. BSM also assumes the possibility of a perfect hedge between an option and its underlying asset, and a risk-free asset (e.g. Treasury bonds). This requires that investors have the ability to rebalance their hedge portfolio of the underlying asset and the risk-free asset instantaneously to fully replicate the option. Although the level of risk aversion of investors does not appear in the model, the model has a simplistic assumption: the holder of the option can borrow and lend infinitely at the risk-free rate. Short selling is also possible in the model. It also assumes that there exists no transaction cost or taxes. More importantly, it is assumed that the return on the underlying asset follows a Brownian motion (i.e. the price follows a continuous geometric path). This means that returns are assumed to be normally distributed (therefore the price follows a log-normal distribution). Also this process is based on a constant volatility. Another basic hypothesis is that all relevant information at any time is incorporated in the price of securities at that time; explicitly based on the EMH, as discussed below.20 No matter how sophisticated the models based on MPT are, they all suffer from one or more of the following assumptions: (1) markets are efficient; (2) investors are rational; (3) one of the popular market indices can be used as a proxy for market portfolio; (4) investors are price takers and cannot affect security prices by placing orders; (5) that there is unlimited liquidity and unlimited capital; and (6) the widespread adoptions of MPT and subsequent risk management techniques based on it have no consequence on risk management techniques themselves. Additionally, all argue that there is, in essence, a firewall between idiosyncratic and system risk.21 4. Efficient market hypothesis If the MPT provides the most important normative statements in financial economics; the EMH provides the most important positive statement in this field. EMH states that security prices fully reflect all the available and relevant information in the market. The origin of EMH goes back to the turn of the previous century, but it was Paul Samuelson (1965) and later on Eugene Fama (1970) who presented the idea in a clear framework. Samuelson (1965) argued that changes in future security prices must be random (i.e. cannot be forecasted) if all the available information is already incorporated in the current prices, that is if they are properly anticipated. If a part of relevant information is not fully incorporated in asset prices (i.e. the price of an asset is too low or too high), in order to make profit informed investors would buy or sell the asset so that the equilibrium prices reflect the expectations and information of all market participants. Hence deviations from well-informed investors’ expectations could not last for long. Definition of what constitutes ‘information’ to ‘well-informed investors’ formed the basis for varying forms or strengths of the EMH which led most financial economists to distinguish among three forms of market efficiency: weak, semi-strong and strong. Under the weak form security prices reflect the history of prices and returns. Under the semi-strong form security prices reflect all publicly available information. Finally, under the strong form, prices reflect all possibly relevant information, both public and private. Stout (2003, 637 – 638) argues that the ‘Achilles’ heel’ of the EMH is the process whereby ‘information flows into prices’, which is both obscure, and ‘jury-rigged’ in part because there are at least four price moving market mechanisms. Even more fundamentally, she argues that EMH (especially when combined with CPAM) assumes that all ‘. . . investors share homogenous expectations regarding the likely future returns and risks associated with particular securities’ (2003, 642).22 If we drop the assumption

Journal of Sustainable Finance and Investment

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of homogenous expectations, then risk analysis needs to consider uncertainty in addition to risk, making the whole project far less calculable probabilistically (2003, 646; 1995, 59– 64). If we assume heterogeneous expectations one implication is the informationally efficient markets (under a number of definitions of what that actually means) are not related to fundamental value (or the relation is a weak one). 4.1.

Information asymmetry and bounded rationality

The nature of information is a key point in characterizing an efficient market. For example, Tobin (1958) asserts that one should distinguish between informational and fundamental efficiency, that is, before testing or making an argument about EMH, one should distinguish between high- and low-quality information. To put it differently, one should make it clear if one believes that all information, no matter how accurate or reliable (e.g. rumours) is incorporated in the current asset prices, versus if one believes only that the most accurate information about fundamentals are incorporated. Anecdotal evidence suggests that information asymmetry exists among different market participants.23 This asymmetry leads to other phenomena in financial markets such as moral hazard and adverse selection. If investors are aware of the degree of information asymmetry between themselves and their counterparties, they would adjust their offers based on the perceived magnitude of the asymmetry. In order to prevent taking advantage of private information and to reduce the cost of information asymmetry in the markets, regulatory organizations enforce certain disclosure rules on businesses and FIs. However, the degree to which these regulations are enforced and/or feasible is open to question. Opponents of disclosure rules believe they violate privacy rights. Hedge fund managers in particular are concerned that imposing new disclosure requirements might lead to a leakage of their proprietary investment strategies resulting in a loss of their competitive position in the market. Rational and consistent processing of information by investors is another area that has been under severe criticism from behavioural scientists and psychologists. Williamson (1981), Simon (1997) and Rubinstein (1998) suggest that individuals employ the use of heuristics to make economic decisions rather than a deterministic set of optimization rules. They show that people ‘are concerned only with finding a choice mechanism that will lead to pursue a satisficing path that will permit satisfaction at some specified level of all of its needs’ Simon (1957, 270). In other words, people use available information to make choices with satisfactory-enough outcomes rather than the most optimal ones.24 4.2.

The joint hypothesis problem

Attempts to test the EMH have been numerous, yet all such testing has run into a number of common roadblocks. One problem, known as the ‘joint hypothesis problem’, is that one must assume a certain model to predict security price returns in order to test whether the dissemination of a piece of information would produce abnormal returns. Therefore, if the result of a test does not support market efficiency, it can either be due to the fact that the market is inefficient or that the model for return prediction is not a proper one. A second problem focuses on the costs of gathering and processing information which can be so high as to prevent investors from gathering it in the first place. This is the familiar transaction cost problem.25 Believing in any form of market efficiency has an immediate implication for investors, traders and MPT itself. If a market is efficient with respect a to certain information set, trading based on that information set cannot be expected to produce above market returns (i.e. it cannot generate ‘alpha’). For instance, if one believes in the semi-strong form of

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market efficiency, then using available public information to forecast future asset prices is meaningless as all the available public information is already incorporated in asset prices. The same reasoning can be used to show the inadequacy of technical analysis based on previous stock returns to predict future return. Therefore, if the information contained in past asset price movements (and levels) is fully, continuously and instantly conveyed in the current asset price, the conditional expectations of future price changes, condition on the available information set, should be equal to zero. In other words, future price changes would be completely random and unpredictable. Thus, the random walk hypothesis of security prices would make sense. 4.3. The random walk hypothesis The application of different forms of stochastic processes and distributions based on the random walk hypothesis is the building block of the modern asset pricing and financial engineering. A random walk argues if all relevant information is incorporated in stock prices then future changes in stock prices are completely random (noise). Therefore, future values of stock prices cannot be predicted. Random walks can have different implications in terms of return distributions. Campbell, Lo, and MacKinlay (1996) summarize different types of the random walk hypothesis into three groups and provide different econometric tests to examine the validity of each group. The first type of the random walk hypothesis is that price increments are independently and identically distributed. The second group of random walk hypothesis relaxes the assumption of identically distributed increments, as the distribution of asset returns can change over time due to innovations in technology, regulation and economy in general. The third group of tests checks the serial correlation between price increments and focus on whether the price increments are uncorrelated. The random walk hypothesis is important because based on it, stock prices cannot be predicted using the current available information (the information is already incorporated in stock prices), and therefore it rules out technical analysis. The random walk hypothesis and EMH facilitated an exponential growth in the field of asset pricing. Financial economists started to work on the pricing of various derivatives and contingent claims which was previously difficult or impossible in large part due to lack of real-time computing power. A fundamental assumption behind most of the sophisticated stochastic models used for these purposes, which flows directly from the random walk hypothesis, is that stock prices are ‘martingales’, a mathematical term for a random variable whose expected future value is equal to its current value, at any point of time. The second central assumption is the concept of a Brownian motion, which is a random process that grows continuously in time with the property that its change over any time period is normally distributed with a constant mean (equal to zero) and a variance proportional to the length of the time period. Thus, any continuous martingale would just be a transformation of a Brownian motion.26 Therefore, using different tools provided in the field of stochastic calculus, many contingent claims with a complicated future payoff formula can be price based on the price of the underlying securities. As an example, the famous BSM formula is a simple modification of Itˆo process in which the drift is an assumed constant expected stock return, based on the risk-return trade-off of the stock prices, and the diffusion coefficient is an assumed constant standard deviation for stock price movement.27 The inception of financial stochastic calculus and continuous-time models sparked a boom in the derivative market, the rise of futures on indices and currencies, as well as index options. MacKenzie (2006) describes how fast trades/market makers chose the Black – Scholes equation as their main reference to price derivatives and to take advantage of arbitrage opportunities.

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Irrational behaviour and inconsistent preferences

Despite the fact that the efficient market and the random walk hypotheses are the building blocks of the modern financial economics which have facilitated an exponential growth in the financial industry, certain scientific and anecdotal evidence raised doubts about its accuracy. In this section, the results of three groups of studies from the literature of economics and behavioural finance – inconsistent preferences, seasonality and irrational bubbles – are presented. In the next section, two other important behavioural factors – herding and feedback loops – that violate the assumption of MPT will be discussed. Utility theory,28 which is the base of many important economic theories such as MPT, has been criticized by different studies in the field of psychology and behavioural economics for its inconsistencies with observed investors’ behaviours. Utility theory, a model of rational choice, states that people in their economic decisions prefer one risky investment choice over another based on its expected utility of pay outs, where the expected utility of pay outs is the sum of the utility of each probable pay out times the probability of its occurrence. However, Nobel laureates Kahneman and Tversky based on psychological studies in 1979 provided an alternative to expected utility theory that explains people’s behaviour more realistically. Founded on empirical evidence, they argue that it is possible that a person with a certain utility function when encountering with the same investment opportunity (same probable payouts and same probability distribution) forms different expected utility for this opportunity if information is presented to him differently. The theory describes how different people evaluate potential losses and gains and the different ways they are decided when they are presented with two equal investment opportunities with the same payoffs, one expressed in terms of possible losses and the other in possible gains. Therefore, Kahneman and Tversky (1979) provide several classes of choice problems in which final outcomes and investors’ preferences systematically violate the axioms of expected utility theory. Moreover, findings about what are called ‘seasonality effects’ also provide evidence that is not consistent with the MPT and EMH. Seasonality exists when the market experiences regular and predictable changes which recur at certain times during a day, a quarter or a year, etc. If markets are efficient then seasonality should not exist as any trend that investors have information about should disappear if the prices fully reflect all available information. In contrast, different types of seasonality exist in different markets.29 Behavioural Finance studies attribute these anomalies to various human errors in reasoning and information processing such as overconfidence in one’s own knowledge and decision making and cognitive biases. Aside from the seasonality studies conducted by behavioural finance researchers, the existence of irrational bubbles in capital markets is counted as the evidence of market inefficiency in some research (Malkiel 2003). Market bubbles happen when market participants drive asset prices above their value in relation to some system of asset valuation. This system of asset valuation usually views values as based on fundamentals. Pro-efficient market hypothesis researchers, however, raise doubt about whether existence of bubbles undermines the validity of EMH. For example, Fama (1991) states irrational bubbles in stock prices can be indistinguishable from rational time-varying expected returns. Nevertheless, the fact that historically these bubbles burst and asset prices move towards the ‘fundamental values’ shows that even if there exist investors with true knowledge about the real value of assets they did not have the ability or will, to take action that would adjust asset prices in a shorter time. Nevertheless, during financial crises one observes rapid security price changes over short periods of time. If the EMH holds that one should expect investors to have received a large amount of information regarding sudden dramatic changes in the underlying assets’ value, the very event and its nature suggests otherwise. Another piece of evidence contradicting the EMH is the historical evidence of increases in stock prices

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Figure 2. Co-movement of newly added stocks to S&P 500 index after inclusion in the index. This figure shows changes in co-movement of stocks added to the S&P 500 index and of stocks with matching characteristics. The results presented are based on a bivariate regression with daily returns of stocks added to the index against the index and the return on the rest of the market. The sample includes 153 stocks added to S&P 500, for which rolling 12-month coefficients are computed and the averages are plotted. Source: Barberis, Shleifer, and Wurgler (2005).

followed by their inclusion in a major index. That is, why should a seemingly irrelevant definitional change cause a movement in stock prices? Yet, Wurgler (2010) shows that new indexincluded stocks show a high correlation between their returns and the return of other stocks in the index although this correlation did not exist before its inclusion in the index. More on index inclusion and also other evidence related to irrationality of investors including herding and market panic follows. Figure 2 provides the evidence.

6.

Herding and feedback loops

MacKenzie (2006) asserts that academic financial economics did more than positively analyse markets, it normatively altered them. It was an ‘engine’, an active force transforming its environment, not merely a ‘camera’ passively recording it. In particular, an implicit assumption of many financial economics models, most especially MPT, is that individual investors make their investment decisions independently. In addition to the assumption of rationality another important but implicit assumption is that the market is so big that all investors are price takers and therefore the theory does not need to account for its own widespread adoption and application. Recent studies (Devenow and Welch 1996; Avery and Zemsky 1998; Hirshleifer and Teoh 2003; Cipriani and Guarino 2005, among others) have documented a crowd effect or herd behaviour among investors in capital markets. Investors are influenced by other investors’ decisions (as Poincare´ pointed out a century ago, as noted in a footnote above) and therefore they may herd, that is, they may converge in behaviour; or they may cascade, that is, they may ignore their private information signals in their investment decisions. For example, Hirshleifer and Teoh (2003) show that not only is the behaviour of investors, firms and analysts affected directly by herding or cascading, but also there

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is a group of investors, for example, short-sellers, that change their behaviour, knowingly to protect against or take advantage of herding or cascading by others. One can easily imagine an infinite set of possibilities in this situation, as Keynes’ well-known comment about early 20th British newspaper based ‘beauty contests’ as similar to the stock market.30 A famous example of herding/cascading is the decision of bank depositors to withdraw their deposits because they believe the bank is about to become insolvent (bank-run). A bank run generates its own momentum as more depositors withdraw their money. Of course, a bank run on one bank can be contagious and can lead to a panic in which other banks suffer runs at the same time. Indeed, the freezing of the inter-bank markets in 2008 was an internal bank run, not of depositors on banks (Northern Rock in the UK aside, a retail development), but rather a refusal to transact inter-bank ‘wholesale’ business as no bank trusted any others’ books or statements about its liquidity. Another example is the decision to issue debt, dividend or to go public (the IPO decision) by some firms in the same industry around the same time since these firms believe that the market is ‘hot’ and they can time the market. Baker and Wurgler (2004) show that as market sentiments change over time and investors value dividend/debt differently at different times, firms time the market in order to ‘cater’ to investors’ sentiments. Another example is the herd behaviour in the analysis of securities or analyst behaviour. Herding becomes more serious when the cost to obtain private signals is high or there is a reputational incentive especially among less experienced and less skilful analysts.31 Welch (2000) shows that analysts’ forecasts are correlated with the prevailing consensus forecast, even if the consensus forecast aggregate information poorly. Another form of herding in investment decisions, as mentioned before, which is also a fallacy of MPT, is that MPT does not account for its widespread adoption among investors. An epidemic imitation of a certain investment strategy by a large group of market players can lead to a ‘superportfolio’ for which the realized result can fall far from the expected. In this situation each investor is not solely a price taker any more but rather each investor is a part of a super-portfolio that has a direct impact on security prices and liquidity. In the case of the failure of Long-Term Capital Management (LTCM), aside from being over-leveraged, an important factor was the wide spread imitation prior to LTCM’s collapse of its arbitrage strategies by other market players after they found out about the outstanding past success of LTCM. The widely used value-at-risk models and ‘portfolio insurance strategy’ based on the asset pricing literature (option pricing specifically) is another important example in this regard. The key failure of these strategies is that they are based on models that ignore their own impact on the market. Bookstaber (2007), for example, asserts that the source of 1987 crash was ‘market illiquidity, an unintended by-product of the new and wildly successful portfolio insurance strategy’. Portfolio insurance is an implication of BSM model adopted in order to prevent the downward risk of holding a security by replicating a put option that consists of constantly adjusting in the underlying security based on its market price. The idea is based on a key and yet widely ignored assumption in the BSM model: you cannot affect either stock or option prices by placing orders (Black 1990). However, what happened in the real world is that an initial decline in security prices encourages portfolio insurers to sell stocks and futures based on the formula provided by the portfolio insurance strategy. Sale orders placed at the same time by a large number of portfolio managers following the same insurance strategy led to a widespread liquidity ‘dry-out’ in the market and therefore, ironically, ‘portfolio insurance’ caused portfolio failure for many fund managers. In the pursuit of ‘alpha’ and pushed by a hugely competitive market, most practitioners and theorists alike ignored the age-old fallacy of composition problem. We believe this is a common occurrence. In a similar vein, Bhide´ (2010, 145) suggests that BSM’s method of estimating volatility, an unobserved variable, using historical data alone, led to what could be called a model-herding effect. When all or most adopt the same methods, in this case BSM, the result is all have the

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same volatility measure and thus everyone has the same option pricing. There is a lack of diversity of opinion ‘. . . that is supposed to be the hallmark of a decentralized market . . .’ Another ironic result of the MPT and EMH is the wide use of indices as investment vehicles. Indices based on MPT should provide a better reward to risk proportion as they are much better diversified than single stocks. As Wurgler (2010) shows the indexed-based investing strategies and products have risen rapidly over the last few decades. However, the efficient market view, as MacKenzie (2006) puts it, seems to have created an anomaly itself as index members are subject to an evident mispricing. Evidence shows that indices are so popular that their members’ price movements are not entirely due to new information; rather market demand can move them independently. Wurgler counts two impacts that index inclusion bring for a stock. First is initial inclusion impact; the second the continuation in inclusion impact. The increase in price followed by the inclusion of a stock in a major index is not the result of new information about the stock’s value (fundamentals) but it is because of the increase in demand for that stock by investors, a result of index inclusion. This impact is documented in several studies (Shleifer 1986; Lynch and Mendenhall 1997; Petajisto, 2011, among others). Wurgler (2010) shows that the mispricing is not limited to the time of inclusion. After inclusion stocks start co-moving with other index participants and as a result over time index members can slowly drift away from the rest of the market. This phenomenon has an important real implication for the market. As fund managers face pressure to be evaluated based on a benchmark, or to chase an index return, the resultant growth in demand to include an index in their portfolio will create a feedback loop that exacerbates index members’ detachment from the rest of the market. Therefore, rather than be a representative of a market, an index (to the degree that it is adopted and mirrored by others) can become a super-portfolio whose movement is not related to the fundamentals and therefore expose its holders (and others) to an enormous systematic risk, or as Wurgler states, a ‘high-frequency risk’. This impact shows that index-based strategies as suggested by MPT can have different consequence that one expects based on MPT. 7.

Systematic risk calculation and neglected risks

The previous sections summarized some of the main criticisms on MPT’s and EMH’s assumptions. However, even if these problems are ignored, there are significant studies that raise doubt over the accuracy of how FIs assess systematic (market) risk, even from within the MPT framework. As mentioned previously, risks associated with each asset are of two types, idiosyncratic risk and systematic risk. The idiosyncratic risk, or firm-specific risk, can be eliminated through sufficient amount of diversification. However, the systematic part of an asset’s risk cannot be diversified away. The return on each portfolio is then proportional to the amount of systematic risk to which the portfolio is exposed. By definition, systematic risk is the risk to which all equities (and other assets) in the market are exposed to that is independent of firm. At its extreme, as in the 2007– 2008 crisis, this can be failure or breakdown of the market as a whole, or specific, critical parts of it. Systemic risk is also related to cascading failure caused by interdependencies in a market.32 Identifying and measuring systematic risk is not only an important issue for pension funds and other institutional investors, but also is a central concern of policymakers. The recent financial crisis shows that governments, and indeed most economists (including financial economists), did not have a clear understanding of the extent and magnitude of systematic risk. The basic requirement to reduce the risks of financial crises is the ability to define and measure those risks explicitly (Lo 2009). It is not possible, at least at a low cost, for an institutional investor to identify and assess different components of systematic risk because to do so the investor needs to be able to gather and process data about security positions of all market participants,

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Figure 3. Correlations of various indices to S&P 500. The vertical axis represents the degree of correlation with S&P 500. The horizontal axis represents time (year). This figure shows that major indices become more correlated with the S&P index (and therefore with each other as well) during the crisis. A correlation of one for an index means that its movements are exactly correlated with those of S&P 500. The top left chart represents two indices that consist of firms with different market size (mid-size and small size). The chart on the top right include indices that represent different market sectors (industrial, transportation and utility) together with Russell 2000 that represents top largest 2000 firms in the US economy (i.e. over 90% of the US stock market). The bottom two charts demonstrate some of the major international indices’ co-movement with S&P 500. One can observe that during the recent crisis, diversification in stock markets across different firm sizes, industries or geographical locations does not necessarily have the benefits suggested by MPT as their correlations have approached to 1. The shaded part shows the period of the 2007–2009 financial crises.

in addition to their leverage, interdependencies and liquidity in close to real time. Also given the complexity of the global financial market, its interrelatedness and also the lack of an authority that oblige all market participants to reveal all their information, it is impossible even for governments to establish such an organization.33 Therefore even within the MPT framework, ignoring its deficiencies outlined in previous sections, having an accurate and more or less real-time grasp of correlations among different securities and the magnitude and impact of systematic risk is not conceivable. This problem is magnified by the interdependencies unwittingly caused by the widespread adoption of MPT and subsequent developments in risk management techniques. These increased interdependencies prior to significant market-wide crises have increased asset correlations significantly, both within and across asset classes due to the very methods of risk management itself. Figure 3 shows how the correlation of different asset classes, represented by different market indices, has moved away from 0 and closer to 1 during the recent financial crisis.

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This problem is illustrated when commenting on the financial crisis Harry Markowitz said, ‘Diversifying sufficiently among uncorrelated risks can reduce portfolio risk toward zero. But financial engineers should know that’s not true of correlated risks’ (Crovitz 2008). Markowitz did not address why he thinks that correlated risks across or within asset classes have become so important. Before the financial crisis Myron Sholes commented: ‘Models fail because they fail to incorporate the inter-relationships that exist in the real world’.34 It is exactly at this intersection between models and theories and the real world that an institutional economic approach sheds critically important light. For example, ‘financial engineers’ in some cases did not know, in other cases could not know, and in many cases may well have known but were incentivized not to ‘know’ or not to consider, the systemic risks and implications of their own collective actions. For varying combinations of these reasons, conflicts of interests within firms and within the investment chain (i.e. as an internal and external governance problem) characterized the financial crisis. To re-emphasize: previously uncorrelated risks can and did become correlated over time by the very success of MPT and the subsequent risk management techniques and products (‘financial engineering’) created in with the goal of managing risk.35 Gennaioli, Shleifer, and Vishney (2012) point to various historical episodes where the strong demand for new low-risk investment vehicles has led to financial innovations by FIs that were believed by investors and even intermediaries to be good substitutes for traditional securities. However, at some point news reveals that these securities/innovations are vulnerable to some unanticipated risks. As a result of this shock and subsequent fire sales, sudden losses are imposed on FIs and investors. An important recent event of this kind is exemplified in securitization of mortgages and widespread trade of collateralized mortgage obligations in the US as a trigger for the market collapse in 2008. Gennaioli, Shleifer, and Vishney (2012) provide a model that predicts how neglecting certain states of the world (local thinking Gennaioli and Shliefer (2010)) and the resulting risk miscalculation will lead to over-usage of these securities relative to what would be possible under rational expectations. This together with market fragility (high sensitivity of investors to negative news) will finally lead to drastically revision in valuations by investors and fire sales. 8.

Conclusion

This line of argument has, we argued, immediate relevance for both the ongoing analyses of the recent financial crisis, as well as for various proposed financial reforms. In light of the recent financial crisis and governments’ ad hoc responses to protect the economy from either another financial crisis or a continuation of the 2007 – 2009 crisis, regulatory authorities around the world have been working on better protection for systematically important financial institutions (SIFIs).36 One thing is apparent: bailouts are extremely costly in the long run. This costliness is not only in terms of the money that governments need to pay out of tax-payers’ pockets to help a failed SIFI (which may or may not be fully paid back), but more importantly in terms of the moral hazard created by the bailouts.37 Moral hazard per se increases the overall risk of the financial system as more SIFIs become exposed to the risk of non-viability and in need of more government interventions and more bailouts, in other words, a worsening situation in the long run. To prevent an SIFI’s managers from excessive risk taking most recent regulatory reforms such as Dodd-Frank act and Basel III (effective 2013) propose regulatory changes that bring SIFIs under greater scrutiny, enforce higher capital requirements, and impose certain restrictions on lending and investing activities, among other changes. Moreover, to prevent moral hazard costs associated with the resolution processes, regulators are working on regulatory changes that minimize their intervention and instead make SIFIs face a far more stringent form of oversight from their own stakeholders.38 By understanding that the government is not

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willing to bail them out, investors of an SIFI would, in theory, increase their monitoring efforts and adjust their required rate of return based on the riskiness of the SIFI. This would increase the cost of raising capital for a SIFI when stakeholders observed that it was becoming more risky.39 The hope is that this will discipline the SIFI and prevent excessive risk taking that would otherwise arise for a variety of reasons (e.g. from managers risk-seeking for their personal benefits). In other words, a main purpose of many reforms is to make managers comply with all the instruments that modern risk management provides to them and to not act in a rent-seeking selfish manner. This paper argues that a deeper analysis be considered not only by SIFI’s managers and stakeholders but also by regulators as well. We argue that even if all managers comply with the framework that current risk management practices and MPT provides, they are still exposed to huge amount of risk due to its unrealistic theoretical assumptions, to the intricacies of the real world and, finally, to human nature.

Acknowledgements The authors thank Mark Ansen, Andreas Hoepner, Keith Johnson, Jon Lukumnik, Lynn Stout and Ed Waitzer, for their helpful feedback and commentary. In particular, they wish to thank Andrew Williams for his especially close reading, his tough but important questions and as always his close editing. They acknowledge a grant from the International Centre for Pension Management at the University of Toronto for their generous assistance. Any errors and omissions remain solely those of the authors.

Notes 1. 2. 3. 4.

5.

6. 7. 8.

9. 10. 11.

Indeed, the expansion of asset class investments is parallel to the move from bonds into equities in the 1950s and 1960s: both are logical outcomes of MPT. The Conference Board. The 2010 Institutional Investment Report, 2010, New York, 9, 11. Ibid. (22, 27). While the numbers appear small in relation to total holdings (the largest 200 DC funds’ hedge fund assets were 1.6% of their total assets) this is an important change in the last decade as the inflow of funds is typically magnified through hedge funds’ use of significant leverage, a point we examine below (Ibid, 5). Additional factors have also changed the context of investment, most importantly an increasing shortterm investment horizon in practice which is often at odds with long-term value creation and retirement horizons. The average holding period currently stands at less than 1 year on the New York Stock Exchange, while the impact of high-frequency trading has increased dramatically. The Conference Board, ‘Commission on Public Trust and Private Enterprise, Part 2 – Corporate Governance,’ January 2003; New York Stock Exchange. http://www.nyxdata/com/nysedata/asp/factbook. While fiduciary duty to DB plan participants is clearer to some observers, DC retirement plans and more generally mutual funds have equally clear fiduciary obligations to investors as well. http://www.arnoldporter.com/resources/documents/Advisory– Dodd-Frank_Act_Addresses_ Systemic_Risk_071610.pdf, accessed 10 March 2011. Bhide´ (2010, 118 –119) suggests that one reason most financial theory adopted normal distributions is that they were able to have, in Fama’s word, ‘traction’. That is, Le´vy distributions (based on Mandelbrot’s work) made ‘no headway’ as predictions could not be done in a systematic way, leading Bhide´ ton concludes that ‘. . . flawed, but mathematically convenient assumptions of normal distributions . . . shaped the practice of finance’. In statistics risk is associated with undetermined outcomes when the probability distribution of outcomes is known. The word ‘uncertainty’, however, is used when the probability distribution of outcomes is not known. For example, Monte Carlo simulations’ and scenario tests’ accuracy depend on both the number of simulations and how the analysis view outliers. Investor’s irrational behaviour and preferences, herding and cascading and seasonal habits will be discussed later.

34 12.

13. 14.

15. 16. 17.

18.

19.

20. 21. 22. 23. 24. 25.

M. Beyhaghi and J.P. Hawley This formulation was originally proposed by Louis Bachelier in the early 20th century. It was strongly criticized by Henri Poincare´ who argued that the independence assumption is empirically false, that investors watch each others’ actions, which causes both feedback loops and under some conditions herding effects. See, for example, ‘Non-normality of market returns’, J.P. Morgan Asset Management (available at www.jpmorgan.com/insight), and Taleb (2007, 20, 150). A good example of this is the ‘pack mentality’ of hedge funds (correlations of returns) from 2004 to 2010. Jenny Strasburg and Susan Pulliam, ‘Pack mentality grips hedge funds’ The Wall Street Journal. online.wsj.com/article (accessed, January 14, 2011). The concepts of relative and absolute risk aversion are introduced in the seminal work of Arrow (1965). These axioms are part of a more general framework, known as expected utility theory. The seminal work of Von Neumann and Morgenstern (1944) provide a through explanation. Later on in this paper a more psychologically realistic alternative to this theory (prospect theory) will be discussed it is under von Neumann (not Neumann). CAPM assumes that only one source of risk (market risk) is common to everyone. Instead of finding correlation between two assets in your portfolio separately, you only need to find the correlation between each asset to the market. Note that a fundamental difference between CAPM, as a single-factor model, and multifactor models is that CAPM is based on an equilibrium model under conditions of risk, while multifactor models, are based on likely systematic risk candidates, economists identify using intuition. In a significant set of variations of increasing complexity, Fama and French’s model show that two empirically determined variables, size and book-to-market equity do a very good job explaining the cross-section of average returns. They argue that if stocks are priced rationally, systematic differences in average returns are due to differences in risk. So size and book-to-market must proxy for sensitivity to common risk factors in returns. Fama and French (1993) use the cross-section regressions of Fama and MacBeth (1973) where the cross-section of stock returns is regressed on the interested variables. However, they use time-series analysis in which a well-specified asset-pricing model must produce intercepts (alphas) that are not significantly differently from 0. In their study, the returns to be explained are 25 stock portfolios formed on the basis of size and book-to-market equity. Testing intercepts are jointly equal to zero is based on Gibbons, Ross and Shanken (1989) who suggest an F-test under the assumption that the returns and explanatory variables are normal and the true intercepts are 0. Over the years different factors have been added to Fama and French (1992, 1993) model by different researchers. As a famous example, Carhart (1997) suggests momentum factor in his study. Also market microstructure factors including liquidity and information asymmetry costs are suggested to be included in the asset-pricing models (e.g. Brennan, Chordia and Subrahmanyam 1998; Easley, Havildkjaer and O’Hara 2002; Amihud 2002). A portfolio can be benchmarked against a market index for instance Standard and Poor’s 500 index or Russell 1000 index to see if it has performed better than a market or not. The idea is that a good performance of a portfolio may not essentially be based on the portfolio manager’s ability rather it might be due to desirable market conditions. Therefore, the excess return on a benchmark will represent the component of the portfolio’s return that was generated by the portfolio manager (this excess return after adjustment for risk is called alpha). For more discussion see Roll (1977). A market should represent all possible investment choices an investor can make. Therefore, one can argue that the reason that financial models fail in practice, is not because of their inaccuracy but it is because of choosing the wrong proxy for market (a.k.a. market identification problem). Some of these conditions are relaxed or eliminated in more recent models, for example models that assume volatility is not constant but rather follow a different stochastic process. As we argue below, widespread adoption has led to tipping points and feedback loops which have paradoxically led the practice of risk management itself to contribute to increased risk. She notes that Sharpe and Litner (originators of CAPM) published subsequent work on investor heterogeneity. Among this anecdotal evidence, large number of legal cases relating to insider trading can be counted. This evidence per se is a proof against (at least) the strong form of market efficiency. Not only do economic agents not make the essentially optimal decisions based on the information they received, but their decisions are affected by the mechanism with which the information is presented to them. More on this will be discussed below. Grossman and Stiglitz (1980), also Fama (1991) provide a thorough analysis of the difficulties in using market efficiency tests and some abnormalities that cannot be explained through EMH. For instance,

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26. 27. 28. 29.

30.

31. 32. 33. 34. 35. 36.

37. 38. 39.

35

the positive autocorrelation of daily or weekly returns, price swings from fundamental values, reversals in winners and losers (overreactions) also known as momentum and firm size effects that are frequently observed in data are not consistent with the EMH. Levy’s theorem (for more information see Shreve 2004, 158). Ibid (153– 158); the original is in Black and Scholes (1973). Also known as Expected Utility Theory (see Kahneman and Tversky, 1979). Examples that include predictable transaction volume or price changes are weekend effects on stock returns (French 1980, etc.), on exchange rates (Levi 1988) and in money markets (Eisemann and Timme 1984; Flannery and Protopapadakis 1988). Also December and January effects (turn-of-theyear effect) have been observed continuously over time (Rozeff and Kinney 1976). Kamstra, Kramer, and Levi (2000) also provide evidence that a repetitive anomaly can be detected around daylight saving time changes. Keynes (1936) refers to a newspaper promotion in which the object was to select the contestant that most other participants would choose as the most beautiful. Thus, the ‘beauty contest’ was not about which woman was in each individual’s opinion the most beautiful, but rather a wager on what the average opinion was of who was most beautiful. In an iterative game (e.g. the stock market) this can become the average opinion of the average opinion ad infinitum, in short, a cascading and/or herding effect. When cost of obtaining information is low, more likely, people collect information themselves. When it is high, most likely they herd. Cascading effects are dynamic over time and condition dependent, making measurement extremely difficult. As conditions evolve (e.g. innovations in communications and financial instruments; changes in politics regimes) they feedback into cascading tendencies. Even if real-time comprehensive data could be collected, its analysis would need to have perfect understandings both of present conditions and those likely to occur in the future. Myron Scholes, speech at NYU/IXIS conference on hedge funds, New York, September 2005. One reason that there has been correlation among asset classes is due to model standardization among major market actors, which makes investments that were previously less correlated, more correlated, increasing downside risk and negating to some degree the whole point of diversification. Including but not limited to the European Union (Capital Requirements Directive), United States (Dodd-frank Wall Street Reform and Consumer Protection Act) and G-20 (Basel III Accord). Under Bank of International Settlement’s Financial Stability Board’s ‘Guidance to Assess the Systemic Importance of Financial Institutions, Markets and Instruments: Initial Considerations’ reported to G20 Finance Ministers and Governors in October 2009, A SIFI is defined as a financial institution who’s ‘failure or malfunction causes widespread distress, either as a direct impact or as a trigger for broader contagion’. That is when stakeholders of a SIFI believe that the government will bail it out at the time of non-viability they do not have an incentive anymore to incur the cost of monitoring the SIFI. As a result there is no effective market mechanism to prevent SIFI’s managers from excessive risk taking. For more information see Basel Committee on Banking Supervision’s ‘Proposal to Ensure the Loss Absorbency of Regulatory Capital at the Point of Non-Viability’, August 2010. a.k.a. market discipline, see Flannery and Sorescu (1996) for a detailed definition. Balasubramnian and Cyree (2011) show that market discipline has weakened after the US government started to bail-out financial institutions such as Long-term Capital Management in 1998.

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Applied Financial Economics, 2014 Vol. 24, No. 11, 753–762, http://dx.doi.org/10.1080/09603107.2014.904487

Momentum strategy and credit risk Su-Lien Lua,*, Kuo-Jung Leeb and Chia-Chang Yuc a

Graduate Institute of Finance, National Pingtung University of Science and Technology, Neipu Pingtung 912, Taiwan b Department of Commerce Automation and Management, National Pingtung Institute of Commerce, Pingtung City 900, Taiwan c Graduate Institute of Finance, National Pingtung University of Science and Technology, Pingtung, Taiwan

The article first focused on the traditional momentum strategies, and the distanceto-default of the KMV (Kealhofer, McQuown and Vasicek) model was later applied as the proxy of credit risk. Then, based on the credit risk, two factors January effect and business cycle were added to investigate the momentum effect on them and credit risk. Empirical results indicated that investment portfolios had the momentum effects by traditional momentum strategies. After the credit risk was added, when the high-credit-risk group applied the momentum strategies in the mid- and long-term holding periods, significant excess return occurred. For low- and medium-credit-risk groups, the momentum profits only exist in 36– months holding periods. In addition, when credit risk was taken as the basis, and the January effect was included, the study found that positive momentum profits only took place in the low- and medium-credit-risk groups in 36-months holding period. Finally, when credit risk was taken as the basis, and business cycle was included, momentum profits took place during the recession period. Consequently, we found that momentum strategy has different influence on three credit-risk groups. Investors should consider the credit-risk characteristics of their investment portfolio when employing the momentum strategy. Keywords: credit risk; momentum strategy; January effect; business cycle JEL Classification: G00; G11; G14

I. Introduction Fama (1970) adopted the traditional expected utility theory as a basic assumption and proposed the efficient market hypothesis (EMH) to assert the existence of market efficiency. Since that time, researchers have adopted Fama’s hypothesis as a basis for research. However, numerous studies have found that the market does not provide full support for the EMH and may either exhibit overreaction (Howe, 1986) or under-reaction (Moskowitz and Grinblatt, 1999). The majority of these studies refer to this phenomenon as a market anomaly.

Because an increasing number of anomalies have been identified, various scholars have expressed concern and suspicion regarding the EMH. Therefore, studies have attempted to employ contrarian and momentum strategies to explain market anomalies that cannot be elucidated using the EMH. The contrarian strategy, first proposed by DeBondt and Thaler (1985), highlighted the possibility of overreactions in the stock market that would cause overestimation of share prices. Thus, they suggested that investors adopt a contrarian strategy and buy shares that showed inferior performance during the previous period (i.e., a loser portfolio) and conduct short selling of shares that exhibited

*Corresponding author. E-mail: [email protected]

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754 superior performance during the previous period (i.e., a winner portfolio). However, Jegadeesh and Titman (1993) proposed an investment strategy divergent from that of the contrarian strategy, that is, the momentum strategy, which maintains that the stock market tends to underreact. They suggested that investors buy shares that exhibited superior performance during the previous period (i.e., a winner portfolio) and conduct short selling for shares that showed inferior performance during the previous period (i.e., a loser portfolio) to obtain optimal excess returns. The results derived by Jegadeesh and Titman (1993) indicated that returns for the winner is significantly greater than that for the loser during any given investment period. Because investment portfolio returns may not comprehensively reflect market information, an anomaly, where by the strong stay strong and the weak stay weak, is prevalent in the stock market. Further researchers also found similar evidence, such as Lee and Swaminathan (2000), Jegadeesh and Titman (2001), Chordia and Shivakumar (2002), and George and Hwang (2004). Since the global financial crisis in 2008 and the recent European debt crisis, credit-risk management issues have gradually gained increasing attention. Credit risk refers to the risk of potential losses caused by a breach of contract or credit-rating change regarding transaction counterparty. According to the 1996 definition provided by the Bank for International Settlements (BIS), credit risk refers to the inability of transaction counterparty to conduct timely fulfilment of obligations or responsibilities. Because the financial market is rapidly evolving, novel financial tools are continuously emerging and participants in the market encounter increasingly complex credit-risk patterns or forms. According to the risk assessment methods stipulated by the BIS, either external rating methods (e.g., credit ratings) or internal rating models (e.g., the KMV (Kealhofer, McQuown and Vasicek) model)1 can be adopted to evaluate credit risk. Recently, investor awareness has increased regarding credit-rating mechanisms, which enables them to more clearly and accurately observe risk changes. Therefore, scholars have investigated whether implementing the momentum strategy for credit-ratings results in excess returns. Avramov et al. (2007) contended that the momentum strategy produces excess returns for shares with poor credit ratings, whereas returns for shares with superior credit ratings are nonsignificant. The majority of previous studies have focused solely on the momentum strategy. Only Avramov et al. (2007) adopted credit ratings to conduct indirect measurements of the relationship between credit risk and the momentum strategy. However, credit ratings are a type of external

S.-L. Lu et al. rating; thus, a lag exists in relevant evaluations of credit risks. The primary contribution of this study is the adoption of a structural model (i.e., the KMV model) among credit-risk models to measure and evaluates credit risk, which should facilitate more precise understandings of this type of risk. The rapid economic growth of the Asian countries has been a focus of interest for academics and policy makers recently. Taiwan is an economically powerful region, supplying much of the world with their exports and teaching the rest of the world new management techniques. World Bank (1993) showed that Taiwan has achieved high economic growth since the post-World War II due to the guide of government policies, such as interest-rate deregulation, deregulation of foreign exchange rate, liberalization of capital movement, and deregulation of financial institutions. Thus, this study examined momentum strategy returns in Taiwan’s stock market. Subsequently, we considered credit risk to investigate the influence of different types of credit risk on momentum strategy returns. Previous research has identified significant correlations between the momentum strategy and numerous factors such as economic or business cycles and the January effect (DeBondt and Thaler, 1985; Jegadeesh and Titman, 1993; Chordia and Shivakumar, 2002; Avramov and Chordia, 2006; Avramov et al., 2007). Thus, this study adopted credit risk as a basis to examine the influence of these two factors on the momentum strategy. The empirical results of this study confirmed the existence of a momentum strategy in Taiwan’s stock market. Investors can obtain excess returns using investment strategies involving selling loser and buying winner portfolios. Then, we adopted the distance-to-default (DD) of the KMV model as a proxy indicator for credit risk, and further separated investment portfolios into three credit-risk groups. The results imply momentum profits are highest for the high-credit-risk group, which are most speculative stocks. The findings validate the investment theory. If a portfolio offers higher return, then it has to bear higher risk. Considering the January effect, the pay-offs in January are negative albeit statistically insignificant for low-creditrisk group in short- and midterm holding periods. But the momentum pay-offs convert to positive in 36-months holding periods. For the medium-credit-risk group, the momentum pay-offs in January are significant and positive in short-term holding periods. For high-credit-risk group, momentum profit in January or non-January months are positive but insignificant for short-term holding periods. Thus, the January effect only has positive and significant effect on momentum strategy for low- and medium-credit-risk groups in 36-months holding periods.

1  KMV model was based on option pricing theory and named with the first letters of three KMV founder, Kealhofer, McQuown and Vasicek (KMV). The basic idea is to use stock to show the options nature, through the stock market and its volatility as well as the value of corporate debt data to estimate the likelihood of corporate defaults, i.e., the expected probability of default (EDF).

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Table 1. Momentum strategy returns for different credit risk Credit-risk group (L = Lowest credit risk, M = Medium credit risk, H = Highest credit risk) L Holding period k=1 k=3 k=6 k=9 k = 12 k = 24 k = 36

M

H

Winner

Loser

W–L

Winner

Loser

W–L

Winner

Loser

W–L

0.010 (0.000)* 0.031 (0.000)* 0.074 (0.000)* 0.120 (0.000)* 0.167 (0.000)* 0.442 (0.000)* 0.701 (0.000)*

0.011 (0.000)* 0.043 (0.000)* 0.094 (0.000)* 0.145 (0.000)* 0.194 (0.000)* 0.339 (0.000)* 0.400 (0.000)*

0.000 (0.715) −0.013 (0.014)* −0.020 (0.047)* −0.025 (0.151) −0.030 (0.216) 0.103 (0.077) 0.301 (0.018)*

0.014 (0.000)* 0.044 (0.000)* 0.104 (0.000)* 0.151 (0.000)* 0.208 (0.000)* 0.513 (0.000)* 0.778 (0.001)

0.013 (0.000)* 0.048 (0.000)* 0.092 (0.000)* 0.157 (0.001)* 0.207 (0.001)* 0.425 (0.000)* 0.473 (0.003)

0.001 (0.727) −0.004 (0.522) 0.012 (0.454) −0.006 (0.812) 0.002 (0.959) 0.088 (0.165) 0.306 (0.042)*

0.017 (0.000)* 0.065 (0.000)* 0.161 (0.000)* 0.225 (0.000)* 0.299 (0.000)* 0.797 (0.000)* 1.201 (0.000)*

0.017 (0.000)* 0.054 (0.000)* 0.101 (0.000)* 0.156 (0.000)* 0.220 (0.000)* 0.328 (0.000)* 0.275 (0.004)*

0.000 (0.913) 0.011 (0.190) 0.060 (0.004)* 0.069 (0.042)* 0.079 (0.019)* 0.469 (0.000)* 0.925 (0.000)*

Notes: p-Values are in parentheses. The ‘*’ denotes significance at the 5% level. The ‘W – L’ is Winners’ returns minus Losers’ returns.

Reviewing Table 1, the findings validate the positive momentum profits for these two risk groups are due to the January effect. The findings are inconsistent with previous researches that suggest negative momentum profits in January due to the consideration to tax. In Taiwan, the Chinese New Year is always in January, which is the major festival. Thus, the momentum profits are positive in January. Considering the effect of business cycle, the momentum pay-off is significant for three credit-risk groups in a recession economic period. These findings imply that investors earn positive returns by applying momentum strategy when face recession period. The findings imply that Taiwanese market is more inefficient during recession period. Thus, this article investigates momentum effects for three credit-risk groups by including January effect and economy states. This issue is seldom discussed in previous searches. This article found that the momentum strategy returns are significantly affected by credit risk. Investors should consider the credit-risk characteristics of their investment portfolio when employing the momentum strategy. This remainder of this article is organized as follows: Section I presents a discussion of the motivation for this study; Section II provides a review of studies associated with momentum strategy; Section III details the formal methodology; Section IV presents a description of the sample data and provides an analysis of the empirical results; and finally, Section V offers the findings and provides a conclusion.

II. Literature Review In general, the investment strategy can be divided into two categories: one is contrarian strategy, another is the momentum strategy. The contrarian strategy is buying past losers and selling past winners to earn abnormal returns. This contrarian strategy is suggested by DeBondt and Thaler (1985, 1987). They show that investors earn higher returns by buying stocks that performed poor over 3–5 holding years than buying stocks that performed well over the same period. The De Bondt and Thaler results can be explained by overreacting to information. Further researches such as Howe (1986) and Iihara et al. (2004) also found similar results. The other investment strategy is momentum strategy, which is buying past winners and selling past losers to earn large pay-offs. This strategy is documented by Jegadeesh and Titman (1993). They found that excess returns is available from the momentum investment portfolio of buying performed-well stocks and selling performed-poor stocks in the past 3–12 months. The followup scholars have also found similar results, such as Lee and Swaminathan (2000), Jegadeesh and Titman (2001), Chordia and Shivakumar (2002), and George and Hwang (2004). Moskowitz and Grinblatt (1999) investigate industry momentum effect. They find that industry momentum investment strategy earn high profits. That is, there exists extremely strong industry influence. Foster and Kharazi (2008) study the profitability of momentum strategies in TSE returns for the 1997–2002 period.

S.-L. Lu et al.

756 They find evidence in support for excess return in momentum strategy. Jiang et al. (2005) and Zhang (2006) suggest that momentum profits are affected by information uncertainty that is proxies of firm size, firm age, return volatility, cash-flow volatility, and analyst forecast dispersion. However, Avramov et al. (2007) find that the above variables do not capture the momentum profits. They apply credit rating to capture the momentum profits and find that momentum profits are restricted to high-credit-risk firms and are nonexistent for low-creditrisk firms. However, Kraussl (2005) suggested that credit-rating agencies tend to lag financial market developments. Thus, credit ratings are published by rating agencies that may lag in evaluating credit risk. This article applies the KMV model to evaluate credit risk, which is more elaborate than that applied in Avramov et al. (2007). The KMV model is a particular application of Merton’s model (Merton, 1974), in which the equity of the firm is a call option on the underlying value of the firm with a strike price equal to the face value of the firm’s debt. Kealhofer and Kurbat (2001) argued that KMV models capture all of the information in traditional agency ratings and well-known accounting variables. Duffie et al. (2007) showed that KMV models have significant predictive power in estimating default probabilities over time. Reinganum (1981), Blume and Stambaugh (1983) and Keim (1983), suggest that the losers can earn large January returns whereas winners do not. DeBondt and Thaler (1985) find that large positive excess returns can be earned by the loser portfolio in January. Jegadeesh and Titman (1993) and Avramov et al. (2007) suggest that momentum profits are negative in January. The momentum strategy may be different in January or non-January. Therefore, this article adopted credit risk as a basis to examine the influence of January effect on the momentum strategy. The empirical results are consistent with Jegadeesh and Titman (1993) only for low-credit-risk group in medium-holding period. However, the positive and significant momentum profits are found for low- and mediumcredit-risk groups in 36-months holding periods that is different from previous findings. Chordia and Shivakumar (2002) suggest that momentum profits are significantly affected by business cycle. They find that pay-offs are large during expansions and nonexistent during recession. Avramov and Chordia (2006) show that momentum strategy is related to business cycle variables, such as expansion and recession. Thus, this article adopted credit risk as a basis to examine the influence of phases of the economy on the momentum strategy. This article finds that positive momentum profits only exist during recession period, which is inconsistent with Chordia and Shivakumar (2002).

III. Methodology Construction of momentum portfolio Momentum portfolios are constructed as Jegadeesh and Titman (1993). We rank all stocks based on their cumulative return over the formation period, 6 months. We assign stocks to two groups based on their prior 6-month’s returns. The top 30% are denoted as winners and the bottom 30% as losers. Then we construct an investment portfolio by buying the winner stocks and selling the loser stocks. The two portfolios are equally weighted at formation and held for k months (k = 1, 3, 6, 9, 12, 24, 36). We denote that holding periods: 1 and 3 months is short term, 6 and 9 months is midterm, and 12, 24, and 36 months is long term. Credit-risk model In this article, we examine credit risk based on the KMV model, which is extended from Merton’s (1974) model. The Merton model assumed that the equity value of a firm satisfies Vs ¼ VA N ðd1 Þ  Fert N ðd2 Þ

(1)

where Vs is the market value of the firm’s equity, VA is the market value of the firm’s asset, F is the face value of firm’s debt, r is the risk-free rate, and N(.) is the cumulative standard normal distribution. d1 and d2 is given by

 ln VFA þ r þ 12 σ 2A t p ffiffiffiffiffiffi ffi d1 ¼ σ 2A t d 2 ¼ d1 

qffiffiffiffiffiffiffi σ 2A t

where t is the horizon of the face value of the firm’s debt and σA is the volatility of the underlying firm’s asset. Using the Ito’s lemma, the relation between the volatility of the firm (σ E ) and its equity are   VA N ðd1 Þσ A σE ¼ Vs

(2)

The KMV model basically uses nonlinear Equations 1 and 2, to translate the value and volatility of a firm’s equality into an implied probability of default. There are three steps to implement the KMV model. The first step is to estimate the volatility of equity, σ E , from historical stock-return data. The second step is to choose a forecasting horizon and measure the face value of the firm’s debt. The third step is to simultaneously solve Equations 1 and 2 numerically for values of VA and σ A . Thus the DD can be calculated as

Momentum strategy and credit risk DD ¼

EðVA Þ  DPT σA

757 (3)

where EðVA Þ is the expected value of the firm’s asset. DPT is the distance point of default, which is defined as current liability plus half of long-term liability.

IV. Data and Empirical Results Analysis Data The sample includes common stocks listing in Taiwan stock market. There are 12 industries: Foods, Plastics, Textiles, Electric. Machinery, Electrical and Cable, Chemical and Biotech, Iron and Steel, Electronics, Building and Cons., Shipping and Trans., Trading and Consumer, and Others, and includes 439 firms. The sample period was from 2000 to 2011. The data was obtained from the Taiwan Economic Journal (TEJ) database. We exclude stocks that were missing data during the formation and holding periods. These firms are listed in Table A. Empirical results analysis Momentum profits. We followed the approach of Jegadeesh and Titman (1993) to calculate average monthly returns. We sort firms in each month based on the past 6 month’s returns. The top 30% are denoted as winners and the final 30% losers. Then we construct a zero-investment portfolio by buying the winner stocks and selling the loser stocks. The portfolios are equally weighted at formation and held for k subsequent months (k = 1, 3, 6, 9, 12, 24, 36). The evidence in Table 2 shows momentum profitability for short-, mid-, and long-term holding periods. Momentum profits are prominent over all holding periods, which points to a violation of Taiwan’s market efficiency. That is, investors can obtain excess returns using investment strategies involving selling loser and buying winner portfolios.

Momentum profitability and credit risk. Credit risk is a crucial influential factor for corporate and investor decision making, and investors cannot acquire expected profits and earnings if a corporation breaches contract. Therefore, we implement momentum strategies by conditioning on both credit risk and cumulative 6-month formation period returns. We first consider three credit-risk groups according to DD of the KMV model. The ‘L, M, H’ are the low-, medium-, and high-credit-risk groups, respectively. We sort first on credit risk and then on past returns to estimate momentum pay-offs for different credit-risk groups. Table 1 presents the momentum profits of three creditrisk groups. We find that momentum strategy returns are strongly dependent upon credit risk. The excess returns produced by the momentum strategy are applicable only for investment portfolios involving mid- and long-term investments with high credit risks, H. Conversely, no significant excess returns can be obtained when this strategy is applied to medium- or low-credit-risk investment portfolios, M and L. Furthermore, negative returns may be generated when the momentum strategy is employed for short- and midterm holding periods. However, the excess returns gained through implementing the momentum strategy are significantly positive when the holding period exceeds 36 months. Thus, the momentum strategy profit patient investors. The findings support the investment theory, ‘no pains, no gains’. That is, if a portfolio offers higher return, then it has to bear higher risk. Momentum profits, credit risk, and January effect. To mitigate taxes, investors in the US stock market reinvest funds in the stock market in January after selling shares in December, which generates a substantial increase in share prices during January, known as the January effect. Several studies have proposed that Taiwan’s stock market also exhibits a January effect, because returns in January tend to exceed those of other months and may influence investment strategy returns. Table 3 suggests momentum profitability in January and non-January months for three credit-risk groups. The investment portfolios with high credit risk are not influenced by the effect; returns are significantly positive in

Table 2. Momentum profitability for different hold periods Holding period

k=1

k=3

k=6

k=9

k = 12

k = 24

k = 36

Winner

0.222 (0.000)* −0.136 (0.000)* 0.358 (0.000)*

0.423 (0.000)* −0.205 (0.000)* 0.628 (0.000)*

0.665 (0.000)* −0.250 (0.000)* 0.915 (0.000)*

0.875 (0.000)* −0.279 (0.001)* 1.154 (0.000)*

1.096 (0.000)* −0.300 (0.001)* 1.395 (0.000)*

1.895 (0.001)* −0.336 (0.012)* 2.231 (0.000)*

2.496 (0.009)* −0.390 (0.040)* 2.886 (0.003)*

Loser W–L (Winner–Loser)

Notes: p-Values are in parentheses. The ‘*’ denotes significance at the 5% level.

S.-L. Lu et al.

758 Table 3. Momentum strategy returns, credit risk, and January effects

Credit risk group (L = Lowest credit risk, M = Medium credit risk, H = Highest credit risk) L Holding period

January/ Non-January

k=1

January Non-January

k=3

January Non-January

k=6

January Non-January

k=9

January Non-January

k = 12

January Non-January

k = 24

January Non-January

k = 36

January Non-January

M

H

Win

Lose

W–L

Win

Lose

W–L

Win

Lose

W–L

0.029 (0.000)* 0.009 (0.000)* 0.085 (0.000)* 0.018 (0.000)* 0.108 (0.000)* 0.048 (0.000)* 0.127 (0.000)* 0.127 (0.000)* 0.167 (0.000)* 0.144 (0.000)* 0.442 (0.000)* 0.361 (0.000)* 0.701 (0.000)* 0.541 (0.000)*

0.018 (0.009)* 0.010 (0.000)* 0.070 (0.000)* 0.029 (0.000)* 0.111 (0.000)* 0.070 (0.000)* 0.146 (0.000)* 0.115 (0.000)* 0.194 (0.000)* 0.186 (0.000)* 0.339 (0.000)* 0.341 (0.000)* 0.400 (0.000)* 0.421 (0.000)*

0.011 (0.057) −0.001 (0.535) 0.015 (0.051) −0.011 (0.009)* −0.003 (0.837) −0.022 (0.016)* −0.019 (0.297) 0.012 (0.463) −0.026 (0.277) −0.041 (0.036)* 0.103 (0.077) 0.020 (0.623) 0.301 (0.018)* 0.120 (0.148)

0.059 (0.000)* 0.011 (0.000)* 0.135 (0.000)* 0.021 (0.008)* 0.158 (0.000)* 0.051 (0.006) 0.163 (0.000)* 0.154 (0.000)* 0.208 (0.000)* 0.202 (0.000)* 0.513 (0.000)* 0.438 (0.000)* 0.778 (0.001)* 0.613 (0.000)

0.028 (0.004)* 0.010 (0.000)* 0.101 (0.000)* 0.024 (0.005)* 0.131 (0.000)* 0.052 (0.015)* 0.163 (0.001)* 0.105 (0.002)* 0.207 (0.001)* 0.157 (0.000)* 0.425 (0.000)* 0.330 (0.000)* 0.473 (0.003)* 0.484 (0.003)*

0.031 (0.004)* 0.001 (0.583) 0.035 (0.003)* −0.003 (0.603) 0.027 (0.158) −0.001 (0.947) −0.000 (0.991) 0.049 (0.110) 0.002 (0.959) 0.045 (0.181) 0.088 (0.165) 0.108 (0.132) 0.306 (0.042)* 0.129 (0.413)

0.073 (0.000)* 0.014 (0.000)* 0.183 (0.000)* 0.032 (0.002)* 0.220 (0.000)* 0.091 (0.001)* 0.244 (0.000)* 0.164 (0.000)* 0.299 (0.000)* 0.273 (0.000)* 0.797 (0.000)* 0.623 (0.000)* 1.201 (0.000)* 0.904 (0.000)*

0.055 (0.003)* 0.011 (0.000)* 0.160 (0.000)* 0.015 (0.047)* 0.180 (0.000)* 0.027 (0.118) 0.170 (0.000)* 0.107 (0.002)* 0.220 (0.000)* 0.168 (0.000)* 0.328 (0.000)* 0.355 (0.000)* 0.275 (0.004)* 0.344 (0.010)*

0.018 (0.054) 0.002 (0.493) 0.023 (0.182) 0.017 (0.099) 0.040 (0.002)* 0.064 (0.021)* 0.074 (0.028)* 0.057 (0.122) 0.079 (0.019)* 0.106 (0.017)* 0.469 (0.000)* 0.268 (0.048)* 0.925 (0.000)* 0.560 (0.010)*

Notes: p-Values are in parentheses. The ‘*’ denotes significance at the 5% level. The ‘W – L’ is Winners’ returns minus Losers’ returns.

January or non-January. For the medium- or low-creditrisk investment portfolio group, M and L, significantly positive returns for January are derived only if the holding period achieves 36 months. Specifically, for the lowcredit-risk group in midterm holding period, the momentum strategy is unbeneficial and may result in negative returns. These findings are consistent with Jegadeesh and Titman (1993), momentum profits are negative in January. Consequently, the momentum strategy in January and non-January for three credit-risk groups has different influences. The positive momentum profits in January only exist in low- and medium-credit-risk group in 36months holding period. The findings also validate the empirical result in Table 1 that is the positive momentum profits for the two risk groups may be due to the January effect. Momentum profits, credit risk, and business cycle. Subsequently, this study considered the influence of economic or business cycles on the momentum strategy. An

economic or business cycle refers to the long-term economic development process of a country, during which economic activities are more frequent in certain periods, creating economic expansion. Subsequently, when economy reaches its peak, the economic activities become slow and negative growth may occur, which will cause an economic recession. After the economy reaches its nadir, it begins to revive and subsequently enters a new period of expansion. These expansions and recessions are generally periodic or cyclical, and economic or business cycles commonly influence investors’ investment strategies. The Taiwanese government has established a monitoring indicator as a criterion for identifying periods of economic expansion and recession more accurately. For this study, we denoted economic expansion and recession periods as over 32 and under 22 points, respectively, according to the monitoring indicator definition. Finally, the researchers examined whether economic or business cycles affected investors’ momentum strategies. Because a growth-cycle concept is adopted for Taiwan’s

Momentum strategy and credit risk

759

Table 4. Momentum strategy returns, credit risk, and business cycle Credit risk group (L = Lowest credit risk, M = Medium credit risk, H = Highest credit risk) L Holding period

Economics state

k=1

Expansion Recession

k=3

Expansion Recession

k=6

Expansion Recession

k=9

Expansion Recession

k = 12

Expansion Recession

M

H

Win

Lose

W–L

Win

Lose

W–L

Win

Lose

W–L

0.024 (0.000)* −0.009 (0.001)* 0.007 (0.644) 0.035 (0.012)* 0.120 (0.000)* −0.033 (0.005)* 0.015 (0.000)* −0.008 (0.001)* 0.217 (0.000)* −0.056 (0.004)*

0.024 (0.000)* −0.014 (0.000)* −0.002 (0.907) 0.049 (0.011)* 0.157 (0.000)* −0.049 (0.002)* 0.019 (0.000)* −0.011 (0.000)* 0.287 (0.000)* −0.082 (0.000)*

0.000 (0.859) 0.004 (0.005)* 0.009 (0.025)* −0.014 (0.065) −0.037 (0.014)* 0.015 (0.089) −0.004 (0.020)* 0.002 (0.039)* −0.070 (0.011)* 0.026 (0.058)

0.034 (0.000)* −0.015 (0.000)* 0.009 (0.684) 0.051 (0.020)* 0.183 (0.000)* −0.054 (0.003)* 0.020 (0.000)* −0.013 (0.000)* 0.332 (0.000)* −0.087 (0.001)*

0.029 (0.000)* −0.019 (0.000)* 0.002 (0.924) 0.044 (0.027)* 0.176 (0.000)* −0.091 (0.000)* 0.019 (0.000)* −0.017 (0.000)* 0.327 (0.000)* −0.134 (0.000)*

0.005 (0.148) 0.003 (0.342) 0.007 (0.172) 0.007 (0.566) 0.007 (0.663) 0.036 (0.016)* 0.001 (0.538) 0.005 (0.029)* 0.005 (0.875) 0.047 (0.021)*

0.039 (0.000)* −0.017 (0.000) 0.015 (0.533) 0.071 (0.014) 0.230 (0.000)* −0.055 (0.004)* 0.023 (0.000)* −0.017 (0.000)* 0.431 (0.000)* −0.099 (0.000)*

0.038 (0.000)* −0.023 (0.000)* −0.003 (0.896) 0.059 (0.017) 0.206 (0.000)* −0.085 (0.000)* 0.019 (0.000)* −0.022 (0.000)* 0.375 (0.000)* −0.172 (0.000)*

0.001 (0.766) 0.006 (0.114) 0.018 (0.055) 0.012 (0.170) 0.024 (0.209) 0.030 (0.119) 0.004 (0.072) 0.005 (0.041)* 0.056 (0.144) 0.073 (0.003)*

Notes: p-Values are in parentheses. The ‘*’ denotes significance at the 5% level. The ‘W – L’ is Winners’ returns minus Losers’ returns.

economic or business cycle, cyclical fluctuations following long-term trends were excluded; that is, 24- and 36month holding periods were not considered. Therefore, we excluded these holding periods when calculating the accumulated returns of all other holding periods during economic expansions and recessions. Table 4 suggests momentum profitability in expansion and recession periods for three credit-risk groups. For investment portfolios with low credit risk, adopting the momentum strategy in an expanding economy is detrimental because returns are significantly negative. For three credit-risk groups, excess returns can be obtained by adopting the momentum strategy in a recession period, especially for medium-credit-risk group in mid- and longterm holding periods. That is, investors can earn excess return by buying the winner portfolio and selling loser portfolio during the recession period. Consequently, this study found that credit risk and business cycle exert significant influences on momentum strategy returns.

V. Conclusions This article establishes momentum profitability for different credit risk. We establish three credit-risk groups by DD of the KMV model. We investigate relationship between

momentum profitability and credit risk. The empirical findings are based on a sample of 439 listing firms of 12 industries in Taiwan from 2000 to 2011. The winner and loser portfolios are comprised mainly of three credit-risk groups. Momentum profitability is statistically significant and economically large for highcredit-risk portfolio. That is, the evidence shows that momentum profits strongly depend on credit risk. The momentum profitability is significant especially for long-term holding periods. It implies that patient investors are benefited by the momentum strategy. It is nonexistent among medium- and low-credit-risk groups, but not in longterm holding periods, which may be due to the January effect. We investigate momentum profits in January for three credit-risk groups. The January effect has only positive and significant effect on the low- and medium-credit-risk group for 36-months holding period, which is also found in Table 2. Finally, this study considers the influence of business cycle on the momentum profits. The low-credit-risk portfolio has negative momentum profits in an expanding economic period. However, investors earn significant momentum profits during the recession period. Thus, we find that momentum profits strongly depend on credit risk. We recommend that if investors employ the momentum strategy during investments, they should further consider the credit-risk characteristics of their investment portfolios.

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Appendix Table A. The codes and names of sample companies Industry

Company’s name and code

Observations

1436 FUI ; 1805 KPT Industries ; 2501 Cathay Real Estate ; 2504 Goldsun ; 2505 Kuo Yang Const. ; 2506 39 Pacific Construction ; 2509 Chainqui ; 2511 Prince Housing ; 2515 BES Engineering ; 2516 New Asia ; 2520 Kindom Const. ; 2524 King’s Town Cons ; 2527 Hung Ching ; 2530 Delpha Const. ; 2534 Hung Sheng Const. ; 2535 Da Cin Const. ; 2536 Hung Poo ; 2537 WE & WIN ; 2538 Kee Tai Properties ; 2542 Highwealth ; 2543 Hwang Chang ; 2545 Huang Hsiang ; 2546 Kedge ; 2841 TLC ; 5324 SDC ; 5505 Howang ; 5506 Evergreen Cons ; 5508 Yung Shin Const. ; 5511 Te Chang Const. ; 5512 Rich Dev. ; 5514 Sun Fon Const. ; 5515 Chien Kuo ; 5516 Sun-Sea ; 5519 LongDa ; 5520 Lihtai ; 5521 Kung Sing Eng ; 5522 Farglory ; 5523 Hung Tu Cons. ; 5529 Well Glory Chemical & 1701 China Chemical ; 1704 LCYCIC ; 1707 Grape King ; 1708 Sesoda ; 1709 Formosan Union Chem ; 30 Biotech 1710 Oriental Union Chem ; 1711 Everlight Chemical ; 1712 Sinon ; 1713 Cathay Chemical ; 1714 Ho Tung Chemical ; 1717 Eternal Chemical ; 1718 China Man-Made Fiber ; 1720 Standard Chemical ; 1721 Sunko Ink ; 1722 Taiwan Fertilizer ; 1723 China Steel Chem ; 1724 T. N. C. Industrial ; 1725 Yuan Jen Ent. ; 1726 Yung Chi Paint ; 1727 Chung Hwa Chem ; 1730 Farcent ; 1731 Maywufa ; 1732 Mao Bao ; 4111 Chi Sheng Chemical ; 4702 Allied ; 4703 Yang Hwa ; 4706 Tah Kong Chem. ; 4707 Pan Asia Chem. ; 4711 Yong Shun ; 4712 Nan Tsan Electric. 1503 Shihlin Electric ; 1504 Teco ; 1506 Right Way ; 1507 Yungtay Eng. ; 1512 Jui Li ; 1513 Chung-Hsin 28 Machinery Electric ; 1514 Allis Electric ; 1515 Rexon ; 1517 Lee Chi ; 1519 Fortune Electric ; 1521 Daioku ; 1522 TYC Brother ; 1524 Gordon Auto ; 1525 Kian Shen ; 1527 Basso ; 1528 Anderson ; 1529 Luxe ; 1530 Awea Mechantronic ; 1531 Kaulin Mfg. ; 1535 China Ecotek ; 1538 JF ; 1540 Roundtop Machinery ; 4502 Yuan Feng ; 4503 Gold Rain ; 4506 Golden Friends ; 4510 Kao Fong Machinery ; 4513 Falcon ; 4523 Taiwan Calsonic Electric and 1603 China W&C ; 1604 Sampo ; 1605 Walsin ; 1608 Hua Eng ; 1609 Ta Ya Elec. ; 1611 China Electric ; 13 Cable 1612 Hong Tai Electric ; 1613 Tai-I Electric ; 1614 Sanyo ; 1615 Dah San Elec. ; 1616 Evertop Wire ; 1617 Jung Shing Wire ; 1618 Hold-Key Electronics 1437 GTM Corp. ; 1471 Solytech ; 2301 Lite-On Technology ; 2302 Rectron ; 2303 United Microelec. ; 156 2308 Delta Electronics ; 2311 ASE ; 2312 Kinpo ; 2313 Compeq Mfg. ; 2314 Microelec. Tech. ; 2315 Mitac ; 2316 Wus Printed Circuit ; 2317 Hon Hai Precision ; 2321 Tecom ; 2323 CMC Magnetics ; 2324 Compal Electronics ; 2325 Siliconware Prec. ; 2327 Yageo ; 2328 Pan International ; 2329 Orient Semi ; 2330 TSMC ; 2331 Elitegroup ; 2332 D-Link ; 2337 Macronix ; 2338 Taiwan Mask ; 2340 Opto Tech ; 2344 Winbond ; 2345 Accton ; 2347 Synnex ; 2348 Veutron ; 2349 Ritek ; 2351 SDI Corp. ; 2352 Qisda Corp. ; 2353 Acer Inc. ; 2354 Foxconn Technology ; 2355 Chin-Poon ; 2356 Inventec ; 2357 Asustek ; 2359 Solomon Technology ; 2360 Chroma ; 2362 Clevo ; 2363 Silicon Integrated ; 2364 Twinhead ; 2365 KYE Systems ; 2367 Unitech PCB ; 2368 Gold Circuit ; 2369 Lingsen Precision ; 2371 Tatung Co. ; 2373 Aurora ; 2374 Ability ; 2375 Teapo ; 2376 Gigabyte ; 2377 Micro-Star ; 2379 Realtek ; 2380 Avision ; 2381 Arima ; 2382 Quanta Computer ; 2383 Elite Material ; 2384 Wintek ; 2385 Chicony ; 2387 Sunrex ; 2388 VIA Technologies ; 2390 Everspring ; 2392 Foxlink ; 2393 Everlight Elec ; 2395 Advantech ; 2397 DFI Inc. ; 2399 Biostar Microtech ; 2401 Sunplus ; 2402 Ichia ; 2404 United Integrated ; 2405 Shuttle ; 2413 Universal Micro ; 2414 Unitech Computer ; 2415 Cx Tech ; 2417 AverMedia ; 2419 Hitron Tech. ; 2420 Zippy ; 2421 Sunonwealth ; 2423 Good Will Instrument ; 2424 Lung Hwa Electronics ; 2425 Chaintech ; 2426 Tyntek ; 2427 Mercuries Data ; 2428 Thinking Electronic ; 2430 Tsann Kuen Ent ; 2431 Lien Chang ; 2433 Aurora Systems ; 2434 Mospec ; 2436 Weltrend ; 2437 Ralec ; 2438 Enlight ; 2439 Merry ; 2459 Audix ; 2460 Gem Terminal ; 2461 K Laser ; 2462 Taiwan Line Tek ; 2464 Mirle Automation ; 2465 Leadtek ; 2466 COSMO ; 2467 C Sun ; 2468 Fortune Information ; 2471 Ares International ; 2473 Springsoft ; 2474 Catcher ; 2476 G-Shank ; 2478 Ta-I Tech ; 2480 Stark ; 2481 Pan Jit ; 2482 Uniform Industrial ; 2483 Excel Cell Elec ; 2484 Siward Crystal ; 2485 Zinwell ; 2489 Amtran ; 2492 Walsin Technology ; 3024 Action Electronics ; 3026 Holy Stone ; 3027 Billion ; 3029 Zero One ; 3035 Faraday ; 3037 Unimicron ; 3041 Ali Corp. ; 3052 Apex Science & Eng ; 5302 Syntek Design ; 5304 Dbtel ; 5305 Lite-On Semi. ; 5306 KMC ; 5310 CGS International ; 5314 Myson ; 5315 United Radiant ; 5317 Kaimei ; 5326 Episil Technologies ; 5328 Hua Jung Components ; 5340 Baotek ; 5344 Vate Tech. ; 5345 Team Young ; 5346 Powerchip ; 5347 Vanguard ; 5348 Transystem ; 5349 Boardtek ; 5351 Etron ; 5353 Tailyn ; 5355 Gia Tzoong ; 5356 Sirtec ; 5371 Coretronic ; 5381 Uniplus ; 5383 Kenly ; 5384 Genuine ; 5388 SerComm ; 5398 Inalways ; 5403 Dimerco Data System ; 5410 LEO Systems ; 5443 Gallant Precision ; 6190 Wonderful ; 6191 Global Brands ; 9912 Associated Inds CN Foods 1201 Wei Chuan Foods ; 1210 Great Wall ; 1213 Oceanic Beverage ; 1215 Charoen Pokphand ; 1216 Uni- 21 President ; 1217 AGV Products ; 1218 Taisun Enterprise ; 1219 Fwusow Industry ; 1210 Great Wall ; 1225 Formosa Oilseed ; 1227 Standard Foods ; 1229 Lien Hwa Industrial ; 1231 Lian Hwa Foods ; 1232 TTET Union ; 1233 Ten Ren Tea ; 1234 Hey-Song ; 1235 Shin Tai ; 1236 Hunya Foods ; 1702 Namchow Chemical ; 4205 HYFICL ; 4207 Taiwan Fructose

Building & Cons.

(continued )

S.-L. Lu et al.

762 Table A. Continued Industry

Company’s name and code

Observations

1532 China metal Products ; 2002 China Steel ; 2006 Tung Ho Steel ; 2007 Yieh Hsing ; 2008 Kao Hsing 26 Chang ; 2009 First Copper ; 2010 Chun Yuan Steel ; 2012 Chun Yu Works ; 2013 China Steel Struct. ; 2014 Chung Hung Steel ; 2015 Feng Hsin Iron ; 2017 Quintain Steel ; 2020 Mayer Steel ; 2022 Tycoons Group ; 2023 Yieh Phui ; 2025 Chien Shing Stainles ; 2027 Ta Chen ; 2028 Wei Chih Steel ; 2029 Sheng Yu Steel ; 2030 Froch ; 2031 Hsin Kuang Steel ; 2032 Sinkang ; 2033 Chia Ta ; 2034 YC Inox ; 5007 San Shing ; 5009 Gloria Material 22 Plastics 1301 Formosa Plastics ; 1303 Nan Ya Plastics ; 1304 USI Corp. ; 1305 China General ; 1307 San Fang Chemical ; 1308 Asia Polymer ; 1309 Taita Chemical ; 1310 T. S. M. C. ; 1312 Grand Pacific Petro ; 1313 UPC Technology ; 1314 China Petrochem ; 1315 Tah Hsin ; 1316 Sun Yad ; 1319 Tong Yang ; 1321 Ocean Plastics ; 1323 Yon Yu Plastics ; 1324 Globe Industries ; 1325 Universal Inc. ; 1326 Formosa Chem & Fibre ; 1715 Achem Technology ; 4303 HSINLI ; 4304 Sunvic Shipping and 2603 Evergreen Marine ; 2605 Sincere Navigation ; 2606 U-Ming Marine ; 2607 EITC ; 2608 Kerry TJ ; 18 Trans 2609 Yang Ming Marine ; 2610 China AirLines ; 2611 Tze Shin ; 2612 CMT ; 2613 China Container ; 2614 Eastern Media Intl ; 2615 Wan Hai Lines ; 2617 Taiwan Navigation ; 2618 EVA Airways ; 5601 Taiwan Allied ; 5603 Sea & Land ; 5604 Chung Lien Trans. ; 5607 Farglory FTZ Holding Textiles 1402 FENC ; 1409 Shinkong Synthetic ; 1410 Nan Yang D&F ; 1413 Hung Chou Chemical ; 1414 Tung 41 Ho Textile ; 1416 Kwong Fong ; 1417 Carnival ; 1418 Tong Hwa Synthetic ; 1419 Shinko. Textile ; 1423 Reward Wool ; 1434 Formosa Taffeta ; 1439 Chuwa Wool ; 1440 Tainan Spinning ; 1441 Tah Tong Textile ; 1443 Lily Textile ; 1444 Lealea ; 1445 Universal Textile ; 1446 Hong Ho Precision ; 1447 Li Peng ; 1451 Nien Hsing Textile ; 1452 Hong Yi Fiber ; 1453 Ta Jiang ; 1454 Taiwan Taffeta ; 1455 Zig Sheng ; 1456 I-Hwa Industrial ; 1457 Yi Jinn ; 1459 Lan Fa Textile ; 1460 Everest Textile ; 1463 Chyang Sheng ; 1464 De Licacy ; 1465 Wisher ; 1467 Tex-Ray ; 1468 Chang Ho Fibre ; 1470 Evertex ; 1472 Tri Ocean Textile ; 1473 Tainan Enterprises ; 1474 Honmyue Ent. ; 4401 Toung Loong Textile ; 4402 Fu Ta ; 4406 Hsin Sin ; 4414 Roo Hsing Trading and 2901 Shin Shin ; 2903 Far Eastern Dept. ; 2905 Mercuries & Associ. ; 2906 Collins ; 2908 Test Rite ; 2910 11 Consumer Tonlin Dept. Store ; 2911 Les Enphants ; 2912 Pre. Chain Store ; 2913 Taiwan Tea ; 2915 Ruentex ; 5902 Tait M&D Others 1107 Chien Tai Cement ; 1435 C. F. CORP. ; 2514 Long Bon ; 2904 POCS ; 5312 Formosa Optical ; 8705 34 Tong Lung Metal ; 8905 Eagle Cold Storage ; 8906 Forward Graphic ; 8913 Hwa-Hsia Leasing ; 8916 Kwong Lung ; 8923 China Times Publish. ; 9902 Tidehold ; 9904 Pou Chen ; 9905 Great China Metal ; 9906 Corner ; 9907 Ton Yi ; 9910 Feng Tay ; 9911 Taiwan Sakura ; 9914 Merida ; 9917 Taiwan Secom ; 9919 Kang Na Hsiung ; 9921 Giant ; 9924 Taiwan Fu Hsing ; 9925 SKS ; 9927 Thye Ming ; 9928 CTV ; 9929 Choice ; 9930 China Hi-Ment ; 9933 CTCI ; 9934 Globe Union ; 9935 Ching Feng Home ; 9940 Sinyi Realty ; 9941 Taiwan Acceptance ; 9945 Ruentex Develop Total 439

Iron and Steel

Quantitative Finance, Vol. 10, No. 8, October 2010, 855–869

Multivariate models for operational risk KLAUS BO¨CKERy and CLAUDIA KLU¨PPELBERG*z yRisk Integration, Reporting & Policies, UniCredit Group, Munich Branch, c/o HypoVereinsbank AG, Arabellastrasse 12, D-81925 Mu¨nchen, Germany zCenter for Mathematical Sciences and Institute for Advanced Study, Technische Universita¨t Mu¨nchen, 85747, Garching, Germany (Received 22 November 2007; in final form 14 September 2009) Bo¨cker and Klu¨ppelberg [Risk Mag., 2005, December, 90–93] presented a simple approximation of OpVaR of a single operational risk cell. The present paper derives approximations of similar quality and simplicity for the multivariate problem. Our approach is based on the modelling of the dependence structure of different cells via the new concept of a Le´vy copula. Keywords: Dependence model; Le´vy copula; Multivariate dependence; Multivariate Le´vy process; Operational risk; Pareto distribution; Regular variation; Subexponential distribution

1. Introduction The Basel II accord (Basel Committee on Banking Supervision 2004), which should have been fully implemented by year-end 2007, imposes new methods of calculating regulatory capital that apply to the banking industry. Besides credit risk, the new accord focuses on operational risk, defined as the risk of losses resulting from inadequate or failed internal processes, people and systems, or from external events. Choosing the advanced measurement approach (AMA), banks can use their own internal modelling technique based on bank-internal and external empirical data. A required feature of AMA is to allow for explicit correlations between different operational risk events. More precisely, according to Basel II, banks should allocate losses to one of eight business lines and to one of seven loss event types. Therefore, the core problem here is the multivariate modelling encompassing all different risk type/business line cells. For this purpose, we consider a d-dimensional compound Poisson process S ¼ (S1(t), S2(t), . . . , Sd (t))t0 with cadlag (right continuous with left limits) sample paths. Each component has the representation Si ðtÞ ¼

N i ðtÞ X k¼1

Xik ,

t  0,

*Corresponding author. Email: [email protected]

where Ni ¼ (Ni(t))t0 is a Poisson process with rate i40 (loss frequency) and ðXik Þk2N is an i.i.d. sequence of positive random variables (loss severities), independent of the Poisson process Ni. The bank’s total operational risk is then given by the stochastic process S þ ðtÞ :¼ S1 ðtÞ þ S2 ðtÞ þ    þ Sd ðtÞ,

t  0:

Note that S þ is again a compound Poisson process (see proposition 3.2). A fundamental question is how the dependence structure between different cells affects the bank’s total operational risk. The present literature suggests modelling dependence by introducing correlation between the Poisson processes (see, e.g., Powosjowski et al. 2002, Frachot et al. 2004, Bee 2005), or by using a distributional copula on the random time points where operational loss occurs, or on the number of operational risk events (Chavez-Demoulin et al. 2005). In all these approaches, each cell’s severities are assumed to be independent and identically distributed (i.i.d.) as well as independent of the frequency process. A possible dependence between severities has to be modelled separately, yielding in the end to a rather complicated model. Given the fact that statistical fitting of a highparameter model seems out of reach by the sparsity of the data, a simpler model is called for. Our approach has the advantage of modelling dependence in frequency and severity at the same time, yielding a model with comparably few parameters. Consequently, with a rather transparent dependence model, we are able

Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2010 Taylor & Francis http://www.informaworld.com DOI: 10.1080/14697680903358222

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K. Bo¨cker and C. Klu¨ppelberg

to model coincident losses occurring in different cells. From a mathematical point of view, in contrast to the models proposed by Chavez-Demoulin et al. (2005), we stay within the class of multivariate Le´vy processes, a class of stochastic processes that has also been well studied in the context of derivatives pricing (see, e.g., Cont and Tankov 2004). Since operational risk is only concerned with losses, we restrict ourselves to Le´vy processes admitting only positive jumps in every component, hereafter called spectrally positive Le´vy processes. As a consequence of their independent and stationary increments, Le´vy processes can be represented by the Le´vy–Khintchine formula, which, for a d-dimensional spectrally positive Le´vy process S without drift and Gaussian component, simplifies to ( Z ) iðz,St Þ iðz,xÞ Eðe Þ ¼ exp t ðe  1Þ ðdxÞ , z 2 Rd , Rdþ

where  is a measure on RdþP¼ ½0, 1Þd called the Le´vy measure of S and ðx, yÞ :¼ di¼1 xi yi for x, y 2 Rd denotes the inner product. Whereas the dependence structure in a Gaussian model is well-understood, dependence in the Le´vy measure  is much less obvious. Nevertheless, as  is independent of t, it suggests itself for modelling the dependence structure between the components of S. Such an approach has been suggested and investigated by Cont and Tankov (2004), Kallsen and Tankov (2006) and Barndorff-Nielsen and Lindner (2007), and essentially models dependence between the jumps of different Le´vy processes by means of so-called Le´vy copulas. In this paper we invoke Le´vy copulas to model the dependence between different operational risk cells. This allows us to gain deep insight into the multivariate behaviour of operational risk defined as a high quantile of a loss distribution and referred to as operational VaR (OpVaR). In certain cases, we obtain closed-form approximations for OpVaR and, in this respect, this paper can be regarded as a multivariate extension of Bo¨cker and Klu¨ppelberg (2005), where univariate OpVaR was investigated. Our paper is organized as follows. After stating the problem and reviewing the state of the art of operational risk modelling in the introduction, we present in section 2 the necessary concepts and recall the results for the single cell model. In section 3.1 we formulate the multivariate model and give the basic results, which we shall exploit later for the different dependence concepts. The total operational risk process is compound Poisson and we give the parameters explicitly, which results in the asymptotic form for total OpVaR. Before doing this we present in section 3.2 asymptotic results for the OpVaR, when the losses of one cell dominate all the others. In sections 3.3 and 3.4 we examine the cases of completely dependent and independent cells, respectively, and derive asymptotic closed-form expressions for the corresponding bank’s total OpVaR. In doing so, we show that, for very heavytailed data, the completely dependent OpVaR, which is

asymptotically simply the sum of the single cell VaRs, is even smaller than for independent OpVaR. As a more general multivariate model we investigate in section 3.5 the compound Poisson model with regularly varying Le´vy measure. This covers the case of single cell processes, whose loss distributions are of the same order and have a rather arbitrary dependence structure. This dependence structure manifests in the so-called spectral measure, which carries the same information of dependence as the Le´vy copula.

2. Preliminaries 2.1. Le´vy processes, tail integrals, and Le´vy copulas Distributional copulas are multivariate distribution functions with uniform marginals. They are used for dependence modelling within the context of Sklar’s theorem, which states that any multivariate distribution with continuous marginals can be transformed into a multivariate distribution with uniform marginals. This concept exploits the fact that distribution functions have values only in [0, 1]. In contrast, Le´vy measures are in general unbounded on Rd and may have a non-integrable singularity at 0, which causes problems for the copula idea. Within the class of spectrally positive compound Poisson models, the Le´vy measure of the cell process Si is given by i ([0, x)) ¼ iP(Xi  x) for x 2 [0, 1). It follows that the Le´vy measure is a finite measure with total mass i([0, 1)) ¼ i and, therefore, is in general not a probability measure. Since we are interested in extreme operational losses, we prefer (as is usual in the context of general Le´vy process theory) to define a copula for the tail integral. Although we shall mainly work with compound Poisson processes, we formulate definitions and some results and examples for the slightly more general case of spectrally positive Le´vy processes. Definition 2.1 (tail integral): Let X be a spectrally positive Le´vy process in Rd with Le´vy measure . Its tail integral is the function  : ½0, 1d ! ½0, 1, satisfying for x ¼ (x1, . . . , xd),

(1) ðxÞ ¼ ð½x1 , 1Þ      ½xd , 1ÞÞ, x 2 [0, 1)d, where ð0Þ ¼ limx1 #0,...,xd #0 ð½x1 , 1Þ      ½xd , 1ÞÞ (this limit is finite if and only if X is compound Poisson); (2)  is equal to 0 if one of its arguments is 1; and (3) ð0, . . . , xi , 0, . . . , 0Þ ¼ i ðxi Þ for ðx1 , . . . , xd Þ 2 Rdþ , where i ðxi Þ ¼ i ð½xi , 1ÞÞ is the tail integral of component i.

Definition 2.2 (Le´vy copula): A d-dimensional Le´vy copula of a spectrally positive Le´vy process is a measure b : ½0, 1d ! ½0, 1 with marginals, defining function C which are the identity functions on [0, 1]. The following is Sklar’s theorem for spectrally positive Le´vy processes.

Theorem 2.3 (Cont and Tankov 2004, theorem 5.6): Let  denote the tail integral of a d-dimensional spectrally

857

Multivariate models for operational risk positive Le´vy process, whose components have Le´vy measures 1, . . . , d. Then there exists a Le´vy copula b : ½0, 1d ! ½0, 1 such that, for all x1, . . . , xd 2 [0, 1], C

function F. Then F (or sometimes F ) is said to be subexponential (F 2 S) if

If the marginal tail integrals 1 , . . . , d are continuous, then this Le´vy copula is unique. Otherwise, it is unique on b is a Le´vy copula Ran 1      Ran d . Conversely, if C and 1 , . . . , d are marginal tail integrals of spectrally positive Le´vy processes, then (1) defines the tail integral of a d-dimensional spectrally positive Le´vy process and 1 , . . . , d are tail integrals of its components.

The interpretation of subexponential distributions is therefore that their i.i.d. sum is likely to be very large because of one of the terms being very large. The attribute subexponential refers to the fact that the tail of a subexponential distribution decays slower than any exponential tail, i.e. the class S consists of heavy-tailed distributions and is therefore appropriate to describe typical operational loss data. Important subexponential distributions are Pareto, lognormal and Weibull (with shape parameter less than 1). As a useful semiparametric class of subexponential distributions, we introduce distributions whose far out right tails behave like a power function. We present the definition for arbitrary functions, since we shall need this property not only for distribution tails, but also for quantile functions such as, for example, in proposition 2.13.

b 1 ðx1 Þ, . . . , d ðxd ÞÞ: ðx1 , . . . , xd Þ ¼ Cð

ð1Þ

The following two important Le´vy copulas model extreme dependence structures. Example 2.4 (complete (positive) dependence): Let S(t) ¼ (S1(t), . . . , Sd (t), t  0, be a spectrally positive Le´vy process with marginal tail integrals 1 , . . . , d . Since all jumps are positive, the marginal processes can never be negatively dependent. Complete dependence corresponds to a Le´vy copula bk ðxÞ ¼ minðx1 , . . . , xd Þ, C

lim

PðX1 þ    þ Xn 4 xÞ

x!1 PðmaxðX1 , . . . , Xn Þ 4 xÞ

f ðxtÞ ¼ t , x!1 f ðxÞ

ðx1 , . . . , xd Þ ¼ minð1 ðx1 Þ, . . . , d ðxd ÞÞ, on

lim

d

fx 2 ½0, 1Þ :

Example 2.5 (independence): Let S(t) ¼ (S1(t), . . . , Sd (t)), t  0, be a spectrally positive Le´vy process with marginal tail integrals 1 , . . . , d . The marginal processes are independent if and only if the Le´vy measure  of S can be decomposed into ðAÞ ¼ 1 ðA1 Þ þ    þ d ðAd Þ,

for some (all ) n  2:

Definition 2.7 (regularly varying functions): Let f be a positive measureable function. If for some  2 R

implying for the tail integral of S

with all mass concentrated 1 ðx1 Þ ¼    ¼ d ðxd Þg.

¼ 1,

A 2 ½0, 1Þd ,

ð2Þ

with A1 ¼ {x1 2 [0, 1) : (x1, 0, . . . , 0) 2 A}, . . . , Ad ¼ {xd 2 [0, 1) : (0, . . . , xd) 2 A}. Obviously, the support of  is the coordinate axes. Equation (2) implies for the tail integral of S ðx1 , . . . , xd Þ ¼ 1 ðx1 Þ1x2 ¼¼xd ¼0 þ    þ d ðxd Þ1x1 ¼¼xd1 ¼0 :

It follows that the independence copula for spectrally positive Le´vy processes is given by b? ðxÞ ¼ x1 1x ¼¼x ¼1 þ    þ xd 1x ¼¼x ¼1 : C 2 1 d d1 2.2. Subexponentiality and regular variation As Bo¨cker and Klu¨ppelberg (2005), we work within the class of subexponential distributions to model high severity losses. For more details on subexponential distributions and related classes, see Embrechts et al. (1997, appendix A3). Definition 2.6 (subexponential distributions): Let (Xk)k2N be i.i.d. random variables with distribution

t 4 0,

ð3Þ

then f is called regularly varying with index . Here we consider loss variables X whose distribution tails are regularly varying. Definition 2.8 (regularly varying distribution tails): Let X be a positive random variable with distribution tail FðxÞ :¼ 1  FðxÞ ¼ PðX 4 xÞ for x40. If for F relation (3) holds for some   0, then X is called regularly varying with index  and denoted by F 2 R . The quantity  is also called the tail index of F. Finally, we define R :¼ [0R. Remark 1 (a) As already mentioned, R  S. (b) Regularly varying distribution functions have representation FðxÞ ¼ x LðxÞ for x  0, where L is a slowly varying function (L 2 R0) satisfying limx!1 L(xt)/L(x) ¼ 1 for all t40. Typical examples are functions that converge to a positive constant or are logarithmic such as, for example, L() ¼ ln(). (c) The classes S and R,   0, are closed with respect to tail-equivalence, which for two distribution functions (or also tail integrals) is defined as limx!1 FðxÞ=GðxÞ ¼ c for c 2 (0, 1). (d) We introduce the notation FðxÞ GðxÞ as x ! 1, meaning that the quotient of the right-hand and left-hand sides tends to 1, i.e. limx!1 GðxÞ=FðxÞ ¼ 1. (e) In definition 2.8 we have used a functional approach to regular variation. Alternatively, regular variation can be reformulated in terms of

858

K. Bo¨cker and C. Klu¨ppelberg vague convergence of the underlying probability measures, and this turns out to be very useful when we consider in section 3.5 below multivariate regular variation (see, e.g., Resnick 2006, chapter 3.6). This measure theoretical approach will be used in section 3.4 to define multivariate regularly varying Le´vy measures.

Distributions in S but not in R include the heavy-tailed Weibull distribution and the lognormal distribution. Their tail decreases faster than tails in R, but less fast than an exponential tail. The following definition will be useful. Definition 2.9 (rapidly varying distribution tails): Let X be a positive random variable with distribution tail FðxÞ :¼ 1  FðxÞ ¼ PðX 4 xÞ for x40. If ( 0, if t 4 1, FðxtÞ ¼ lim x!1 FðxÞ 1, if 0 5 t 5 1, then F is called rapidly varying, denoted by F 2 R1 .

2.3. Recalling the single cell model Now we are in the position to introduce an LDA model based on subexponential severities. We begin with the univariate case. Later, when we consider multivariate models, each of its d operational risk processes will follow the univariate model defined below. Definition 2.10 (subexponential compound Poisson (SCP) model): (1) The severity process. The severities (Xk)k2N are positive i.i.d. random variables with distribution function F 2 S describing the magnitude of each loss event. (2) The frequency process. The number N(t) of loss events in the time interval [0, t] for t  0 is random, where (N(t))t0 is a homogenous Poisson process with intensity 40. In particular, PðNðtÞ ¼ nÞ ¼ pt ðnÞ ¼ et

ðtÞn , n!

n 2 N0 :

(3) The severity process and the frequency process are assumed to be independent. (4) The aggregate loss process. The aggregate loss S(t) in [0, t] constitutes a process SðtÞ ¼

NðtÞ X k¼1

Xk ,

t  0:

Of main importance in the context of operational risk is the aggregate loss distribution function, given by Gt ðxÞ ¼ PðSðtÞ  xÞ ¼

1 X n¼0

pt ðnÞF n ðxÞ,

x  0, t  0, ð4Þ

with pt ðnÞ ¼ PðNt ¼ nÞ ¼ et

ðtÞn , n!

n 2 N0 ,

and F() ¼ P(X P k  ) is the distribution function of Xk, and F n ðÞ ¼ Pð nk¼1 Xk  Þ is the n-fold convolution of F with F1 ¼ F and F 0 ¼ I[0,1). Now, OpVaR is just a quantile of Gt. The following defines the OpVaR of a single cell process, the so-called stand alone VaR. Definition 2.11 (operational VaR (OpVaR)): Suppose Gt is a loss distribution function according to equation (4). Then, operational VaR up to time t at confidence level , VaRt(), is defined as its -quantile VaRt ðÞ ¼ Gt ðÞ,

 2 ð0, 1Þ,

where Gt ðÞ ¼ inffx 2 R : Gt ðxÞ  g, 0551, is the (left continuous) generalized inverse of Gt. If Gt is strictly increasing and continuous, we may write VaRt ðÞ ¼ G1 t ðÞ. In general, Gt()—and thus also OpVaR—cannot be analytically calculated so that one depends on techniques like Panjer recursion, Monte Carlo simulation, and fast Fourier transform (FFT) (see, e.g. Klugman et al. 2004). Recently, based on the asymptotic identity Gt ðxÞ tFðxÞ as x ! 1 for subexponential distributions, Bo¨cker and Klu¨ppelberg (2005) have shown that for a wide class of LDA models closed-form approximations for OpVaR at high confidence levels are available. For a more natural definition in the context of high quantiles we express VaRt() in terms of the tail FðÞ instead of F(). This can easily be achieved by noting that 1=F is increasing, hence     1 1 F ðÞ ¼ inffx 2 R : FðxÞ  g ¼: , 0 5  5 1: 1 F ð5Þ Bo¨cker and Klu¨ppelberg (2005) have shown that   1 ð1 þ oð1ÞÞ ,  " 1, Gt ðÞ ¼ F 1  t or, equivalently, using (5),         1 1 1 t ð1 þ oð1ÞÞ , ¼ 1 1 F Gt

ð6Þ

 " 1:

In the present paper we shall restrict ourselves to situations where the right-hand side of (6) is asymptotically equivalent to F {1  (1  )/t}) as  " 1. That this is not always the case for F 2 S is shown in the following example. Example 2.12: Consider ð1=F Þ ð yÞ ¼ expð y þ y1" Þ for some 05"51 with y ¼ 1/(1  ), i.e.  " 1 equivalent to y ! 1. Then ð1=F Þ ð yÞ ¼ expð yð1 þ oð1ÞÞÞ, but ð1=F Þ ð yÞ=ey ¼ expð y1" Þ ! 1 as y ! 1. This situation typically occurs when F 2 R0 , i.e. for extremely heavytailed models. The reason is given by the following equivalences, which we will often use throughout this paper. We present

Multivariate models for operational risk a short proof that can be ignored by those readers interested mainly in the OpVar application. Proposition 2.13 (1) (regular variation) Let 40. Then (i) F 2 R , ð1=F Þ 2 R1= , (ii) FðxÞ ¼ x LðxÞ for x  0 , ð1=F Þ ðzÞ ¼ e for z  0, where L and L e are slowly z1= LðzÞ varying functions, (iii) FðxÞ GðxÞ as x ! 1 , ð1=F Þ ðzÞ ð1=GÞ ðzÞ as z ! 1. (2) (rapid variation) If F, G 2 R1 such that FðxÞ GðxÞ as x ! 1, then ð1=F Þ ðzÞ ð1=GÞ ðzÞ as z ! 1. Proof (1) Proposition 1.5.15 of Bingham et al. (1987) ensures that regular variation of 1=F is equivalent to regular variation of its (generalized) inverse and provides the representation. Proposition 0.8(vi) of Resnick (1987) gives the asymptotic equivalence. (2) Theorem 2.4.7(ii) of Bingham et al. (1987) applied to 1=F ensures that ð1=F Þ 2 R0 . Furthermore, tail equivalence of F and G implies that ð1=F Þ ðzÞ ¼ ð1=GÞ ðzð1 þ oð1ÞÞÞ ¼ ð1=GÞ ðzÞ ð1 þ oð1ÞÞ as z ! 1, where we have used that the convergence in definition 2.8 is local uniformly. œ Theorem 2.14 (analytical OpVaR model): Consider the SCP model.

for

the

(ii) The severity distribution tail belongs to R for 40, i.e. FðxÞ ¼ x LðxÞ for x  0 and some slowly varying function L if and only if      t 1= e 1 VaRt ðÞ L ,  " 1, ð8Þ 1 1 e where Lð1=ð1  ÞÞ 2 R0 .

(i) This is a consequence of Bo¨cker and Klu¨ppelberg (2005) in combination with proposition 2.13. (ii) By definition 2.11, VaRt() ¼ G (). In our SCP model we have Gt ðxÞ  tFðxÞ as x ! 1. From proposition 2.13 it follows that         1 1 1 t 1 1 F Gt  1=   t e t ,  " 1, L ¼ 1 1 and the result follows.

We refrain from giving more information on the e (which can be found in relationship between L and L Bingham et al. (1987) as it is rather involved and plays no role in our paper. When such a model is fitted statistically, e are usually replaced by constants then L and L (Embrechts et al. 1997, chapter 6). In that case, L   e   as in the following example. To indicate results in L that the equivalence of theorem 2.14(ii) does not extend to subexponential distribution tails in R1 we refer to example 3.11. We can now formulate the analytical VaR theorem for subexponential severity tails. A precise result can be obtained for Pareto-distributed severities. Pareto’s law is the prototypical parametric example for a heavy tailed distribution and suitable for operational risk modelling (see, e.g., Moscadelli 2004). Example 2.15 (Poisson–Pareto LDA): The Poisson– Pareto LDA is an SCP model, where the severities are Pareto distributed with  x , x 4 0, FðxÞ ¼ 1 þ 

with parameters , 40. Here, OpVaR can be calculated explicitly and satisfies " #     t 1=  t 1= VaRt ðÞ  1  ,  " 1: 1 1

ð9Þ

SCP

(i) If F 2 S \ ðR [ R1 Þ, then VaRt() is asymptotically given by       1 1 1 VaRt ðÞ ¼ F 1 ,  " 1: 1 t Gt ð7Þ

Proof

859

œ

3. Multivariate loss distribution models 3.1. The Le´vy copula model The SCP model of the previous section can be used for estimating OpVaR of a single cell, sometimes referred to as the cell’s stand alone OpVaR. Then, a first approximation to the bank’s total OpVaR is obtained by summing up all different stand-alone VaR numbers. Indeed, the Basel Committee requires banks to sum up all their different operational risk estimates unless sound and robust correlation estimates are available (Basel Committee on Banking Supervision 2004, paragraph 669(d)). Moreover, this ‘simple-sum VaR’ is often interpreted as an upper bound for total OpVaR, with the implicit understanding that every other (realistic) cell dependence model necessarily reduces overall operational risk. However, it is well recognized (Bo¨cker and Klu¨ppelberg 2008, table 5.3) that simple-sum VaR may even underestimate total OpVaR when severity data is heavy-tailed, which, in practice, it is (see, e.g., Moscadelli 2004). Therefore, to obtain a more accurate and reliable result, one needs more general models for multivariate operational risk. Various models have been suggested. Most of them are variations of the following scheme. Fix a time horizon t40, and model the accumulated losses of each

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K. Bo¨cker and C. Klu¨ppelberg

operational risk cell i ¼ 1, . . . , d by a compound Poisson random variable Si(t). Then, in general, both the dependence of the loss sizes in different cells as well as the dependence between the frequency variables Ni(t) is modelled by appropriate copulas, where for the latter, one has to take the discreteness of these variables into account. Considering this model as a dynamic model in time, it does not constitute a multivariate compound Poisson model but leads outside the well-studied class of Le´vy processes. This can easily be seen as follows: since a Poisson process jumps with probability 0 at any fixed time s40, we have for any jump time s of Nj() that P(DNi(s) ¼ 1) ¼ 0 for i 6¼ j, hence any two such processes almost surely never jump at the same time. However, as described in section 2.1, dependence in multivariate compound Poisson processes—as in every multivariate Le´vy process—means dependence in the jump measure, i.e. the possibility of joint jumps. Finally, from a statistical point of view such a model requires a large number of parameters, which, given the sparsity of data in combination with the task of estimating high quantiles, will be almost impossible to fit. We formulate a multivariate compound Poisson model and apply Sklar’s theorem for Le´vy copulas. Invoking a Le´vy copula allows for a small number of parameters and introduces a transparent dependence structure in the model; we present a detailed example in section 3 of Bo¨cker and Klu¨ppelberg (2008). Definition 3.1 (multivariate SCP model): The multivariate SCP model consists of the following. (1) Cell processes. All operational risk cells, indexed by i ¼ 1, . . . , d, are described by an SCP model with aggregate loss process Si, subexponential severity distribution function Fi and Poisson intensity i40. (2) Dependence structure. The dependence between different cells is modelled by a Le´vy copula. More precisely, let i : ½0, 1Þ ! ½0, 1Þ be the tail integral associated with Si, i.e. i ðÞ ¼ i Fi ðÞ, for b : ½0, 1Þd ! ½0, 1Þ be a Le´vy i ¼ 1, . . . , d and let C copula. Then b 1 ðx1 Þ, . . . , d ðxd ÞÞ ðx1 , . . . , xd Þ ¼ Cð

defines the tail integral of the d-dimensional compound Poisson process S ¼ (S1, . . . , Sd). (3) Total aggregate loss process. The bank’s total aggregate loss process is defined as S þ ðtÞ ¼ S1 ðtÞ þ S2 ðtÞ þ    þ Sd ðtÞ,

þ

with tail integral (

 ðzÞ ¼ 

d

ðx1 , . . . , xd Þ 2 ½0, 1Þ :

d X i¼1

t  0,

xi  z

)!

, z  0: ð10Þ

The following result states an important property of the multivariate SCP model.

Proposition 3.2: Consider the multivariate SCP model of definition 3.1. Its total aggregate loss process Sþ is compound Poisson with frequency parameter þ

þ ¼ lim  ðzÞ z#0

and severity distribution þ

Fþ ðzÞ ¼ 1  F ðzÞ ¼ 1 

þ

 ðzÞ , þ

z  0:

Proof: Projections of Le´vy processes are Le´vy processes. For every compound Poisson process with intensity 40 and only positive jumps with distribution function F, the tail integral of the Le´vy measure is given by ðxÞ ¼ FðxÞ, x40. Consequently,  ¼ ð0Þ and FðxÞ ¼ ðxÞ=. We apply this relation to the Le´vy process S þ and obtain the total mass þ of S þ, which ensures that S þ is compound Poisson with the parameters as stated. œ Note that S þ does not necessarily define a onedimensional SCP model because F þ need not be subexponential, even if all components are. This has been investigated for sums of independent random variables in great detail (see, e.g., the review paper of Goldie and Klu¨ppelberg (1998, section 5)). For dependent random variables we present in examples 3.10 and 3.11 two situations where F þ 2 S \ (R [ R1). In that case we can apply (7) to estimate total OpVaR, which shall now be defined precisely. Definition 3.3 (total OpVaR): Consider the multivariate SCP model of definition 3.1. Then, total OpVaR up to time t at confidence level  is the -quantile of the total þ aggregate loss distribution Gþ t ðÞ ¼ PðS ðtÞ  Þ: þ VaRþ ðÞ, t ðÞ ¼ Gt

 2 ð0, 1Þ,

ðÞ ¼ inffz 2 R : Gþ with Gþ t t ðzÞ  g for 0551. Our goal in this paper is to investigate multivariate SCP models and find useful approximations in a variety of dependence structures. 3.2. Losses dominant in one cell Before we discuss Le´vy copula dependence structures we formulate a very general result for the situation, where the losses in one cell are regularly varying and dominate all others. Indeed, the situation of the model is such that it covers arbitrary dependence structures, including also the practitioner’s models described above. Assume for fixed t40 for each cell model a compound Poisson random variable. Dependence is introduced by an arbitrary correlation or copula for (N1(), . . . , Nd () and an arbitrary copula between the severity distributions F1() ¼ P(X1  ), . . . , Fd () ¼ P(Xd  ). Recall that the resulting model (S1(t), . . . , Sd (t))t0 does NOT constitute a multivariate compound Poisson process and so is not captured by the multivariate SCP model of definition 3.1. We want to calculate an approximation for the tail P(S1(t) þ S2(t)4x) for large x and total OpVaR for high levels . We formulate the result in arbitrary dimension.

861

Multivariate models for operational risk Theorem 3.4: For fixed t40 let Si(t) for i ¼ 1, . . . , d have compound Poisson distributions. Assume that F1 2 R for 40. Let 4 and suppose that E [(Xi)]51 for i ¼ 2, . . . , d. Then regardless of the dependence structure between (S1(t), . . . , Sd (t)), 1

PðS1 ðtÞ þ    þ Sd ðtÞ 4 xÞ EN1 ðtÞPðX 4 xÞ, x ! 1,   1 þ ¼ VaR1t ðÞ,  " 1: VaRt ðÞ F1 1  EN1 ðtÞ ð11Þ Proof:

Consider d ¼ 2. Note first that

We have to find conditions such that we can interchange the limit for x ! 1 and the infinite sum. This means that we need uniform estimates for the two ratios on the righthand side for x ! 1. We start with an estimate for the second ratio: lemma 1.3.5 of Embrechts et al. (1997) applies, giving for arbitrary "40 and all x40 a finite positive constant K(") so that 1 i¼1 Xi 4 xÞ 1 PðX 4 xÞ

k,m¼1

0

 Að1  ÞðþaÞ þ xþ ðxÞ m c  E ½ðX2j Þ ÞKð"Þð1 þ "Þk :

P 2 X1i þ m j¼1 Xj 4 xÞ Pk Pð i¼1 X1i 4 xÞ Pm P Pð ki¼1 X1i 4 xð1  ÞÞ Pð j¼1 X2j 4 xÞ  þ : P P Pð ki¼1 X1i 4 xÞ Pð ki¼1 X1i 4 xÞ i¼1

1 1 i¼1 Xi 4 xð1  ÞÞ PðX 4 xð1  PðX1 4 xð1  ÞÞ PðX1 4 xÞ

ÞÞ

ð14Þ

0

Now note that xþ  tends to 0 as x ! 1. Furthermore, we have 1 X

k,m¼0 1 X

¼

Pk

Pk

PðS1 ðtÞ þ S2 ðtÞ 4 xÞ PðX1 4 xÞ 1 X PðN1 ðtÞ ¼ k, N2 ðtÞ ¼ mÞðK0 ð"Þð1 þ "Þk 

 Kð"Þð1 þ "Þk :

PðN1 ðtÞ ¼ k, N2 ðtÞ ¼ mÞ ¼ 1, PðN1 ðtÞ ¼ k, N2 ðtÞ ¼ mÞk 1 X k¼1

PðN1 ðtÞ ¼ kÞk ¼ ENk ðtÞ 5 1:

Consequently, the rhs of (14) converges. By Pratt’s Lemma (see, e.g., Resnick (1987, Ex. 5.4.2.4)), we can interchange the limit and infinite sum on the rhs of (12) and obtain 1 PðS1 ðtÞ þ S2 ðtÞ 4 xÞ X ¼ PðN1 ðtÞ ¼ kÞk ¼ EN1 ðtÞ: x!1 PðX1 4 xÞ k¼1

lim

ð13Þ

1 Regular variation of the distribution Pkof X1 implies regular variation of the distribution of i¼1 Xi with the same index . We write for the first term

Pð

j¼1

k,m¼1

For the first ratio we proceed as in the proof of lemma 2 of Klu¨ppelberg et al. (2005). For arbitrary 0551 we have Pð

The so-called c inequality (see, e.g., Loe´ve (1978, p. 157)) applies, giving " ! # m X 2 E  mc EðX2j Þ , Xj for c ¼ 1 or c ¼ 2 , as   1 or 41. We combine these estimates and obtain in (12) for x  x040,

P P 2 Pð ki¼1 X1i þ m j¼1 Xj 4 xÞ  Pk Pð i¼1 X1i 4 xÞ P Pð ki¼1 X1i 4 xÞ : ð12Þ  PðX1 4 xÞ

Pk

j¼1

j¼1

1

1 X PðS1 ðtÞ þ S2 ðtÞ 4 xÞ ¼ PðN1 ðtÞ ¼ k, N2 ðtÞ ¼ mÞ PðX1 4 xÞ k,m¼1

Pð

for some {050 5  }. By Markov’s inequality, we obtain for the numerator " ! # ! m m X X  2 2 P : Xj Xj 4 x  ðxÞ E

PðX1 4 xÞ : Pk Pð i¼1 X1i 4 xÞ

For the first ratio we use the same estimate as above and obtain for all x40 the upper bound K0 (")(1 þ ")k. For the second ratio, using the so-called Potter bounds (Bingham et al. 1987, theorem 1.5.6(iii)) for every chosen constants a40, A41, we obtain an upper bound A(1  )(þa) uniformly for all x  x0  0. The third ratio is less than or equal to 1 for all k and x. As the denominator of the second term of the rhs of (13) is 0 regularly varying, it can be bounded below by x(þ )

The result for d42 follows by induction. Approximation (11) holds by theorem 2.14 (1). œ Within the context of multivariate compound Poisson models, the proof of this result simplifies. Moreover, since a possible singularity of the tail integral in 0 is of no consequence, it even holds for all spectrally positive Le´vy processes. We formulate this as follows. Proposition 3.5: Consider a multivariate spectrally positive Le´vy process and suppose that 1 2 R . Furthermore, assume that, for all i ¼ 2, . . . , d, the integrability condition Z x i ðdxÞ 5 1 ð15Þ x1

for some 4 is satisfied. Then þ

 ðzÞ ¼ 1: z!1 1 ðzÞ lim

ð16Þ

862

K. Bo¨cker and C. Klu¨ppelberg

Moreover, 1 VaRþ t ðÞ VaRt ðÞ,

 " 1,

ð17Þ

i.e. total OpVaR is asymptotically dominated by the standalone OpVaR of the first cell. Proof: We first show that (16) holds. From equation (15) it follows that, for i ¼ 2, . . . , d, lim z i ðzÞ ¼ 0:

ð18Þ

z!1

Since 5, we obtain from regular variation for some slowly varying function L, invoking (18), 

i ðzÞ z i ðzÞ ¼ 0, ¼ lim  z!1 1 ðzÞ z!1 z LðzÞ lim

i ¼ 2, . . . , d,

because the numerator tends to 0 and the denominator to 1. (Recall that z"L(z) ! 1 as z ! 1 for all "40 and L 2 R0.) We proceed by induction. For d ¼ 2 we have by the decomposition as in (13) þ

example 3.11 below that the following does NOT hold in general for x ! 1 (equivalently, "1): Fi ðxÞ ¼ oðF1 ðxÞÞ¼)VaRit ðÞ ¼ oðVaR1t ðÞÞ,

We now study two very basic multivariate SCP models in more detail, namely the completely dependent and the independent one. Despite their extreme dependence structure, both models provide interesting and valuable insight into multivariate operational risk.

3.3. Multivariate SCP model with completely dependent cells Consider a multivariate SCP model and assume that its cell processes Si, i ¼ 1, . . . , d, are completely positively dependent. In the context of Le´vy processes this means that they always jump together, implying also that the expected number of jumps per unit time of all cells, i.e. the intensities i, must be equal,

þ

2 ðzÞ :¼  ðzÞ  1 ðzð1  "ÞÞ þ 2 ðz "Þ, z 4 0, 0 5 " 5 1: It then follows that lim sup z!1

þ 2 ðzÞ

1 ðzÞ

þ

1 ðzð1  "ÞÞ 2 ðz "Þ 1 ðz "Þ þ lim z!1 1 ðz "Þ 1 ðzÞ 1 ðzÞ ð19Þ ¼ ð1  "Þ :  lim

z!1

Similarly, 2 ðzÞ  1 ðð1 þ "ÞzÞ for every "40. Therefore, þ

 ðzÞ 1 ðð1 þ "ÞzÞ lim inf 2  lim ¼ ð1 þ "Þ : z!1 1 ðzÞ z!1 1 ðzÞ

ð20Þ

þ

Assertion (16) follows for 2 from (19) and (20). þ þ This implies that 2 2 R . Now replace 1 by 2 and þ þ 2 by 3 and proceed as above to obtain (16) for general dimension d. Finally, theorem 2.14(i) applies, giving (17). œ This result is mostly applied in terms of the following corollary, which formulates a direct condition for (19) and (20) to hold. œ Corollary 3.6: Consider a multivariate spectrally positive Le´vy process and suppose that 1 2 R . Furthermore, assume that, for all i ¼ 2, . . . , d, lim

i ðzÞ

z!1 1 ðzÞ

¼ 0:

Then (16) and (17) hold. Hence, for arbitrary dependence structures, when the severity of one cell has a regularly varying tail dominating those of all other cells, total OpVaR is tail-equivalent to the OpVaR of the dominating cell. This implies that the bank’s total loss at high confidence levels is likely to be due to one big loss occurring in one cell rather than an accumulation of losses of different cells regardless of the dependence structure. From our equivalence results of proposition 2.13 and theorem 2.14 this is not a general property of the completely dependent SCP model. We shall see in

i ¼ 2, . . . , d:

 :¼ 1 ¼    ¼ d :

ð21Þ

The severity distributions Fi, however, can be different. Indeed, from example 2.4 we infer that, in the case of complete dependence, all Le´vy mass is concentrated on fðx1 , . . . , xd Þ 2 ½0, 1Þd : 1 ðx1 Þ ¼    ¼ d ðxd Þg, or, equivalently, fðx1 , . . . , xd Þ 2 ½0, 1Þd : F1 ðx1 Þ ¼    ¼ Fd ðxd Þg:

ð22Þ

Until further notice, we assume for simplicity that all severity distributions Fi are strictly increasing and continuous so that Fi1 ðqÞ exists for all q 2 [0, 1). Together with (22), we can express the tail integral of S þ in terms of the marginal 1 , ( )! d X þ xi  z  ðzÞ ¼  ðx1 , . . . , xd Þ 2 ½0, 1Þd : ¼ 1

i¼1

(

x1 2 ½0, 1Þ : x1 þ

z  0:

d X i¼2

Fi1 ðF1 ðx1 ÞÞ

z

)!

,

P Set Hðx1 Þ :¼ x1 þ di¼2 Fi1 ðF1 ðx1 ÞÞ for x1 2 [0, 1) and note that it is strictly increasing and therefore invertible. Hence, þ

 ðzÞ ¼ 1 ðfx1 2 ½0, 1Þ : x1  H1 ðzÞgÞ ¼ 1 ðH1 ðzÞÞ, z  0:

ð23Þ

Now we can derive an asymptotic expression for total OpVaR. Theorem 3.7 (OpVaR for the completely dependent SCP model): Consider a multivariate SCP model with completely dependent cell processes S1, . . . , Sd and strictly increasing and continuous severity distributions Fi. Then Sþ is compound Poisson with parameters þ ¼ 

þ

and F ðÞ ¼ F1 ðH1 ðÞÞ:

ð24Þ

863

Multivariate models for operational risk þ

If furthermore F 2 S \ ðR [ R1 Þ, total OpVaR is asymptotically given by VaRþ t ðÞ

d X i¼1

VaRit ðÞ,

 " 1,

ð25Þ

Assume that ci 6¼ 0 for 2  i  b  d and ci ¼ 0 for e be the i  b þ 1  d. For F1 ðxÞ ¼ x LðxÞ, x  0, let L function as in theorem 2.14(ii). Then VaRþ t ðÞ

where VaRit ðÞ denotes the stand-alone OpVaR of cell i. Proof: Expression (24) immediately follows from (21) and (23),   Proof: þ 1 1 þ  ¼ lim  ðzÞ ¼ lim  F1 ðH ðzÞÞ ¼  F1 lim H ðzÞ ¼ : z!0

z!0

z!0

þ

If F 2 S \ ðR [ R1 Þ, we may use (7) and the definition of H to obtain    1 1 VaRþ ðÞ H F 1  t 1 t     1 1 ¼ F11 1  þ    þ Fd1 1  t t VaR1t ðÞ þ    þ VaRdt ðÞ,

 " 1:

Example 3.8 (identical severity distributions): Assume that all cells have identical severity distributions, i.e. F :¼ F1 ¼  ¼ Fd. In this case we have H(x1) ¼ dx1, for x1  0 and, therefore, z þ  ðzÞ ¼ F , z  0: d þ

If furthermore F 2 S \ ðR [ R1 Þ, it follows that F ðÞ ¼ Fð =d Þ, and we obtain   1 þ VaRt ðÞ dF 1  ,  " 1: t We can derive very precise asymptotics in the case of dominating regularly varying severities. Proposition 3.9: Assume that the conditions of theorem 3.7 hold. Assume further that F1 2 R with 40 and that, for all i ¼ 2, . . . , d, there exist ci 2 [0, 1) such that lim

Fi ðxÞ

x!1 F1 ðxÞ

¼ ci :

ð26Þ

c1= VaR1t ðÞ i

i¼1

b X

c1= i

i¼1



t 1

1=   e 1 L , 1

 " 1:

From theorem 2.14(ii) we know that      t 1= e 1 1 L ,  " 1, VaRt ðÞ 1 1

e where L½1=ð1  Þ 2 R0 . Note: If all ci ¼ 0 hold for i ¼ 2, . . . , d, then corollary 3.6 applies. So assume that ci 6¼ 0 for 2  i  b. From (26) and Resnick (1987, proposition 0.8(vi)), we obtain Fi ð1  ð1=zÞÞ c1= i F1 ð1  ð1=zÞÞ as z ! 1 for i ¼ 1, . . . , d. This yields for x1 ! 1, Hðx1 Þ ¼ x1 þ

œ Theorem 3.7 states that for the completely dependent SCP model, total asymptotic OpVaR is simply the sum of the asymptotic stand-alone cell OpVaRs. Recall that this is similar to the new proposals of Basel II, where the standard procedure for calculating capital charges for operational risk is just the simple-sum VaR. Or stated another way, regulators implicitly assume complete dependence between different cells, meaning that losses within different business lines or risk categories always happen at the same instants of time. This is often considered as the worst-case scenario, which, however, in the heavy-tailed case can be grossly misleading. The following example describes another regime for completely dependent cells.



b X

¼ x1 þ ¼ x1

d X i¼2

d X i¼2

b X i¼1

Fi1 ð1  F1 ðx1 ÞÞ 1 c1= i F1 ð1  F1 ðx1 ÞÞð1 þ oi ð1ÞÞ

c1= i ð1 þ oð1ÞÞ,

P where we have c1 ¼ 1. Defining C :¼ bi¼1 c1= i , then H(x1) Cx1 as x1 ! 1, and hence H1(z) z/C as z ! 1, which implies by (23) and regular variation of F1 þ

 ðzÞ ¼ 1 ðH1 ðzÞÞ  F1 ðz=CÞ  C F1 ðzÞ, þ

z ! 1:

Obviously, F ðzÞ ¼ C F1 ðzÞ 2 R and theorem 3.7 applies. By (8) together with the fact that all summands from index b þ 1 on are of lower order, (25) reduces to     1 1 þ VaRt ðÞ F1 1  þ    þ Fb 1  t t   1 F1 1   t C     b X  t 1= e 1 L c1= ,  " 1: i 1 1 i¼1 œ

An important example of proposition 3.9 is the Pareto case. Example 3.10 (Pareto-distributed severities): Consider a multivariate SCP model with completely dependent cells and Pareto-distributed severities as in example 2.15. Then we obtain for the ci   Fi ðxÞ i ¼ , i ¼ 1, . . . , b, lim x!1 F1 ðxÞ 1 lim

Fi ðxÞ

x!1 F1 ðxÞ

¼ 0,

i ¼ b þ 1, . . . , d,

864

K. Bo¨cker and C. Klu¨ppelberg

for some 1  b  d. This, together with proposition 3.9, leads to !  !  b b X X i z  þ 1þ i z , z ! 1: F ðzÞ   1 i¼1 i¼1 1

Recall from example 2.5 that, in the case of independence, all mass of the Le´vy measure  is concentrated on the axes. Hence,

Finally, from (9) and (25) we obtain the total OpVaR as

ðf0g  ½z, 1Þ      f0gÞ ¼ ð½0, 1Þ  ½z, 1Þ      ½0, 1ÞÞ, .. .. . . ðf0g  f0g      ½z, 1ÞÞ ¼ ð½0, 1Þ  ½0, 1Þ    

VaRþ t ðÞ

b X i¼1

VaRit ðÞ

  b X  t 1= i , 1 i¼1

 " 1:

We conclude this session with an example showing that corollary 3.6 does not hold for every general dominating tail.

ð½z, 1Þ  f0g      f0gÞ ¼ ð½z, 1Þ  ½0, 1Þ      ½0, 1ÞÞ,

 ½z, 1ÞÞ,

and we obtain þ

Example 3.11 (Weibull severities): Consider a bivariate SCP model with completely dependent cells and assume that the cells0 severities are Weibull distributed according to pffiffiffiffiffiffiffiffi pffiffiffi F1 ðxÞ ¼ expð x=2Þ and F2 ðxÞ ¼ expð xÞ, x 4 0:

 ðzÞ ¼ ð½z, 1Þ  ½0, 1Þ      ½0, 1ÞÞ þ    þ ð½0, 1Þ      ½0, 1Þ  ½z, 1ÞÞ

Note that F1,2 2 S \ R1 . Equation (27) immediately implies that F2 ðxÞ ¼ oðF1 ðxÞÞ. We find that Hðx1 Þ ¼ 32 x1 , þ implying that F 2 S \ R1 , since pffiffiffiffiffiffiffi þ ð28Þ F ðzÞ ¼ expð z=3Þ, z 4 0:

Theorem 3.12 (OpVaR for the independent SCP model): Consider a multivariate SCP model with independent cell processes S1, . . . , Sd. Then Sþ defines a onedimensional SCP model with parameters

ð27Þ

It is remarkable that, in this example, the total severity (28) is heavier tailed than the stand-alone severities (27), i.e. F1,2(x) ¼ o(F þ(x)) as x ! 1. However, from    1 2 and VaR1t ðÞ 2 ln t    1 2 VaR2t ðÞ ln ,  " 1, t

VaR2t ðÞ 1 ¼ : "1 VaR1t ðÞ 2

Nevertheless, equation (25) of theorem 3.7 still holds,    1 2 3 ln ¼ VaR1t ðÞ þ VaR2t ðÞ, t

þ ¼ 1 þ    þ d and 1 þ F ðzÞ ¼ þ ½1 F1 ðzÞ þ    þ d Fd ðzÞ, 

 " 1:

z  0:

ð30Þ

If F1 2 S \ ðR [ R1 Þ and for all i ¼ 2, . . . , d there exist ci 2 [0, 1) such that lim

lim

ð29Þ

Now we are in the position to derive an asymptotic expression for total OpVaR in the case of independent cells.

Fi ðxÞ

x!1 F1 ðxÞ

we find that the stand-alone VaRs are of the same order of magnitude:

VaRþ t ðÞ

¼ 1 ðzÞ þ    þ d ðzÞ:

¼ ci ,

ð31Þ

then, setting C ¼ 1 þ c22 þ    þ cdd, total OpVaR can be approximated by   1 VaRþ ðÞ F 1  ,  " 1: ð32Þ t 1 C t Proof: From proposition 3.2 we know that S þ is a compound Poisson process with parameters þ (here following from (29)) and F þ as in (30), from which we conclude þ

F ðzÞ 1 C ¼ þ ½1 þ c2 2 þ    þ cd d  ¼ þ 2 ð0, 1Þ, z!1 F1 ðzÞ   lim

i.e. 3.4. Multivariate SCP model with independent cells Let us now turn to a multivariate SCP model where the cell processes Si, i ¼ 1, . . . , d, are independent and so any two of the component sample paths almost surely never jump together. Therefore, we may write the tail integral of S þ as þ

 ðzÞ ¼ ð½z, 1Þ  f0g      f0gÞ þ    þ ðf0g      f0g  ½z, 1ÞÞ, z  0:

þ

F ðzÞ

C F1 ðzÞ, þ

z ! 1:

ð33Þ

þ

In particular, F 2 S \ ðR [ R1 Þ and S þ defines a onedimensional SCP model. From (7) and (33), total OpVaR follows as     1 1 þ 1  ðÞ F 1  VaRþ ,  " 1: F 1 t þ t C t œ

865

Multivariate models for operational risk Example 3.13 (multivariate SCP model with independent cells): (1) Assume that ci ¼ 0 for all i  2, i.e. Fi ðxÞ ¼ oðF1 ðxÞÞ, i ¼ 2, . . . , d. We then have C ¼ 1 and it follows from (32) that independent total OpVaR asymptotically equals the stand-alone OpVaR of the first cell. In contrast to the completely dependent case (see proposition 3.9 and example 3.11), this holds for the class S \ (R [ R1) and not only for F1 2 R. (2) Consider a multivariate SCP model with independent cells and Pareto-distributed severities so that the constants ci of theorem 3.12 are given by   Fi ðxÞ i ¼ , i ¼ 1, . . . , b lim x!1 F1 ðxÞ 1 and lim

Fi ðxÞ

x!1 F1 ðxÞ

¼ 0,

i ¼ b þ 1, . . . , d,

for some b  1. Then C ¼

b   X i i¼1

1

i ,

þ

z ! 1:

VaRþ t ðÞ

 i¼1 i i 1

!1=

b X ðVaRit ðÞÞ ¼ i¼1

!1=

(a) Let  be a Le´vy measure of a spectrally positive Le´vy process in Rdþ . Assume that there exists a function b : (0, 1) ! (0, 1) satisfying b(t) ! 1 as t ! 1 and a Radon measure  on E, called the limit measure, such that lim tð½0, bðtÞ xc Þ ¼ ð½0, xc Þ,

t!1

It follows that Pb

We also recall that a Radon measure is a measure that is finite on all compacts. Finally, henceforth all operations and order relations of vectors are taken componentwise. As already mentioned in remark 1(e), multivariate regular variation is best formulated in terms of vague convergence of measures. For spectrally positive Le´vy processes we work in the space of non-negative Radon measures on E. From lemma 6.1 of Resnick (2006, p. 174), however, it suffices to consider regions [0, x]c for x 2 E which determine the convergence, and this is how we formulate our results. Definition 3.15 (multivariate regular variation)

and the distribution tail F satisfies   b b 1 X z  1 X þ þ i 1 þ i i z , F ðzÞ ¼ þ  i¼1 i  i¼1 t

useful sources for insight into multivariate regular variation. To simplify notation we denote E :¼ [0, 1]dn{0}, where 0 is the zero vector in Rd. Then we introduce for x 2 E the complement

yi c ½0, x :¼ E n ½0, x ¼ y 2 E : max 4 1 : 1id xi

,  " 1,

where VaRit ðÞ denotes the stand-alone OpVaR of cell i according to (9). For identical cell frequencies  :¼ 1 ¼  ¼ b this further simplifies to !1=   b  t 1= X þ  ,  " 1: i VaRt ðÞ 1 i¼1 Example 3.14 (continuation of example 3.11): Consider a bivariate SCP model with independent cells and Weibulldistributed severities according to (27). According to theorem 3.12 we have C ¼ 1 and independent total OpVaR is asymptotically given by    1 2 1 ,  " 1: ðÞ VaR ðÞ 2 ln VaRþ t t t

3.5. Multivariate SCP models of regular variation As the notion of regular variation has proved useful for one-dimensional cell severity distributions, it seems natural to exploit the corresponding concept for the multivariate model. The following definition extends regular variation to the Le´vy measure, which is our natural situation; the books of Resnick (1987, 2006) are

x 2 E:

ð34Þ

Then we call  multivariate regularly varying. (b) The measure  has a scaling property: there exists some 40 such that for every s40 ð½0, sxc Þ ¼ s ð½0, xc Þ,

x 2 E,

ð35Þ

c

i.e. ([0, ] ) is homogeneous of order , and  is called multivariate regularly varying with index . Remark 1 (a) In (34), the scaling of all components of the tail integral by the same function b() implies lim

i ðxÞ ¼ cij 2 ½0, 1, ðxÞ

x!1 j

for 1  i, j  d. We now focus on the case where all i are tail-equivalent, i.e. cij40 for some i, j. In particular, we then have marginal regular variation i 2 R with the same tail index, and thus for all i ¼ 1, . . . , d lim ti ðbðtÞ xÞ ¼ ð½0, 1      ðx, 1

t!1

 ½0, 1      ½0, 1Þ

¼ i ðx, 1 ¼ ci x ,

x 4 0,

ð36Þ

for some ci40. (b) Specifically, if 1 is standard regularly varying (i.e. with index  ¼ 1 and slowly varying function L  1), we can take b(t) ¼ t.

866

K. Bo¨cker and C. Klu¨ppelberg

(c) There exists also a broader definition of multivariate regular variation that allows for different i in each marginal (see theorem 6.5 of Resnick 2006, p. 204). However, we have already dealt with the situation of dominant marginals and, hence, the above definition is the relevant one for us. From the point of view of dependence structure modeling, multivariate regular variation is basically a special form of multivariate dependence. Hence, a natural question in this context is how multivariate regular variation is linked to the dependence concept of aLe´vy copula. Theorem 3.16 (Le´vy copulas and multivariate regular variation): Let  be a multivariate tail integral of a spectrally positive Le´vy process in Rdþ . Assume that the marginal tail integrals i are regularly varying with index . Then the following assertions hold.

b is a homogeneous function of (1) If the Le´vy copula C order 1, then  is multivariate regularly varying with index . (2) The tail integral  is multivariate regularly varying b is with index  if and only if the Le´vy copula C regularly varying with index 1; i.e.

b ðx1 , . . . , xd ÞÞ Cðt ¼ gðx1 , . . . , xd Þ, ðx1 , . . . , xd Þ 2 ½0, 1Þd , t!1 Cðt b ð1, . . . , 1ÞÞ lim

and g(sx) ¼ sg(x) for x 2 [0, 1)d.

ð37Þ

Proof b we can write the Le´vy (1) For any Le´vy copula C, c measure ([0, x] ) for x 2 E as

ð½0, xc Þ ¼ fy 2 E : y1 4 x1 or    or yd 4 xd g ¼

d X i¼1

þ

i ðxi Þ 

d X

i1 ,i2 ,i3 ¼1 i1 5i2 5i3

d X

i1 ,i2 ¼1 i1 5i2

b i ðxi Þ, i ðxi ÞÞ Cð 1 1 2 2

b i ðxi Þ, i ðxi Þ, i ðxi ÞÞ þ    Cð 1 1 2 2 3 3

b i ðxi Þ, . . . , i ðxi ÞÞ: þ ð1Þd1 Cð 1 1 d d

The homogeneity allows interchange of the factor t b which, together with marginal regular with C, variation as formulated in (36), yields the limit as in (34): c

lim t ð½0, bðtÞ x Þ

t!1

¼

d X i¼1

i ðxi , 1 

d X

i1 , i2 ¼1 i1 5 i2

b i ðxi , 1, i ðxi , 1Þ þ    Cð 1 1 2 2

b i ðxi , 1, . . . , i ðxi , 1Þ þ ð1Þd1 Cð 1 1 d d

¼ fy 2 E : y1 4 x1 or    or yd 4 xd g ¼ ð½0, xc Þ,

x 2 E:

ð38Þ

(2) This follows from the same calculation as in the proof of (1) by observing that asymptotic interb is possible if and only change of the factor t with C if (37) holds. œ

Remark 2

(a) For definition (37) of the multivariate regular variation of arbitrary functions, we refer to Bingham et al. (1987, appendix 1.4). (b) The general concept of multivariate regular variation of measures with possibly different marginals requires different normalizing functions b1(), . . . , bd () in (34). In that case marginals are usually transformed into standard regular variation with  ¼ 1 and L  1. In this case the scaling property (35) in the limit measure  always scales with  ¼ 1. This is equivalent to all marginal Le´vy processes being one-stable. In this situation the multivariate measure  defines a function (x1, . . . , xd) that models the dependence between the marginal Le´vy measures, and which is termed a Pareto Le´vy copula by Klu¨ppelberg and Resnick (2008) as well as by Bo¨cker and Klu¨ppelberg (2009). Furthermore, according to corollary 3.2 of Klu¨ppelberg and Resnick (2008), the limit measure  is the Le´vy measure of a one-stable Le´vy process in Rdþ if and only if all marginals are one-stable and the Pareto Le´vy copula is homogeneous of order 1 (see theorem 3.16 for the classical Le´vy copula). In the context of multivariate regular variation this approach seems to be more natural than the classical Le´vy copula with Lebesgue marginals. We now want to apply the results above to the problem of calculating total OpVaR. Assume that the tail integral  is multivariate regularly varying according to (34), implying tail equivalence of the marginal severity distributions. We then have the following result. Theorem 3.17 (OpVaR for the SCP model with multivariate regular variation): Consider an SCP model with multivariate regularly varying cell processes S1, . . . , Sd with index  and limit measure  in (34). Assume further that the severity distributions Fi for i ¼ 1, . . . , d are strictly increasing and continuous. Then, S þ is compound Poisson with 1 ð39Þ F1 ðzÞ 2 R , z ! 1, þ P where þ ðz, 1 ¼ fx 2 E : di¼1 xi 4 zg for z40. Furthermore, total OpVaR is asymptotically given by   1 VaRt ðÞ F1 1  ,  " 1: ð40Þ t 1 þ ð1, 1 þ

F ðzÞ þ ð1, 1

Proof: First recall that multivariate regular variation of  implies regular variation of the marginal tail integrals, i.e. i 2 R for all i ¼ 1, . . . , d. In analogy to relation

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Multivariate models for operational risk Compound Poisson with 1/2−stable jumps, Clayton Levy copula (θ =0.3) 12 0.6 ∆X

10

0.2

8 X, Y

0.4

0 6

0

0.1

0.2

0.3

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0.5 t

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1

0

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0.3

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0.5 t

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1

0

0.1

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1

0

0.1

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0.3

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0.6

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1

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

8

4 ∆Y

6 2

4 2

0

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

0 0

1

Compound Poisson with 1/2−stable jumps, Clayton Levy copula (θ =2) 1.5

1

∆X

0.8

1

0.6 0.4 0.2

X, Y

0

0.2 ∆Y

0.5

0.15 0.1 0.05

0

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

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0

1

X, Y

∆X

Compound Poisson with 1/2−stable jumps, Clayton Levy copula (θ =10) 0.5 0.15 0.45 0.1 0.4 0.35

0.05

0.3

0

0.25 0.2 0.15 ∆Y

0.15 0.1

0.05

0.05 0

0.1

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

0

Figure 1. Two-dimensional LDA Clayton-1/2-stable model (the severity distribution belongs to R1/2) for different dependence parameter values. Left column: compound processes. Right column: frequencies and severities. Upper row:  ¼ 0.3 (low dependence). Middle row:  ¼ 2 (medium dependence). Lower row:  ¼ 10 (high dependence).

(3.8) of Klu¨ppelberg and Resnick (2008), we can calculate þ the limit measure þ of the tail integral  by ( ) d X þ þ xi 4 z lim t  ðbðtÞ zÞ ¼  ðz, 1 ¼  x 2 E : t!1

(

i¼1

¼ z  x 2 E : þ

þ

d X i¼1

)

xi 4 1 ,

i.e.  and thus F are regularly varying of index . Now we can choose b(t) so that limt!1 t1 ðbðtÞÞ ¼ 1

and thus þ

þ

 ðzÞ t ðbðtÞÞ lim ¼ lim ¼ þ ð1, 1: t!1 t1 ðbðtÞÞ z!1 1 ðzÞ Relation (39) follows theorem 2.14(i).

immediately,

and

(40)

by œ

There are certain situations where the limit measure þ(1, 1] and therefore also total OpVaR can be explicitly calculated. In the following example we present a result for d ¼ 2.

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K. Bo¨cker and C. Klu¨ppelberg

Example 3.15 (Clayton Le´vy copula): The Clayton Le´vy copula is for 40 defined as b 1 , . . . , ud Þ ¼ ðu þ    þ u Þ1= , u1 , . . . , ud 2 ð0, 1Þ: Cðu 1 d

In figure 1 we show sample paths of two dependent compound Poisson processes, where the dependence is modelled via a Clayton Le´vy copula for different parameter values. With increasing dependence parameter b is homogenous of  we see more joint jumps. Note that C order 1. Hence, from theorem 3.16, if i 2 R for some 40 the Le´vy measure is multivariate regularly varying with index . To calculate þ(1, 1] we follow section 3.2 of Klu¨ppelberg and Resnick (2008). According to remark 1(a) we can set lim

2 ðxÞ

x!1 1 ðxÞ

¼ c 2 ð0, 1Þ,

ð41Þ

i.e. we assume that both tail integrals are tail-equivalent. Choosing 1 ðbðtÞÞ t1 we have lim t1 ðbðtÞx1 Þ ¼ 1 ðx1 , 1 ¼ x 1

t!1

and 2 ðbðtÞx2 Þ 1 ðbðtÞÞ 2 ðux2 Þ 2 ðuÞ ¼ lim ¼ cx 2 : u!1 2 ðuÞ 1 ðuÞ

lim t2 ðbðtÞx2 Þ ¼ lim

t!1

t!1

Then from (38) we obtain for d ¼ 2     1= ð½0, ðx1 , x2 Þc Þ ¼ x x2  , 1 þ cx2  ½x1 þ c x1 4 0, x2 4 0:

By differentiating we obtain the density 0 for 0551 (the completely positive dependent case ( ! 1) and the independent case ( ! 0) are not covered by the following calculation) as 0 ðx1 , x2 Þ ¼ c 2 ð1 þ Þ x1ð1þÞ1 x1 2   !1=2 x2 ,  1 þ c x1 x1 4 0, x2 4 0: We can then write þ ð1, 1 ¼ fðx1 , x2 Þ 2 E : x1 þ x2 4 1g Z 1Z 1 0 ðx1 , x2 Þ dx2 dx1 ¼ ðð1, 1  ½0, 1Þ þ 0

¼ 1 ð1, 1 þ ¼1þ

Z

Z 1Z 0

1

0

1þc

0

1x1

1 

1x1

 ðx1 , x2 Þ dx2 dx1

 !1=1 1 x11 dx1 , 1 x1



and substituting  ¼ (1/x1)  1 we obtain Z1 þ  ð1, 1 ¼ 1 þ  ð1 þ c  Þ1=1 ð1 þ Þ1 d 0 Z1 1= ¼1þc ð1 þ s Þ1=1 ðc1= þ s1= Þ1 ds: 0

ð42Þ

 1/1

, y40, is the density of a Since g(y) :¼ (1 þ y ) positive random variable Y, we finally arrive at þ ð1, 1 ¼ 1 þ c1= E ½ðc1= þ Y1= Þ1  ¼: 1 þ c1= Cð, Þ:

Then an analytical approximation for OpVaR follows together with expression (40),   1 þ VaRt ðÞ F1 1  ,  " 1: ð43Þ 1 ð1 þ c1= Cð, ÞÞ t Note that VaRþ t ðÞ increases with C(, , c, 1, 2). For  ¼ 1 the constant C(1, ) ¼ 1 implies that total OpVaR for all Clayton parameters in the range 0551 is given by   1 þ VaRt ðÞ F1 1  1 ð1 þ cÞ t   1 ,  " 1, ¼ F1 1  ð1 þ c2 2 Þ t which is (independent of the dependence parameter ) equal to the independent OpVaR of theorem 3.12. Note also the relation c ¼ 2/1c2 between the different constants in (31) and (41). Furthermore, þ(1, 1] is greater than or less than 2, as  is greater than or less than 1. If  ¼ 1 we can solve the integral in (42) (similarly to example 3.8 of Bregman and Klu¨ppelberg (2005)) and obtain þ ð1, 1 ¼

c11=  1 : c1=  1

Acknowledgement and disclaimer Figure 1 was produced by Irmingard Eder. The opinions expressed in this paper are those of the author and do not reflect the views of UniCredit Group.

References Aue, F. and Kalkbrenner, M., LDA at work., 2006, Preprint, Deutsche Bank, Available online at: www.gloriamundi.org Barndorff-Nielsen, O. and Lindner, A., Le´vy copulas: dynamics and transforms of Upsilon type. Scand. J. Statist., 2007, 34, 298–316. Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards, 2004 (Basel Committee on Banking Supervision: Basel).

Multivariate models for operational risk Bee, M., Copula-based multivariate models with applications to risk management and insurance. Preprint, University of Trento, 2005. Available online at: www.gloriamundi.org Bingham, N.H., Goldie, C.M. and Teugels, J.L., Regular Variation, 1987 (Cambridge University Press: Cambridge). Bo¨cker, K. and Klu¨ppelberg, C., Operational VaR: a closedform approximation. RISK Mag, 2005, December, 90–93. Bo¨cker, K. and Klu¨ppelberg, C., Modelling and measuring multivariate operational risk with Le´vy copulas. J. Oper. Risk, 2008, 3, 3–27. Bo¨cker, K. and Klu¨ppelberg, C., First order approximations to operational risk—dependence and consequences. In Operational Risk Towards Basel III, Best Practices and Issues in Modeling, Management and Regulation, edited by G.N. Gregoriou, 2009 (Wiley: New York). in press. Bregman, Y. and Klu¨ppelberg, C., Ruin estimation in multivariate models with Clayton dependence structure. Scand. Act. J, 2005, 462–480. Chavez-Demoulin, V., Embrechts, P. and Nesˇ lehova´, J., Quantitative models for operational risk: extremes, dependence and aggregation. J. Bank. Finance, 2005, 30, 2635–2658. Cont, R. and Tankov, P., Financial Modelling With Jump Processes, 2004 (Chapman & Hall/CRC Press: Boca Raton). Embrechts, P., Klu¨ppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance, 1997 (Springer: Berlin). Frachot, A., Roncalli, T. and Salomon, E., The correlation problem in operational risk. Preprint, Credit Agricole, 2004. Available online at: www.gloriamundi.org

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Goldie, C.M. and Klu¨ppelberg, C., Subexponential distributions. In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, edited by R. Adler, R. Feldman, and M.S. Taqqu, pp. 435–459, 1998 (Birkha¨user: Boston). Kallsen, J. and Tankov, P., Characterization of dependence of multivariate Le´vy processes using Le´vy copulas. J. Multivar. Anal., 2006, 97, 1551–1572. Klu¨ppelberg, C. and Resnick, R.I., The Pareto copula, aggregation of risks and the emperor’s socks. J. Appl. Probab., 2008, 45, 67–84. Klu¨ppelberg, C., Lindner, A. and Maller, R., Continuous time volatility modelling: COGARCH versus Ornstein–Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, edited by Y. Kabanov, R. Lipster, and J. Stoyanov, pp. 393–419, 2005 (Springer: Berlin). Klugman, S., Panjer, H. and Willmot, G., Loss Models—From Data to Decisions, 2004 (Wiley: Hoboken, NJ). Loe´ve, M., Probability Theory, Vol. I, 1978 (Springer: Berlin). Moscadelli, M., The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee. Banca D’Italia, Termini di discussione No. 517, 2004. Powosjowski, M.R., Reynolds, D. and Tuenter, J.H., Dependent events and operational risk. Algo Res. Q, 2002, 5, 65–73. Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, 1987 (Springer: New York). Resnick, S.I., Heavy-Tail Phenomena. Probabilistic and Statistical Modeling, 2006 (Springer: New York).

International Journal of Computer Mathematics, 2014 http://dx.doi.org/10.1080/00207160.2014.887274

Numerical analysis for Spread option pricing model of markets with finite liquidity: first-order feedback model A. Shidfara , Kh. Paryaba , A.R. Yazdaniana,b∗ and Traian A. Pirvub a Department

of Mathematics, Iran University of Science and Technology, Tehran, Iran; b Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

(Received 20 February 2013; revised version received 11 May 2013; second revision received 21 August 2013; third revision received 11 October 2013; fourth revision received 24 December 2013; accepted 21 January 2014) In this paper, we discuss the numerical analysis and the pricing and hedging of European Spread options on correlated assets when, in contrast to the standard framework and consistent with a market with imperfect liquidity, the option trader’s trading in the stock market has a direct impact on one of the stocks price. We consider a first-order feedback model which leads to a linear partial differential equation. The Peaceman– Rachford scheme is applied as an alternating direction implicit method to solve the equation numerically. We also discuss the stability and convergence of this numerical scheme. Finally, we provide a numerical analysis of the effect of the illiquidity in the underlying asset market on the replication of an European Spread option; compared to the Black–Scholes case, a trader generally buys less stock to replicate a call option. Keywords: Spread option pricing; price impact; illiquid markets; Peaceman–Rachford scheme 2010 AMS Subject Classifications: 91G20; 35K15; 65M06

1. Introduction Black and Scholes [2] and most of the work undertaken in mathematical finance assume that the market in the underlying asset is infinitely (or perfectly) liquid, such that trading had no effect on the price of underlying asset. In the market with finite liquidity, trading does affect the underlying asset price, regardless of her trading size. The model we consider involves the price impact due to the action of a large trade that may itself impact the price, independent of all the other factors affecting the price dynamics; this is termed price impact. In the presence of such a price impact, the most important issue is how the impact price can affect the replication of an option. This encouraged researchers to develop the Black–Scholes model to models that involve the price impact due to a large trader who is able to move the price by his/her actions. An excellent survey of these research can be found in [12,16,25,41]. In [37], we investigated the effects of the full-feedback model in which price impact is fully incorporated into the model and results in highly nonlinear partial differential equation. Our purpose of this paper is to investigate the effects of imperfect liquidity on the replication of an European Spread option by a typical option trader, when the hedging strategy does not take into account the feedback effect (we term first-order feedback model). We assume that a Spread ∗ Corresponding

author. Email: [email protected]

© 2014 Taylor & Francis

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option is to be hedged and furthermore that the hedger holds the number of stocks dictated by the analytical Black–Scholes delta, rather than the delta from the modified option price. This leads to the linear partial differential equation (PDE), which is somewhat easier to solve than the full-feedback model but still has important and interesting differences from the classical Black–Scholes PDE. Spread option is the simplest example of multi-assets derivative, whose payoff is the difference between the prices of two or more assets; for instance let the prices of two underlying assets at time t ∈ [0, T ] be S1 (t) and S2 (t), then the payoff function of a European Spread option with maturity T is [S1 (T ) − S2 (T ) − k]+ (here k is the strike of the option and the function x + is defined as x + = max(x, 0)). Therefore the holder of a European Spread option has the right but not the obligation to buy the spread S1 (T ) − S2 (T ) at the prespecified price k and maturity T . In general, there is no analytical formula for the price of multi-assets options. The only exception is Margrabe formula for exchange options (Spread options with a strike of zero) [26]. Kirk [23] found an analytical approximation for Spread options with k positive and close to zero. Several Spread options are traded in the markets, e.g. fixed income Spread options, foreign exchange and commodity Spread options. In this work, we focus on commodity Spread options. Spread options, in commodity market, hedge the risk of price fluctuations between input and output products. In order to price them one needs to take into account the characteristics of the commodities prices they are written upon. During the past decades, several stochastic models for commodity prices have been introduced. The first models assumed that the price processes follow a geometric Brownian motion and that all the uncertainty could be summarized by one factor. Models of this type include Cox and Schwartz [6] for pricing commodity-linked securities, Brennan and Schwartz [3], Paddock et al. [30], and Cortazar and Schwartz [4] for valuing real assets. Mean reverting price processes were considered by Schwartz [35]. Most models assumed that there is a single source of randomness driving the prices of the commodities. Since empirical evidence suggests more sources of randomness, several two- and three-factor models were subsequently developed. In their two-factor model, Gibson and Schwartz [13] assumed that the spot price of the commodity and the convenience yield (the difference between the interest rate and the cost of carry) follow a joint stochastic process. Cortazar and Schwartz in [5] took a different approach; they used all the information contained in the term structure of commodity futures prices together with the historical volatilities of future return for different maturities. A good comparison among these models was performed in [35]. Schwartz and Smith [36] modelled the log spot price as the sum of two stochastic factors and they showed that this model is equivalent to the Gibson and Schwartz [13] model. Pascheke and Prokopczuk [31] developed a continuous time factor model which allows for higher-order autoregressive and moving average components. A review of these models is done in [1]. There are several types of commodity Spread options, some of the popular ones are: Crush Spread option. In the agricultural markets, the Chicago Board of Trade the so-called crush spread which exchanges soyabeans (as a unrefined product) with a combination of soyabean oil and soyabean meal (as the derivative products). Johnson et al. [22] studied Spread options in the agricultural markets. Spark Spread option. In the energy markets, spark Spread options are a spread between natural gas and power (electricity). Girma and Paulson [14,15] studied these type of options. Crack Spread options. A Crack Spread represents the differential between the price of crude oil and petroleum products (gasoline or heating oil). The underlying indexes comprise futures prices of crude oil, heating oil and unleaded gasoline. Details of Crack Spread options can be found in the New York Mercantile Exchange Crack Spread Handbook [29]. Our paper is aimed at pricing Crack Spread options. In the oil markets with finite liquidity, trading does affect the underlying assets price. In our study, we are going to investigate the effects of price impact when trading affects only the crude oil price and not the petroleum products. Our model is

International Journal of Computer Mathematics

3

related to the constant convenience yield model of [35]. In their model of the commodity price, the rate of return is affected by a stochastic convenience yield; in our model, due to the liquidity risk, both the rate of return and the volatility of the risky asset are affected by stochastic factors. We study in this work a splitting scheme of the alternating direction implicit (ADI) type associated with a two-dimensional PDE (which characterizes the option price). This method has the desirable stability features of the Crank Nicolson method, but it proceeds in two steps. The first half step is taken implicitly in one space variable and explicitly in the other, while the second half step reverses the explicit and implicit variables. Thus the numerical problem reduces to solving two matrix equations. ADI method goes back to [32] and has been further developed in many works, e.g. [18,19,42] (for financial applications see [7,24,27,28,34]). ADI schemes were not originally developed for multi-dimensional convection–diffusion equations with mixed derivative terms. The problems generated by the cross-derivatives were first discussed in [39,40]. Furthermore, Pospisil and Vecer [33] applied the Peaceman–Rachford and Douglas–Rachford schemes as ADI method; Hout and Foulon [20] investigated four splitting schemes of ADI type: the Douglas scheme, the Craig–Sneyd (CS) scheme, the modified CS scheme and the Hundsdorfer–Verwer scheme; Haentjens [17] investigated the effectiveness of ADI time discretization schemes in the numerical solution of three-dimensional Heston–Hull–White PDE; Dang et al. [8] employed the ADI method based on Hundsdorfer and Verwer (HV) splitting approach for pricing foreign exchange interest rate hybrid derivatives. In this work, we use the Peaceman and Rachford scheme that was introduced first in [32]. Since the resulting multi-dimensional linear PDE has mixed derivative terms we have to adjust the Peaceman and Rachford scheme. This paper is organized as follows: in Section 2, we introduce our problem and discuss the general framework we use. In Section 3 we propose the splitting scheme of the ADI type (subsequently, we discuss the stability and the convergence of the scheme). In Section 4, we carry out several numerical experiments and provide a numerical analysis. Section 5 contains the concluding remarks.

2. The model setup In this section we describe the setup for Spread option pricing. Our model of a financial market, based on a filtered probability space (, F, {Ft }t∈[0,T ] , P) that satisfies the usual conditions, consists of two assets. Their prices are modelled by a two-dimensional Ito-process S(t) = (S1 (t), S2 (t)). All the stochastic processes in this work are assumed to be {Ft }t≥0 adapted. Their dynamics are given by the following stochastic differential equations, in which W (t) = (w1 (t), w2 (t)) is defined a two-dimensional standard Brownian motion with {Ft }t∈[0,T ] being its natural filtration augment by all P-null sets: dSi (t) = μi (t, Si (t)) dt + σi (t, Si (t)) dwi (t); Si (t)

i = 1, 2,

(1)

where w1 and w2 are two correlated Brownian motions with correlation ρ, μi (t, Si (t)) and σi (t, Si (t)) are the expected return and the volatility of stock i in the absence of price impact. It is possible to add a partial price impact for the first stock, i.e. dS1 (t) = μ1 (t, S1 (t))S1 (t) dt + σ1 (t, S1 (t))S1 (t) dw1 (t) + λ(t, S1 ) df (t, S1 , S2 ),

dS2 (t) = μ2 (t, S2 (t))S2 (t) dt + σ2 (t, S2 (t))S2 (t) dw2 (t),

(2)

where λ(t, S1 ) ≥ 0 is an arbitrary function and λ(t, S1 ) df (t, S1 , S2 ) represents the price impact of the investor’s trading. We see that the two-dimensional classical Black–Scholes model is a special case of this model with λ(t, S1 (t)) = 0.

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Our aim is to price a Spread option under the modified stochastic process (2), with the following payoff at maturity T (a call at this case): h(S1 (T ), S2 (T )) = (S1 (T ) − S2 (T ) − k)+ ,

(3)

where k is the strike price. In order to provide a derivation of the pricing PDE considered in this work, we use the well-known generalized Black–Scholes equation (more details in [9]). This leads to the following pricing PDE for the modified stochastic process incorporating the forcing term (2)  2  1 ∂f ∂V 2 2 2 2 2 (t, S1 , S2 ) + (t, S , S ) σ S + λ (t, S )σ S 1 2 1 1 1 2 2 ∂t 2(1 − λ(t, S1 )(∂f /∂S1 )(t, S1 , S2 ))2 ∂S2  2 ∂ V ∂ 2V ∂f 1 (t, S1 , S2 ) + 2ρσ1 σ2 S1 S2 λ(t, S1 ) (t, S1 , S2 ) + σ22 S22 2 (t, S1 , S2 ) 2 ∂S2 2 ∂S1 ∂S2   ∂f 1 (t, S1 , S2 ) σ1 σ2 ρS1 S2 + λ(t, S1 )σ22 S22 + (1 − λ(t, S1 )(∂f /∂S1 )(t, S1 , S2 )) ∂S2   2 ∂ V ∂V ∂V × (t, S1 , S2 ) + r S1 (t, S1 , S2 ) + S2 (t, S1 , S2 ) ∂S1 ∂S2 ∂S1 ∂S2 − rV (t, S1 , S2 ) = 0,

0 < S1 , S2 < ∞, 0 ≤ t < T .

(4) Here r is the riskless rate of the money market. Consistent with standard Black–Scholes arguments, the drift of the modified process μ(t, S(t)) does not appear in the option pricing PDE. In the context of markets with finite liquidity, we can define f (t, S1 , S2 ) to be the number of extra shares traded due to some deterministic hedging strategy, and λ(t, S1 (t)) as some function dependent on how we choose to model the form of price impact. Here, similar to [25], we consider λ(t, S1 (t)) = ˆ S1 (t)), with λ(t, ˆ S1 ) a function such that λ(T ˆ , S1 ) = 0 and ε > 0 the constant price impact ελ(t, coefficient. In the first-order feedback model f (t, S1 , S2 ) in Equation (4) is f (t, S1 , S2 ) =

∂V BS (t, S1 , S2 ), ∂S1

where V BS (t, S1 , S2 ) is the Black–Scholes value (see [16]). This leads to the following linear PDE: 1 ∂V (t, S1 , S2 ) + 2 ∂t 2(1 − λ(t, S1 )(∂ V BS /∂S12 )(t, S1 , S2 ))2   2  2 BS 2 BS ∂ V ∂ V (t, S1 , S2 ) + 2ρσ1 σ2 S1 S2 λ(t, S1 ) (t, S1 , S2 ) × σ12 S12 + λ2 (t, S1 )σ22 S22 ∂S1 ∂S2 ∂S1 ∂S2 ∂ 2V 1 2 2 ∂ 2V 1 (t, S , S σ2 S2 2 (t, S1 , S2 ) + ) + 1 2 2 2 2 ∂S1 ∂S2 (1 − λ(t, S1 )(∂ V BS /∂S12 )(t, S1 , S2 ))  2  2 BS ∂ V 2 2 ∂ V (t, S1 , S2 ) (t, S1 , S2 ) + r × σ1 σ2 ρS1 S2 + λ(t, S1 )σ2 S2 ∂S1 ∂S2 ∂S1 ∂S2   ∂V ∂V × S1 (t, S1 , S2 ) + S2 (t, S1 , S2 ) ∂S1 ∂S2 ×

− rV (t, S1 , S2 ) = 0,

0 < S1 , S2 < ∞, 0 ≤ t < T .

(5)

International Journal of Computer Mathematics

5

For investigating the treatment of boundary conditions we apply Fichera’s theory [11]. In order to determine the subsets where boundary conditions can be imposed, we need to evaluate the Fichera function. Equation (5) is defined on D where D = {(t, S1 , S2 )|, 0 < t ≤ T , 0 < S1 < ∞, 0 < S2 < ∞}. The corresponding coefficient matrix is ⎛

a11 A = ⎝a21 0

where the components of A are the following:

a12 a22 0

⎞ 0 0⎠ , 0

a11 =

1 (σ12 S12 + λ2 σ22 S22 (VSBS )2 + 2ρσ1 σ2 S1 S2 λVSBS ), 1 S2 1 S2 2 ) 2(1 − λVSBS 1 S1

a12 =

1 ), (σ1 σ2 ρS1 S2 + λσ22 S22 VSBS 1 S2 ) 2(1 − λVSBS S 1 1

a21

1 ), = (σ1 σ2 ρS1 S2 + λσ22 S22 VSBS 1 S2 2(1 − λVSBS ) 1 S1

a22 =

(6)

1 2 2 σ S . 2 2 2

A is a singular matrix everywhere. For the boundaries S1 = 0, S2 = 0 and t = 0 we have the corresponding inward normals n = (1, 0, 0), (0, 1, 0) and (0, 0, 1), and the inward normal on t = T is (0, 0, −1). We let 0 be the subset

of ∂D where An, n = 0. We observe that < An, n >= 0 at all of the boundary points so ∂D = 0 . The Fichera function is  λVSBS 1 S1 S 1 h = rS1 + (σ 2 S 2 + λ2 σ22 S22 (VSBS )2 + 2ρσ1 σ2 S1 S2 λVSBS ) 1 S2 1 S2 )4 1 1 2(1 − λVSBS 1 S1 −

1 (2σ12 S1 + 2λ2 σ22 S22 VSBS V BS + 2ρσ1 σ2 S2 λVSBS + 2ρσ1 σ2 S1 S2 λVSBS ) 1 S1 S2 S 1 S2 1 S2 1 S 1 S2 )2 2(1 − λVSBS 1 S1

1 + λσ22 S22 VSBS ) (ρσ1 σ2 S1 + 2λσ22 S2 VSBS 1 S2 1 S 2 S2 ) 2(1 − λVSBS 1 S1 λVSBS 2 2 BS 1 S 1 S2 (σ1 σ2 ρS1 S2 + λσ2 S2 VS1 S2 ) n1 + )2 (1 − λVSBS 1 S1  1 + rS2 − σ22 S2 − ) (ρσ1 σ2 S2 + λσ22 S22 VSBS 1 S1 S2 ) 2(1 − λVSBS S 1 1 λVSBS 2 2 BS 1 S1 S1 + (σ1 σ2 ρS1 S2 + λσ2 S2 VS1 S2 ) n2 − n3 . )2 (1 − λVSBS 1 S1

−

(7) On S2 = 0 we see that h(S1 , 0, t) = 0 and according to Fichera’s theory no boundary data should be given. Instead the differential equation should hold on S2 = 0. On S1 = 0 we see that h(0, S2 , t) = 0 and according to Fichera’s theory no boundary data should be given. Instead the differential equation should hold on S1 = 0. On t = 0 we see that h(S1 , v, 0) = −1 so we can

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impose the payoff of the option at maturity as initial condition on Equation (5). At t = T the differential equation holds. Remark 2.1 In [37], we have investigated a full-feedback model, where the price impact is fully incorporated into the model. The corresponding equation is 1 ∂V (t, S1 , S2 ) + 2 ∂t 2(1 − λ(t, S1 )(∂ V /∂S12 )(t, S1 , S2 ))2   2  2 2 ∂ ∂ V V (t, S1 , S2 ) + 2ρσ1 σ2 S1 S2 λ(t, S1 ) (t, S1 , S2 ) × σ12 S12 + λ2 (t, S1 )σ22 S22 ∂S1 ∂S2 ∂S1 ∂S2 ∂ 2V 1 2 2 ∂ 2V 1 (t, S , S ) + σ2 S2 2 (t, S1 , S2 ) + 1 2 2 2 2 ∂S1 ∂S2 (1 − λ(t, S1 )(∂ V /∂S12 )(t, S1 , S2 ))  2  2 ∂ V 2 2 ∂ V (t, S1 , S2 ) (t, S1 , S2 ) × σ1 σ2 ρS1 S2 + λ(t, S1 )σ2 S2 ∂S1 ∂S2 ∂S1 ∂S2   ∂V ∂V + r S1 (t, S1 , S2 ) + S2 (t, S1 , S2 ) − rV (t, S1 , S2 ) = 0, 0 < S1 , S2 < ∞, 0 ≤ t < T , ∂S1 ∂S2 ×

V (T , S1 , S2 ) = h(S1 , S2 ),

0 < S1 , S2 < ∞.

(8) The first-order approximation is V (t, S1 , S2 ) = V 0 (t, S1 , S2 ) + εV 1 (t, S1 , S2 ) + o(ε 2 ), where V 0 (t, S1 , S2 ) is the Black–Scholes price for European Spread option, i.e. 

σ22 S22 ∂ 2 V 0 ∂ 2V 0 ∂V 0 ∂V 0 σ 2 S2 ∂ 2 V 0 ∂V 0 + + σ σ S S ρ − rV 0 = 0, + 1 1 + r S + S 1 2 1 2 1 2 ∂t 2 ∂S12 2 ∂S22 ∂S1 ∂S2 ∂S1 ∂S2

V 0 (T , S1 , S2 ) = max(S1 (T ) − S2 (T ) − k, 0),

0 < S1 , S2 < ∞, (9)

and V 1 (t, S1 , S2 ) is the solution of the following problem 

∂V 1 ∂V 1 σ 2 S2 ∂ 2 V 1 σ22 S22 ∂ 2 V 1 ∂ 2V 1 ∂V 1 − rV 1 = G, + r S + S + + σ σ S S ρ + 1 1 1 2 1 2 1 2 ∂t 2 ∂S12 2 ∂S22 ∂S1 ∂S2 ∂S1 ∂S2

V 1 (T , S1 , S2 ) = 0,

0 < S1 , S2 < ∞. (10)

Here 

∂ 2V 0 ∂ 2V 0 G = −ε 2ρσ1 σ2 S1 S2 + σ12 S12 ∂S1 ∂S2 ∂S12



∂ 2V 0 ∂S12

2

+

σ22 S22



∂ 2V 0 ∂S1 ∂S2

2 

.

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International Journal of Computer Mathematics

3. Numerical solution of partial differential equation 3.1

The ADI

In this section, we present a numerical method for solving the pricing partial differential equation 

V (T , x, y) = h(x, y),

σ12 x 2

2

σ22 y2



∂ 2 V BS ∂x∂y

2

∂ 2 V BS +λ + 2ρσ1 σ2 xyλ ∂x∂y   2 BS 1 ∂ 2V 1 ∂ 2V 2 2∂ V + σ22 y2 2 + σ σ ρxy + λσ y 1 2 2 2 ∂y (1 − λ(∂ 2 V BS /∂x 2 )) ∂x∂y ∂x∂y   ∂V ∂V +y − rV = 0, +r x ∂x ∂y

1 ∂V + ∂t 2(1 − λ(∂ 2 V BS /∂x 2 ))2



∂ 2V ∂x 2 (11)

0 < x, y < ∞,

where the functions V := V (t, x, y), V BS := V BS (t, x, y) are defined on [0, T ] × [0, ∞) × [0, ∞) and λ := λ(t, x) on [0, T ] × [0, ∞). For the sake of notation, we write the following operators:

L=

∂ + Ax + Ay + Axy , ∂t

(12)

where 1 Ax V = 2(1 − λ(∂ 2 V BS /∂x 2 ))2



σ12 x 2

+λ

2

σ22 y2



∂ 2 V BS ∂x∂y

2

∂ 2 V BS + 2ρσ1 σ2 xyλ ∂x∂y

∂V − r, ∂x 1 ∂ 2V ∂V Ay V = σ22 y2 2 + ry − r(1 − ), 2 ∂y ∂y  2 BS  2 1 ∂ V 2 2∂ V Axy V = , σ σ ρxy + λσ y 1 2 2 2 BS 2 (1 − λ(∂ V /∂x )) ∂x∂y ∂x∂y



∂ 2V ∂x 2

+ rx

(13)

and 0 ≤  ≤ 1. While symmetry considerations might speak for an  = 21 , it is computationally simpler to use  = 0 or  = 1, i.e. to include the rV − term fully in one of the two operators. Hence, we can write LV = 0,

0 < x, y < ∞, 0 < t < T ,

V (T , x, y) = h(x, y),

0 < x, y < ∞.

(14)

In order to define a numerical solution to the equation, we need to truncate the spatial domain to a bounded area as {(x, y); 0 ≤ x ≤ xmax , 0 ≤ y ≤ ymax }. We follow [21] in choosing the upper bounds of the domain. The upper bounds should be large enough to include the stock price limits within which there is a price impact. Let us introduce a grid of points in the time interval and in

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A. Shidfar et al.

the truncated spatial domain tl = l t,

l = 0, 1, . . . , L, t =

T , L

xm = m x,

m = 0, 1, . . . , M, x =

yn = n y,

n = 0, 1, . . . , N, y =

xmax , M

(15)

ymax . N

For the simplicity of notation, we assume that xmax = ymax and x = y. Functions V (t, x, y) and l BS,l V BS (t, x, y) at a point of the grid will be denoted as Vm,n = V (tl , xm , yn ) and Vm,n = V BS (tl , xm , yn ). Furthermore, let us introduce the approximations l l − Vm−1,n Vm+1,n ∂V (tl , xm , yn ) = + O( 2 x), ∂x 2 x l l − Vm,n−1 Vm,n+1 ∂V (tl , xm , yn ) = + O( 2 y), ∂y 2 y l l l − 2Vm,n + Vm−1,n Vm+1,n ∂ 2V (t , x , y ) = + O( 2 x), l m n ∂x 2 ( x)2

(16)

l l l − 2Vm,n + Vm,n−1 Vm,n+1 ∂ 2V (t , x , y ) = + O( 2 y), l m n ∂y2 ( y)2 l l l l − Vm−1,n+1 − Vm+1,n−1 + Vm−1,n−1 Vm+1,n+1 ∂ 2V (tl , xm , yn ) = + O( 2 x + 2 y). ∂x∂y 4 x y

Let symbols Adx , Ady and Adx dy denote second-order approximations of the operators Ax , Ay and Axy obtained by using Equation (16) into Equation (13). We can use ADI because the differential operator can be split as in Equation (13). The general idea is to split a time step in two and to take one operator or one space coordinate at a time (see more details in [10,38]). In this work, particularly we use the Peacman–Rachford scheme. Taking our inspiration from the Crank–Nicolson method we begin discretizing (11) in the time-direction V l+1 − V l + O( t 2 ), t (Ax + Ay + Axy )V = 21 Ax (V l+1 + V l ) + 21 Ay (V l+1 + V l ) + 21 Axy (V l+1 + V l ) + O( t 2 ). (17) Insert in Equation (12), multiply by t, and rearrange Vt ((l + 1/2) t, x, y) =

(I − 21 tAx − 21 tAy )V l = (I + 21 tAx + 21 tAy )V l+1 + 21 tAxy (V l+1 + V l ) + O( t 3 ), (18) where I denotes the identity operator. If we add 41 t 2 Ax Ay V l on the left side and 41 t 2 Ax Ay V l+1 on the right side then we commit an error which is O( t 3 ) and therefore can be included in that term (I − 21 tAx )(I − 21 tAy )V l = (I + 21 tAx )(I + 21 tAy )V l+1 + 21 tAxy (V l+1 + V l ) + O( t 3 ). (19) Now, we discretize in the space coordinates replacing Ax by Adx , Ay by Ady and Axy by Adx dy (I − 21 tAdx )(I − 21 tAdy )V l = (I + 21 tAdx )(I + 21 tAdy )V l+1 + 21 tAdx dy (V l+1 + V l ) + O( t 3 ) + O( t x 2 ),

(20)

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International Journal of Computer Mathematics

and this gives rise to the Peaceman–Rachford method     t t Adx V l+1/2 = I + Ady V l+1 + α, I− 2 2     t t l I− Ady V = I + Adx V l+1/2 + β, 2 2

(21)

where auxiliary function V l+1/2 links above equations. We have introduced the values α and β to take into account the mix derivative term because it is not obvious how this term should be split. In order to correspond the solution (21) by the solution to Equation (20), we have the requirement that     t 1 t Adx α + I − Adx β = tAdx dy (V l+1 + V l ), (22) I+ 2 2 2 where a discrepancy of order O( t 3 ) may be allowed with reference to a similar term in Equation (19). One of the possible choices for α and β is α=

t Adx dy V l+1 , 2

β=

t Adx dy V l+1/2 . 2

Finally, the Peaceman–Rachford scheme for V in Equation (11) is obtained as follows:     t t t Adx V l+1/2 = I + Ady V l+1 + Adx dy V l+1 , I− 2 2 2     t t t l Ady V = I + Adx V l+1/2 + Adx dy V l+1/2 . I− 2 2 2

(23)

(24)

In a first step we compute V l+1/2 using V l+1 . This step is implicit in direction x. In a second step, defined by Equation (24), we use V l+1/2 to calculate V l . This step is implicit in the direction of y. We need boundary conditions to apply the algorithm, which we consider as follows: • if x = 0 then the payoff function is 0 and so the option price is 0. • if y = 0 then S2 = 0, and so is just the price of the option on one risky asset. Note that due to the use of centred approximations of the derivatives, at x0 = y0 = 0, xM = xmax and yN = ymax , there appear external fictitious nodes x−1 = − x, y−1 = − y, xM+1 = (M + 1) x and yN+1 = (N + 1) y. The approximations in these nodes are obtained by using linear interpolation throughout the approximations obtained in the closest interior nodes of the numerical domain. Thus we have the following relations: l l l = 2V0,n − V1,n , V−1,n l l l = 2Vm,0 − Vm,1 , Vm,−1

l l l VM+1,n = 2VM,n − VM−1,n ,

n = 0(1)N,

l l l Vm,N+1 = 2Vm,N − Vm,N−1 ,

m = 0(1)M,

(25)

and also l l l l l = 4V0,N − 2(V1,N + V0,N−1 ) + V1,N−1 , V−1,N+1 l l l l l , − 2(VM,1 + VN−1,0 ) + VN−1,1 = 4VM,0 VM+1,−1 l l l l l , − 2(V0,1 + V1,0 ) + V1,1 = 4V0,0 V−1,−1

(26)

l l l l l = 4VM,N − 2(VM,N−1 + VM−1,N ) + VM−1,N−1 . VM+1,N+1 l Now all values Vm,n are available. By repeating this procedure for l = L − 1, L − 2, . . . , 0, we obtain Vm,n at all time points and can approximate the price of a Spread option at time t = 0.

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3.2

Stability and convergence of the numerical solution

In this section, we analyse stability of the Peaceman–Rachford method. In this case, we can use the Von Neumann analysis to establish the conditions of stability. This approach was described in Chapter 2.2 of [38]. The Von Neumann analysis is based on calculating the amplification factor of a scheme, g, and deriving conditions under which |g| ≤ 1. For finding the amplification factor, l a simpler and equivalent procedure is to replace Vmn in the scheme by g−l eimθ einφ for each value of l, n and m. The resulting equation can then be solved for the amplification factor. l+1/2 l Replacing Vmn and Vmn by gˆ g−l eimθ einφ and g−l eimθ einφ , respectively, we have   t 1 l+1/2 Adx Vm,n = gˆ g−l eimθ einφ −a1 sin2 θ + b1 i sin θ , 2 2   t l −l imθ inφ 2 1 Ady Vm,n = g e e −a2 sin φ + b2 i sin φ − c1 , 2 2 (27) t 0,l+1/2 −l imθ inφ l+1/2 Adx dy Vm,n = −ˆgg e e c2 sin θ sin φ, 2 t 0,l Adx dy Vm,n = −g−l eimθ einφ c2l sin θ sin φ, 2 where a1 =: a1 (xm , yn , tl+1/2 ) =

BS BS t(σ12 xm2 + λ2 σ22 yn2 (Vxy (xm , yn , tl+1/2 ))2 + 2ρσ1 σ2 xm yn λVxy (xm , yn , tl+1/2 )) 2 BS (x , y , t x 2 (1 − λVxx m n l+1/2 ))

trxm , 2 x tryn , b2 =: b2 (yn ) = 2 y

,

b1 =: b1 (xm ) =

a2 =: a2 (yn ) = c1 =

(28)

tσ22 yn2 , y2

r t , 2

and c2l =: c2 (xm , yn , tl ), c2 (xm , yn , tl ) =

l+1/2

c2

=: c2 (xm , yn , tl+1/2 ),

c2l+1 =: c2 (xm , yn , tl+1 ),

t BS (σ σ ρxm yn + λ(xm , tl )σ22 yn2 Vxy (xm , yn , tl )). BS (x , y , t )) 1 2 2 x y(1 − λ(xm , tl )Vxx m n l (29)

We obtain the amplification factor as g=

1 − a2 sin2 21 φ + b2 i sin φ − c1 − c2l+1 sin θ sin φ (1 + a1 sin2 21 θ − b1 i sin θ )ˆg

,

(30)

where gˆ =

1 + a2 sin2 21 φ − b2 i sin φ + c1 1 − a1 sin2 21 θ + b1 i sin θ − c2

l+1/2

sin θ sin φ

,

(31)

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International Journal of Computer Mathematics

by arranging, we have sin θ sin φ + (b1 sin θ )i] [1 − a1 sin2 21 θ − c2 [1 − a2 sin2 21 φ − c1 − c2l+1 sin θ sin φ + (b2 sin φ)i] l+1/2

g=

[1 + a1 sin2 21 θ − (b1 sin θ )i][1 + a2 sin2 21 φ + c1 − (b2 sin φ)i]

,

(32)

and thus sin θ sin φ)2 + b12 sin2 θ] [(1 − a1 sin2 21 θ − c2 l+1 2 1 [(1 − a2 sin 2 φ − c1 − c2 sin θ sin φ)2 + b22 sin2 φ] l+1/2

|g(θ, φ)|2 =

[(1 + a1 sin2 21 θ )2 + b12 sin2 θ][(1 + a2 sin2 21 φ + c1 )2 + b22 sin2 φ]

.

(33)

ˆ S1 (t)) the coefficients a1 , c2l , c2l+1/2 and c2l+1 are continuous with respect Since λ(t, S1 (t)) = ελ(t, to ε. Therefore the amplification factor g is continuous with respect to ε as well. Thus if |g| < 1 for ε0 = 0 then there is a neighbourhood Bε0 of ε0 such that |g| ≤ 1 for all ε ∈ Bε0 . Moreover a1 > 0 for ε0 = 0 so a1 ≥ 0 for all ε ∈ Bε0 . l+1/2 ˆ 1 , where For ε = 0, according to definitions (28) and (29), a2 = Ca1 , c2l = c2 = c2l+1 = Ca 2 ˆ ˆ C =: C(m, n) = (σ2 n/σ1 m) , C =: C(m, n) = ρσ2 n/2σ1 m. Since |ρ| ≤ 1 it follows that C ≥ 4Cˆ 2 .

(34)

Moreover b1 = ξ a1 , b2 = (ξ n/m)a1 , c1 = (ξ/m)a1 , where ξ =: r/2σ12 m. By replacing the above relations in Equation (33), we find out that lim g(θ, φ)2 =

ξ →0

ˆ 1 sin θ sin φ)2 (1 − Ca1 sin2 1 φ − Ca ˆ 1 sin θ sin φ)2 (1 − a1 sin2 21 θ − Ca 2 (1 + a1 sin2 21 θ)2 (1 + Ca1 sin2 21 φ)2

. (35)

Hence it is enough to find the conditions for which ˆ 1 sin θ sin φ)2 (1 − Ca1 sin2 1 φ − Ca ˆ 1 sin θ sin φ)2 (1 − a1 sin2 21 θ − Ca 2 (1 + a1 sin2 21 θ )2 (1 + Ca1 sin2 21 φ)2

< 1.

(36)

Notice that ˆ 1 sin θ sin φ ≤ a1 | sin2 1 θ| + Ca ˆ 1 | sin θ sin φ| a1 sin2 21 θ + Ca 2 ˆ cos 1 θ sin φ|] ≤ a1 | sin 21 θ|[| sin 21 θ | + 2C| 2

(37)

ˆ ≤ a1 [1 + 2C]. ˆ 1 sin θ sin φ ≥ 0, provided that a1 [1 + 2C] ˆ ≤ 1, and Thus 1 − a1 sin2 21 θ − Ca ˆ 1 sin θ sin φ ≤ Ca1 | sin2 1 φ| + Ca ˆ 1 | sin θ sin φ| Ca1 sin2 21 φ + Ca 2 ˆ cos 1 φ sin θ |] ≤ a1 | sin 21 φ|[C sin 21 φ + 2C| 2

(38)

ˆ ≤ a1 [C + 2C]. ˆ 1 sin θ sin φ ≥ 0, provided that a1 [C + 2C] ˆ ≤ 1. Now we should find Thus 1 − Ca1 sin2 21 φ − Ca the conditions under which ˆ 1 sin θ sin φ)(1 − Ca1 sin2 1 φ − Ca ˆ 1 sin θ sin φ) (1 − a1 sin2 21 θ − Ca 2 (1 + a1 sin2 21 θ )(1 + Ca1 sin2 21 φ)

< 1,

(39)

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A. Shidfar et al.

or equivalently a1 (sin2 21 θ + Cˆ sin θ sin φ + C sin2 21 φ)(−2 + a1 Cˆ sin θ sin φ) < 0.

(40)

If |y| ≤ 1, then for any x ∈ R, xy ≥ −|x|, and by Equation (34) ˆ sin 1 θ sin 1 φ| + 4Cˆ 2 | sin 1 φ|2 sin2 21 θ + Cˆ sin θ sin φ + C sin2 21 φ ≥ | sin 21 θ |2 − 4C| 2 2 2 ˆ sin 1 φ|)2 ≥ 0. = (| sin 21 θ| − 2C| 2

(41)

ˆ Consequently a sufficient condition for the Hence Equation (40) is satisfied if a1 < 2/C. amplification factor to be bounded by 1, i.e. |g(θ , φ)| ≤ 1, is 

1 1 2 , , a1 < A = min 2 ˆ ˆ ˆ C 1 + 2C 4C + 2Cˆ



or

A t ≤ 2 2 , 2 x σ1 · xmax

A t ≤ 2 2 . (42) 2 y σ2 · ymax

Although a1 involves partial derivatives of V BS the first condition can be met for ε ∈ Bε0 . By assuming x = y and xmax = ymax , a sufficient condition for the stability of the scheme is A t . ≤ 2 x 2 max{σ12 , σ22 }xmax

(43)

Thus, the Peaceman–Rachford scheme is stable if the number of steps in the time interval, L, and in the spatial domain, M = N, satisfy inequality (43). This condition is a consequence of the cross-derivative term in the formula for the amplification factor. In the absence of this term, the scheme would be unconditionally stable. The remaining issue we need to address is the convergence of the numerical method. According to [38] the scheme is consistent and hence the scheme is convergent. Numerical results of this convergence are investigated in the next section. Notice that according to [38] the scheme has first-order accuracy in time and second order in space. The Peaceman–Rachford scheme in the absence of the cross-derivative term defines an unconditionally stable scheme with a higher-order of accuracy [O( t 2 ) + O( x 2 )] (dependent on  of Equation (13)). However, in the presence of the mixed derivatives, the accuracy remains [O( t) + O( x 2 )] independent of . Although the higher-order accuracy leads to a more efficient method, the numerical results in next section show the efficiency of the scheme. Modified schemes which overcome this restriction attain a higher-order of accuracy (at least O( t 2 )). Craig and Sneyd [7] developed a ADI scheme the so-called CS scheme for parabolic equation with mixed derivatives to attain a stable second-order ADI scheme; Walfert [40] modified the CS scheme and introduced modified Craig–Sneyd (MCS) to obtain the unconditional stability of second-order ADI schemes in the numerical solution of finite difference discretization of multi-dimensional diffusion problems containing mixed spatialderivative terms; Hundsdorfer [18] and Hundsdorfer and Verwer [19] presented the HV scheme for numerical solution of time-dependent advection–diffusion-reaction equations.

4. Numerical results In this section, we provide numerical results of the partial liquidity effect in the underlying asset market. We fix the values of the parameters of the marginal dynamical equations according to Table 1.

13

International Journal of Computer Mathematics Table 1. Model data together with r = 0.05. S(t0 )

σ

Smin

Smax

100 100

0.15 0.10

0 0

200 200

Asset 1 Asset 2

Table 2. Convergence of the Peaceman–Rachford method introduced in Section 3 for a call exchange option in standard Black–Scholes model, based on different correlation and expiration date.

ρ = 0.1 Margrabe ρ = 0.5 Margrabe ρ = 0.7 Margrabe ρ = 0.9

m

l

T = 0.1

T = 0.3

T = 0.5

T = 0.7

T =1

50 100 200

100 100 200

50 100 200

100 100 200

50 100 200

100 210 200

50 100 200

100 100 200

2.0118 2.1298 2.1555 2.1665 1.5860 1.6544 1.6641 1.6686 1.3525 1.3733 1.3600 1.3528 1.1030 1.0500 0.9846 0.9356

3.6578 3.7181 3.7327 3.7516 2.8475 2.8723 2.8773 2.8900 2.3746 2.3519 2.3391 2.3429 1.8391 1.7175 1.6498 1.6204

4.7508 4.7970 4.8083 4.8422 3.6841 3.7012 3.7053 3.7304 3.0464 3.0192 3.0087 3.0244 2.3029 2.1691 2.1068 2.0918

5.6265 5.6655 5.6750 5.7279 4.3553 4.3692 4.3730 4.4132 3.5849 3.5578 3.5492 3.5782 2.6697 2.5336 2.4766 2.4750

6.7160 6.7487 6.7566 6.8436 5.1912 5.2029 5.2065 5.2737 4.2560 4.2308 4.2242 4.2761 3.1236 2.9897 2.9395 2.9580

Margrabe

Note: Margrabe is the result of Margrabe’s closed formula which appear in italic and the other numbers are the approximation solution of our method. m denotes the number of steps in the spatial domain, while l is the number of time steps. The values of the parameters used for these runs are given in Table 1.

We also assume the following price impact form 

3/2

¯ ε(1 − e−β(T −t) ), S  S1  S, λ= 0, otherwise, where ε is a constant price impact coefficient, T − t is time to expiry, β is a decay coefficient, S and S¯ represent, respectively, the lower and upper limit of the stock price within which there is a impact price. We consider S = 60, S¯ = 140, ε = 0.01 and β = 100 for the subsequent numerical analysis. Choosing a different value for β, S and S¯ will change the magnitude of the subsequent results, however, the main qualitative results remain valid. At maturity T , on the line x + y = K, the BS gamma ∂ 2 V BS /∂x 2 will blow up. However the above choice of λ guarantees that at maturity λ(∂ 2 V BS /∂x 2 ) = 0. Convergence of numerical results. For the investigation of the numerical scheme, since the PDE (4) with λ = 0 is the standard Black–Scholes model, we can compare the numerical results for λ = 0 with the Margrabe’s closed formula while k = 0. We fix the values of the parameters of the marginal dynamical equations according to Table 1, and vary the values of the correlation coefficient ρ. Results of this convergence study are summarized in Table 2. In comparison of the efficiency and accuracy, we can see from the table that the agreement is excellent. We plot the absolute error between our approximation and Margrabe’s closed formula against the correlation in Figure 1. The numerical value of call Spread option in illiquid market is stated in Table 3. The values of the parameters used for these runs are given in Table 1, with different strike price.

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A. Shidfar et al.

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

120 110

120 100

110 100

90

90

80

80 70

70

Figure 1. Absolute errors between our approximation and Margrabe’s closed formula. Data are in given in Table 1 with ρ = 0.7, T = 0.7 year, m = 50 and l = 100.

Table 3. The values of a 0.4 year European call Spread option based on different correlation, and strike price structure.

ρ = 0.1 Excess price ρ = 0.5 Excess price ρ = 0.7 Excess price ρ = 0.9 Excess price

k = −15

k = −5

k = −2

k=0

k=2

k=5

k = 10

k = 20

15.0923 −0.0003 14.7990 −0.0001 14.7084 0.0000 14.6832 0.0000

7.1590 −0.0004 6.2962 −0.0001 5.7964 0.0000 5.2287 0.0001

5.3265 −0.0004 4.3635 −0.0001 3.7719 0.0000 3.0505 0.0001

4.2927 −0.0003 3.3138 −0.0001 2.7073 0.0000 1.9581 0.0000

3.4018 −0.0002 2.4476 0.0000 1.8631 0.0000 1.1513 0.0000

2.3388 −0.0001 1.4902 0.0000 0.9972 0.0000 0.4375 0.0000

1.1263 0.0000 0.5431 0.0000 0.2589 0.0000 0.0085 0.0000

0.1904 0.0000 0.0426 0.0000 0.0055 0.0000 0.0029 0.0000

Note: Excess price shows the difference in call Spread option from Black–Scholes. The values of the parameters used for these runs are given in Table 1 with m = l = 100.

Replicating cost. Now, we are ready to investigate the effects of the partial price impact (firstorder feedback model) on the replication of Spread option. We plot the time 0 difference between the call price in the first-order feedback model and the corresponding Black–Scholes price against the stock price S1 (0) and S2 (0). The difference between the amount borrowed to replicate a call (in the first-order feedback model and the classical no impact model) at time 0 with expiration date T = 0.1, 0.4 and 1 year, are shown, respectively, in Figures 2–4. The figures indicate that, the Spread option price in the first-order feedback model is less than the classical Spread option price. In other words, the trader can borrow less (for a call) or lend less (for a put) to replicate a call or put Spread option. Excess cost. Figure 5 shows the numerical results from the excess replicating costs above the corresponding Black–Scholes price (obtained using the Peacman–Rachford scheme with m = l = 100) for a call as a function of the strike price (with S1 (t0 ) = 100, S2 (t0 ) = 110, σ1 = 0.15, σ2 = 0.10, r = 0.05, ρ = 0.7, T = 0.4 year). As the option becomes more and more in the money

15

International Journal of Computer Mathematics

Shortage amount over black scholes

x 10–5 0 –1 –2 –3 –4 –5 –6 120 110 100 90 80 70

Stock price 2

70

80

90

100

110

120

Stock price 1

Figure 2. The call price difference (first-order feedback model and classical model) as a function of stock price at time 0 against S1 and S2 . K = 5, σ1 = 0.3, σ2 = 0.2, r = 0.05, ρ = 0.7, T = 0.1 and m = l = 100.

Shortage amount over blak scholes

x 10–4 0 –0.5 –1 –1.5 –2 –2.5 120 110

120 100

110 100

90

90

80 Stock price 2

70

80 70

Stock price1

Figure 3. The call price difference (first-order feedback model and classical model) as a function of stock price at time 0 against S1 and S2 . K = 5, σ1 = 0.3, σ2 = 0.2, r = 0.05, ρ = 0.7, T = 0.4 and m = l = 100.

and out of the money, the excess cost decreases and converges monotonically to zero. However, as the option gets more and more out of the money, the trader needs to buy less stock and eventually, when the option is far in the money and out of the money, the investor does not need to buy any share.

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A. Shidfar et al.

x 10–4 Shortage amount over black scholes

0 –0.5 –1 –1.5 –2 –2.5 –3 120 110 100 90 80 70

Stock price2

70

80

110

100

90

120

Stock price1

Figure 4. The call price difference (first-order feedback model and classical model) as a function of stock price at time 0 against S1 and S2 . K = 5, σ1 = 0.3, σ2 = 0.2, r = 0.05, ρ = 0.7, T = 1 and m = l = 100.

0

x 10–4

Shortage amount over black scholes

–0.2

–0.4

–0.6

–0.8

–1

–1.2 –40

–35

–30

–25

–20

–15 –10 Strike price

–5

0

5

10

Figure 5. The replicating cost difference (first-order feedback model and classical model) against the strike price K. S1 (t0 ) = 100, S2 (t0 ) = 110, σ1 = 0.15, σ2 = 0.10, r = 0.05, ρ = 0.7, T = 0.4 year and m = l = 100.

International Journal of Computer Mathematics

17

5. Conclusion In this work, we have investigated a model which incorporates illiquidity of the underlying asset into the classical multi-asset Black–Scholes–Merton framework. We considered the first-order feedback model in which only a large trader affect the underlying price and the trading strategies of other traders do not influence the price. Since there is no analytical formula for the price of a option within this model, we proposed the partial differential equation approach to price options. We applied a standard ADI method (Peaceman–Rachford scheme) to solve the partial differential equation numerically. We also discussed the stability and the convergence of the numerical scheme. By numerical experiment, we investigated the effects of liquidity on the Spread option pricing in the first-order feedback model. As future research we plan to investigate other schemes (including CS, MCS and HV) and their stability. Finally, we found out that the Spread option price in the market with finite liquidity (firstorder feedback model) is less than the Spread option price in the classical Black–Scholes–Merton framework. Consequently one needs to borrow less (for a call) or lends less (for a put) to replicate a call or put in a first-order feedback model.

Acknowledgements Work supported by NSERC grant 5-36700, SSHRC grant 5-26758 and MITACS grant 5-26761 (T.A.P.).

References [1] J. Back and M. Prokopczuk, Commodity price dynamics and derivatives valuation: A Review, Working paper, Zeppelin University, 2012. Available at SSRN 2133158. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81(3) (1973), pp. 637–654. [3] M.J. Brennan and E.S. Schwartz, Evaluating natural resource investments, J. Bus. 58 (1985), pp. 135–157. [4] G. Cortazar and E.S. Schwartz, A compound option model of production and intermediate inventories, J. Bus. 66(4) (1993), pp. 517–540. [5] G. Cortazar and E.S. Schwartz, The valuation of commodity contingent claims, J. Derivatives 1(4) (1994), pp. 27–39. [6] J.C. Cox and E.S. Schwartz, The pricing of commodity linked bonds, J. Financ. 37(2) (1982), pp. 525–539. [7] I.J.D. Craig and A.D. Sneyd, An alternating-direction implicit scheme for parabolic equations with mixed derivatives, Comput. Math. Appl. 16(4) (1988), pp. 341–350. [8] D.M. Dang, C. Christara, K. Jackson, and A. Lakhany, An efficient numerical PDE approach for pricing foreign exchange interest rate hybrid derivatives, preprint (2012). Available at SSRN 2028519. [9] D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, Princeton, 2010. [10] D.J. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley & Sons, Chichester, 2006. [11] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti della Accademia Nazionale dei Lincei. Memorie. Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian) 5 (1) (1956), pp. 1–30. [12] R. Frey, Perfect option hedging for a large trader, Financ. Stoch. 2(2) (1998), pp. 115–141. [13] R. Gibson and E.S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, J. Financ. 45(3) (1990), pp. 959–976. [14] P.B. Girma and A.S. Paulson, Seasonality in petroleum futures spreads, J. Futures Markets 18(5) (1998), pp. 581–598. [15] P.B. Girma and A.S. Paulson, Risk arbitrage opportunities in petroleum futures spreads, J. Futures Markets 19(8) (1999), pp. 931–955. [16] K.J. Glover, P.W. Duck, and D.P. Newton, On nonlinear models of markets with finite liquidity: Some cautionary notes, SIAM J. Appl. Math. 70(8) (2010), pp. 3252–3271. [17] T. Haentjens, ADI finite difference schemes for the Heston–Hull–White PDE. No. 1111.4087, preprint (2011). Available at arXiv.org. [18] W. Hundsdorfer, Accuracy and stability of splitting with stabilizing corrections, Appl. Numer. Math. 42(1) (2002), pp. 213–233. [19] W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-dependent Advection–Diffusion-Reaction Equations, Vol. 33, Springer, Berlin, 2003. [20] K.J. In’t Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Model. 7(2) (2010), pp. 303–320.

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[21] L. Jódar and J.-R. Pintos, A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets, Math. Comput. Simul. 82(10) (2012), pp. 1972–1985. [22] R.L. Johnson, C.R. Zulauf, S.H. Irwin, and M.E. Gerlow, The soybean complex spread: An examination of market efficiency from the viewpoint of a production process, J. Futures Markets 11(1) (1991), pp. 25–37. [23] E. Kirk, Correlation in the energy markets, in Managing Energy Price Risk, Risk Publications and Enron, London, 1995, pp. 71–78. [24] A. Lipton, Mathematical Methods for Foreign Exchange: A Financial Engineer’s Approach, World Scientific Publishing Company Incorporated, Singapore, 2001. [25] H. Liu and J. Yong, Option pricing with an illiquid underlying asset market, J. Econ. Dyn. Control 29(12) (2005), pp. 2125–2156. [26] W. Margrabe, The value of an option to exchange one asset for another, J. Financ. 33(1) (1978), pp. 177–186. [27] S. McKee and A.R. Mitchell, Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative, Comput. J. 13(1) (1970), pp. 81–86. [28] S. McKee, D.P. Wall, and S.K. Wilson, An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms, J. Comput. Phys. 126(1) (1996), pp. 64–76. [29] New York Mercantile Exchange, Crack Spread Handbook, New York Mercantile Exchange, 2011. Available at partners.futuresource.com/marketcenter/pdfs/crack.pdf. [30] J.L. Paddock, D.R. Siegel, and J.L. Smith, Option valuation of claims on real assets: The case of offshore petroleum leases, Quart. J. Econ. 103(3) (1988), pp. 479–508. [31] R. Paschke and M. Prokopczuk, Commodity derivatives valuation with autoregressive and moving average components in the price dynamics, J. Bank. Financ. 34(11) (2010), pp. 2742–2752. [32] D.W. Peaceman and H.H. Rachford Jr, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3(1) (1955), pp. 28–41. [33] L. Pospisil and J. Vecer, PDE methods for the maximum drawdown, J. Comput. Financ. 12(2) (2008), pp. 59–76. [34] C. Randall, PDE Techniques for Pricing Derivatives with Exotic Path Dependencies or Exotic Processes, Lecture Notes, Workshop CANdiensten, Amsterdam, 2002. [35] E.S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, J. Financ. 52(3) (1997), pp. 923–973. [36] E. Schwartz and J.E. Smith, Short-term variations and long-term dynamics in commodity prices, Manage. Sci. 46(7) (2000), pp. 893–911. [37] A. Shidfar, Kh. Paryab, A.R. Yazdanian, and T.A. Pirvu, Numerical analysis for Spread option pricing model in illiquid underlying asset market: full feedback model (under review). [38] J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, Philadelphia, PA, 2007. [39] B.D. Welfert, Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms, Appl. Numer. Math. 57(1) (2007), pp. 19–35. [40] B.D. Welfert, Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms, Appl. Numer. Math. 59(3) (2009), pp. 677–692. [41] P. Wilmott and P.J. Schönbucher, The feedback effect of hedging in illiquid markets, SIAM J. Appl. Math. 61(1) (2000), pp. 232–272. [42] T.P. Witelski and M. Bowen, ADI schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math. 45(2) (2003), pp. 331–351.

Stochastics: An International Journal of Probability and Stochastic Processes, 2013 Vol. 85, No. 5, 917–927, http://dx.doi.org/10.1080/17442508.2012.673616

On dependence of volatility on return for stochastic volatility models Mikhail Martynov and Olga Rozanova* Moscow State University, Leninskie Gory, Main Building, 199991 Moscow, Russia (Received 27 July 2011; final version received 5 March 2012) We study the dependence of volatility on the stock price in the stochastic volatility framework on the example of the Heston model. To be more specific, we consider the conditional expectation of stochastic variance (square of volatility) given stock price log-return f as a function of f and time t. The behaviour of this function depends on the initial stock price log-return distribution density. In particular, we show that the graph of the conditional expectation of the stochastic variance is convex downwards near a special value of the stock price log-return f * ðtÞ. For the Gaussian distribution, this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down. Keywords: stochastic volatility; the Heston model; conditional expectation of stochastic variance AMS Subject Classification: 91B24; 60H10

1.

Introduction

Stochastic volatility (SV) models are quite popular in the field of mathematical finance in recent decades due to a need for reliable quantitative analysis of market data. They are generally used to evaluate derivative securities, such as options. The name derives from the models’ treatment of the volatility of security as a random process. The most popular models are the Heston [13], Stein – Stein [25], Scho¨ble – Zhu [23], Hull – White [14] and Scott [24] models. We refer for reviews to Refs [9,18,19]. The main reason for introducing the SV models is to find a realistic alternative approach to option pricing to capture the timevarying nature of the volatility, assumed to be constant in the Black – Scholes approach. Nevertheless, SV models can be used for the investigation of other properties of financial markets. For example, in Ref. [6] the time-dependent probability distribution of stock price log-returns was studied for the Heston model [13]. In this model, the square of the stock-price volatility, called the stochastic variance, follows a random process known in financial literature as the Cox – Ingersoll – Ross (CIR) process and in mathematical statistics as the square root process. While returns are readily known from financial data, variance is not given directly, so it acts as a hidden stochastic variable. In Ref. [6] the joint probability density function of log-return and stochastic variance was found, then the integration over variance was performed and the probability distribution function of log-return unconditional on variance was obtained. The latter PDF can be directly compared with the Dow-Jones data for the 20-year period of 1982 – 2001 and an

*Corresponding author. Email: [email protected] q 2013 Taylor & Francis

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M. Martynov and O. Rozanova

excellent agreement was found. The tails of the PDF decay slower than what the lognormal distribution predicts (the so-called fat-tails effect). Technically our paper is connected with Ref. [6]. However, we study the dependence of the stochastic variance on fixed log-return, thus, we estimate hidden stochastic variable through the variable that can be easily obtained from financial data. The result strongly depends on initial distribution of log-return and stochastic variance. It is natural that the distributions change their shape with time. In particular, we show that for Gaussian initial distribution of log-return, the plot of the conditional expectation of stochastic variance given return demonstrates the convexity downwards near some special value of log-return; this value depends on time and parameters of the model. 2.

General formulae for the conditional expectation and variance given return

Let us consider the stochastic differential equation system: dF t ¼ Adt þ s dW 1 ; V 0 ¼ v;

dV t ¼ Bdt þ ldW 2 ;

t $ 0; f [ R; v [ R;

F0 ¼ f ;

ð1Þ

where WðtÞ ¼ ðW 1 ðtÞ; W 2 ðtÞÞ is a two-dimensional standard Wiener process, A ¼ Aðt; F t ; V t Þ; B ¼ Bðt; F t ; V t Þ; s ¼ s ðt; F t ; V t Þ; l ¼ lðt; F t ; V t Þ are prescribed functions. The joint probability density Pðt; f ; vÞ of random values F t and V t obeys the Fokker – Plank Equation (e.g. [22])

›P › › 1 ›2 1 ›2 2 ¼ 2 ðAPÞ 2 ðBPÞ þ ðs 2PÞ þ ðl PÞ 2 ›t ›f ›v 2 ›f 2 ›v 2

ð2Þ

with initial condition Pð0; f ; vÞ ¼ P0 ðf ; vÞ;

ð3Þ

determined by initial distributions of F t and V t . If Pðt; f ; vÞ is known, one can find EðV t jF t ¼ f Þ, which is the conditional expectation of V t given F t . This value can be found by the following formula (see, [5]): Ð ð2L;LÞ vPðt; f ; vÞdv : ð4Þ EðV t jF t ¼ f Þ ¼ lim Ð L!þ1 ð2L;LÞ Pðt; f ; vÞdv Let us also define the variance of V t given F t as Ð 2 ð2L;LÞ v Pðt; f ; vÞdv VarðV t jF t ¼ f Þ ¼ lim Ð 2 E 2 ðV t jF t ¼ f Þ: L!þ1 ð2L;LÞ Pðt; f ; vÞdv

ð5Þ

In both these formulae, the improper integrals in the numerator and denominator are assumed to converge. The assumption imposes a restriction on the coefficients A; B; s ; l. Functions EðV t jF t ¼ f Þ and VarðV t jF t ¼ f Þ both depend on f and t. Note that if we choose P0 ðf ; vÞ ¼ dðv 2 v0 ðf ÞÞgðf Þ, where v0 ðf Þ and gðf Þ are arbitrary smooth functions, then EðV t jF t ¼ f Þjt¼0 ¼ v0 ðf Þ. In particular, gðf Þ can be considered as initial distribution of the random value F t . For some classes of systems (1), the conditional expectation EðV t jF t ¼ f Þ was found in Refs [2,3,22] within an absolutely different context.

Stochastics: An International Journal of Probability and Stochastic Processes 919 Let us remark that sometimes it is easier to find the Fourier transform of Pðt; f ; vÞ function with respect to f and v than the function itself. We will get formula allowing to express EðV t jF t ¼ f Þ in terms of Fourier transform of Pðt; f ; vÞ and will apply it for finding the conditional expectation of stochastic variance of the stock price, which depends on known log-return. ^ m; jÞ be the Fourier transform of function Pðt; f ; vÞ with respect Proposition 2.1. Let Pðt; to variables f and v, which is the solution of problem (2), (3), and all integrals from (4) and ^ m; 0Þ and ›j Pðt; ^ m; 0Þ are decreasing in m at infinity faster (5) comverge. Assume that Pðt; than any power. Then, functions EðV t jF t ¼ f Þ and VarðV t jF t ¼ f Þ determined by (4) and (5) can be found as EðV t jF t ¼ f Þ ¼ VarðV t jF t ¼ f Þ ¼

^ iF21 m ½›j Pðt; m; 0ފ ; ^ m; 0ފ F21 ½Pðt;

t $ 0; f [ R;

m

^ m; 0ފÞ2 2 F21 ½›2 Pðt; ^ m; 0ފF21 ½Pðt; ^ m; 0ފ ðFm21 ½›j Pðt; m j m ; 21 ^ 2 ðF ½Pðt; m; 0ÞŠÞ m

ð6Þ ð7Þ

21 where F21 m and Fj mean the inverse Fourier transforms over m and j, respectively. The proof is a simple exercise in the Fourier analysis.

3.

Example: the Heston model

Of course, there is no explicit formula for the joint probability density function Pðt; f ; vÞ for arbitrary system (1). We will consider a particular, but important, case of the Heston model [13]:
 pffiffiffiffiffi Vt dt þ V t dW 1 ; dF t ¼ a 2 ð8Þ 2 dV t ¼ 2gðV t 2 uÞdt þ k

pffiffiffiffiffi V t dW 2 :

ð9Þ

Here a, u, g, k are constants: a is the rate of return of the asset, u . 0 is the long variance, or long run average price variance (as t tends to infinity, the expected value of V t tends to u), g . 0 is the rate at which V t reverts to u, k is the ‘vol of vol’ (volatility of the volatility), this determines the variance of V t . Equation (9) describes the CIR process (known in mathematical statistics as the Feller or square root process) [8,9]. In Ref. [8] it is shown that this equation has a non-negative solution for t [ ½0; þ1Þ when 2gu . k 2 . The first equation describes a log-return F t on the stock price, in assumption that the stock price itself obeys a geometric Brownian motion with volatility s t . The second equation describes the square of volatility s 2t ¼ V t . The Fokker –Planck Equation (2) for the joint density function Pðt; f ; vÞ of log-return F t and stochastic variance V t takes here the following form:  ›Pðt; f ; vÞ ›Pðt; f ; vÞ ›Pðt; f ; vÞ v 2a ¼ gPðt; f ; vÞ þ ðgðv 2 uÞ þ k 2 Þ þ 2 ›t ›v ›f k 2v ›2Pðt; f ; vÞ v ›2Pðt; f ; vÞ þ : þ ›v 2 2 2 ›f 2

ð10Þ

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M. Martynov and O. Rozanova

Now, we can choose different initial distributions for log-return and stochastic variance. Note that it is natural to assume that initially the variance does not depend on return. ^ m; jÞ, the Fourier transform of P with respect to f and v, satisfies the The function Pðt; equation ^ m ; jÞ 1 ^ m; jÞ ›Pðt; ›Pðt; ^ m; jÞ ¼ 0: þ ðm þ im 2 þ 2gj þ ik 2j 2 Þ þ iðguj þ maÞPðt; ›t ›j 2 The first-order PDE (11) can be integrated, the solution has the following form:
2 2ðgu=k 2 Þ 2 2 2 ^ m; jÞ ¼ eðð2imak 2 þg 2uÞtÞ=k 2 k ðið2gj þ mÞ 2 m 2 k j Þ Pðt; q
pffiffiffi  2i arctanðð2k 2j þ igÞ= qÞ ; pffiffiffi *G m; 2t þ q

ð11Þ

ð12Þ ð13Þ

where q ¼ 2ik 2m þ k 2m 2 þ g 2 , G is an arbitrary differentiable function of two variables.

3.1

The uniform initial distribution of log-return

In what follows, we will assume that the mean value of initial log-return is zero, since we can apply a linear shift with respect to F t in (8). We begin with the simplest and almost trivial case. Let us assume that initially the logreturn is distributed uniformly in the interval ð2L; LÞ, (L ¼ const . 0), and volatility is equal to some constant a $ 0. Then, the initial joint density distribution of Ft and Vt is Pð0; f ; vÞ ¼

1 dðv 2 aÞ: 2L

ð14Þ

To simplify further calculations, we will exclude randomness for t ¼ 0, i.e. we will assume a ¼ 0. The respective initial condition for the Fourier transform is ^ m; jÞ ¼ p dðmÞ: Pð0; ð15Þ L The solution of problem (11), (15) takes the form
ððgutÞ=k 2 Þ 4g 2 e2gt ^Pðt; m; jÞ ¼ p dðmÞ : L ð2gegt þ ik 2jðegt 2 1Þ2 It is easy to calculate that ^ m; 0Þ ¼ p dðmÞ; Pðt; L

^ m; 0Þ ¼ ›j Pðt;

p dðmÞiuðe 2gt 2 1Þ: L

ð16Þ

ð17Þ

Finally from (6) and (7), we get EðV t jF t ¼ f Þ ¼ uð1 2 e2gt Þ;

VarðV t jF t ¼ f Þ ¼

uk 2 ð1 2 e2gt Þ2 : 2g

ð18Þ

ð19Þ

Stochastics: An International Journal of Probability and Stochastic Processes 921 It is evident that here there is no dependence on f and the result is the same as we could obtain from the calculation of mathematical expectation and variance of V t from Equation (9). 3.2 The Gaussian initial distribution of log-return Let us assume that initial distribution gðf Þ of log-return is Gaussian law with standard deviation m; m . 0. Then, we have the following initial condition: 1 2 2 Pð0; f ; vÞ ¼ pffiffiffiffi e2ðf =m ÞdðvÞ: m p

ð20Þ

When a ¼ 0, the Fourier transform of initial data (3) with respect to f and v is ^ m; jÞ ¼ e2ððm 2 m2 Þ=4Þ . Pð0; Solution of problem (11), (20) takes the form:

mðm 2 iÞ þ ðg 2 =k 2 Þ 2 m þ k 2g 2 2 ið2gj þ mÞ

gu=k 2



 m 2m 2 g 2u 2 ami 2 2 t * exp 2 4 k !!!2ðð2guÞ=k 2 Þ t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 2j þ ig 2cosh : ð21Þ k 2mðm 2 iÞ þ g 2 2 i arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2mðm 2 iÞ þ g 2


pffiffiffiffi ^ m; jÞ ¼ m p 2 Pðt;

^ m; jÞ exponentially decreases with respect to m. That is why we can We see that Pðt; use formula (6) and obtain (after cumbersome transformations) the following integral expression: Ð Fðt; m; f Þdm Ð ; ð22Þ EðV t jF t ¼ f Þ ¼ 2gu R m; f Þdm Cðt; R where




l ðl coshððltÞ=4Þ þ 2g sinhððltÞ=4ÞÞ sinhððltÞ=4Þ ; Fðt; m; f Þ ¼ Cðt; m; f Þ ðl coshððltÞ=4Þ þ 2g sinhððltÞ=4ÞÞ l 2 ¼ k 2 ð4m 2 þ 1Þ þ 4g 2 :

Cðt; m; f Þ ¼ eð2m

2

ðm 2 2imð4f 24ta21ÞÞ=4

ðð2guÞ=k 2 Þ

;

Let us remark that if a – 0, we can also get a similar formula, but it will be more cumbersome. The limit case as the standard deviation of the Gaussian distribution m ! 0 for (20) is Pð0; f ; vÞ ¼ dðf ÞdðvÞ:

ð23Þ

For this case, formula (22) modifies as follows: the exponential factor in the expression for C takes the form eimðf 2taÞ . 3.3 ‘Fat-tails’ initial distribution of log-return Integral formula, analogous to (22), can be obtained for initial distributions intermediate between uniform and Gaussian ones. For example, as initial distribution of the log-return gðf Þ; we can take the Pearson type VII distribution [1] (so called ‘fat-tails’ power

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M. Martynov and O. Rozanova

distribution). Thus, we get

Pð0; f ; vÞ ¼


2q 1 f2 1þ 2 dðvÞ; lBðq 2 ð1=2Þ; 1=2Þ l

l . 0; q . 0;

^ m; jÞ can be where B is the Beta function. Exact formula for the Fourier transform Pðt; ^ m; jÞ decays as jmj ! 1 sufficiently found for q ¼ 1=2; n; n [ N. For all these cases, Pðt; fast and Proposition 2.1 can be applied for the calculation of EðV t jF t ¼ f Þ. 2 2 For example, for q ¼ 21 the difference with (21) is only in the multiplier e2ððm m Þ=4Þ : it should be changed to ðe2am 2 elm ÞHðmÞ þ elm ; with the Heaviside function H. 3.4 Convexity downward of the graph of conditional expectation of SV and asymptotic behaviour for small time It turns out that if in the Heston model the conditional expectation of SV is considered as a function of the log-return, we will observe a deflection of the plot. The effect appears in numerical calculation of both integrals in (22) with the use of standard algorithms. The numerical calculation of the integrals over an infinite interval is based on the QUADPACK routine QAGI [21], where the entire infinite integration range is first transformed to the segment ½0; 1Š. For example, Figure 1 (left) presents the graph of function Vðt; f Þ :¼ EðV t jF t ¼ f Þ at three consequent moments of time for the following values of parameters: g ¼ 1; k ¼ 1; u ¼ 1; a ¼ 1; m ¼ 1. This behaviour of the volatility plot can be studied by analytical methods as well. Indeed, let us fix log-return f. Then from (22) by expansion of integrand functions into formal series as t ! 0 up to the forth component and by further term-wise integration

0.8

3

0.08

2

0.07

0.6 3

0.06

0.4 2 0.2 –4.5 –3 –1.5

0.05

1 0

1.5

3

4.5

1 –4.5 –3

–1.5

0

1.5

3

4.5

Figure 1. Dependence of Vðt; f Þ (vertical axis) on log-returns f (horizontal axis) at three consequent moments of time (t ¼ 0.06, 0.12, 0.18); Gaussian distribution (left) and power-law (Cauchy) distribution (right).

Stochastics: An International Journal of Probability and Stochastic Processes 923 (series converge at least for small f and large m), we will get that 1 1 Vðt; f Þ ¼ gut 2 g 2ut 2 þ guðg 2 þ 2f 2m 24k 2 2 fm 22k 2 2 m 22k 2 Þt 3 2 6 1 2 2 24 2 guð8gk f m 2 4ðg þ 4m 22aÞk 2m 22f 2 4ðg þ aÞm 22k 2 þ g 3 Þt 4 þ Oðt 5 Þ: 6 ð24Þ Let us justify a possibility to expand Vðt; xÞ into the Taylor series. We should prove that both integrals in the numerator and denominator of (22) can be differentiated with respect to t. Indeed, let t [ Vt ¼ ð2t; tÞ; 0 , t , 1. It can be readily shown that both integrands in (22), F and C, are continuous with respect to m and t on R £ Vt , the derivatives of any order ›nt F, ›nt C, n ¼ 0; 1; . . . are also continuous on R £ Vt . Moreover, j›nt Fj, j›nt Cj can 2c2 m 2 , with positive constants c1 and c2. Therefore, be from Ð estimated Ð nabove by c1
e n › m and › m converge uniformly on Vt . Thus, according to the classical Fd Cd R t R t theorem of calculus the numerator and denominator in (22) can be differentiated on Vt under the integral sign. Since for ð ð ›nt Qðt; m; f Þdmjt¼0 ¼ ›nt Qðt; m; f Þjt¼0 dm; R

R

Ð where Q ¼ C or F, the Taylor coefficients in the expansion of R Qðt; m; f Þdm can be obtained by integration of the respective coefficient of Qðt; m; f Þ of the Taylor series in t with respect to m. The latter integrals can be explicitly calculated. This gives expansion (24). Hence for t ! 0 we find that (up to Oðt 5 Þ)
 1 2 1 3 24 2 2 22 Vðt; f Þ , gut ð1 2 gtÞm k f 2 am t þ ð1 2 gtÞ t 3gum 22k 2f 3 3 6 1 1 1 1 þ ðgt 2 ð1 2 atÞÞugt 3m 22k 2 þ gut 2 g 2ut 2 þ g 3ut 3 2 g 4ut 4 6 2 6 24 is a quadratic trinomial over f with a minimum in point f * ðtÞ ¼

4at þ ð1 2 gtÞm 2 ; 4ð1 2 gtÞ

for t . 0. As m ! 0, the value f * ðtÞ tends to the mean value of the log-return distribution. The effect holds for ‘fat-tails’ power-law initial distributions as well. Nevertheless, this effect weakens as the decay of the distribution at infinity becomes slower. Figure 1 (right) presents the function Vðt; f Þ for three consequent moments of time for the initial standard Cauchy distribution of log-return gðf Þ given by formula gðf Þ ¼

1 1 : p1 þ f 2

The values of parameters are g ¼ 10; k ¼ 1; u ¼ 0:1; a ¼ 10. It seems that the curves are straight lines, but the analysis of numerical values shows that the deflection still persists near the mean value of return. Acting as in the case of the Gaussian initial distribution, one can find the Taylor expansion of Vðt; f Þ as t ! 0, 1 R4 ðf ; g; kÞ 3 R8 ðf ; g; k; aÞ 4 t þ g 2u t þ Oðt 5 Þ; Vðt; f Þ ¼ gut 2 g 2ut 2 2 gu 2 R6 ðf Þ R4 ðf Þ

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M. Martynov and O. Rozanova

where we denote by Rk a polynomial of order k with respect to f. We do not write down these polynomial, let us only note that
 R4 ðf ; g; kÞ 1 k2 , 2 2 g2 8 R6 ðf Þ 2f and R8 ðf ; g; k; aÞ 16f 4 , ð2g 2 2 k 2 Þ 3 R4 ðf Þ as j f j ! 1. It is very interesting to study the asymptotic behaviour of Vðt; f Þ as j f j ! 1 and t ! 1. We do not dwell here on this quite delicate question at all and reserve it for future research. Some hints can be found in Refs [6,11,12]. 3.5 Modifications of the Heston model Let us analyse the situation when the coefficient g from Equation (9) (the rate of relaxation of the mean) depends on time. For some interesting cases of this dependence, one can find the Fourier transform of Pðt; f ; vÞ and formula for EðV t jF t ¼ f Þ. For example, if we set g ¼ 1=ðT 2 tÞ, then we get a Brownian bridge-like Equation (see, [20] describing square of volatility behaviour with start at v0 ¼ a . 0 and end at vT ¼ b $ 0). Here, the solution will be represented in terms of integrals of Bessel functions and the solution is cumbersome. It may seem that the described approach, which helps to find the conditional expectation of SV given fixed log-return in the Heston model, can be successfully applied in other variations of this model. This is true when initial log-return has a uniform distribution. However, this situation is trivial, because the answer does not contain f and is equal to the expectation of SV obtained from the second equation of model. In the case of non-uniform initial distribution of log-return (for instance, Gaussian), formula (6) may be ^ m; jÞ can be found. The cause is that non-applicable, even when explicit expression for Pðt; ^Pðt; m; jÞ increases as jmj ! 1. For example, if we replace Equation (9) with dV t ¼ 2gðV t 2 uÞdt þ kdW 2 ;

g; u; k . 0;

ð25Þ

(the mean-reverting Ornstein – Uhlenbeck process) subject to initial data (20), a ¼ 0, we will get
2
 
 2 2 2 pffiffiffiffi ^ m; jÞ ¼ m pexp k t m 4 2 i k t m 3 2 ut þ k t þ m m 2 þ i u 2 a tm ; Pðt; 2 4 8g 2 4g 2 2 8g 2

whence it follows that the coefficient of m 4 in exponent power is positive when t is positive. This means that integrals from (6) are divergent. 4.

Possible application

As we can see from Section 3.4, in the framework of the Heston model the asset becomes risky if the value of its log-return diverges from some specific value of log-return f * ðtÞ and this effect intensifies with time. Moreover, the effect strongly depends on the rate of decay of the initial distribution of log-return: for distributions decaying fast the dependence of volatility on deviation of the log-return deepens.

Stochastics: An International Journal of Probability and Stochastic Processes 925 40

1

100

1

2 2

20

50 3

3 0

0 –4.5

–3

–1.5

1.5

3

–20

4.5

–4.5

–3

–1.5

1.5

3

4.5

–50

–100

Figure 2. Dependence of Rðt; f Þ (vertical axis) on log-returns f (horizontal axis) at three consequent moments of time (t ¼ 0.06, 0.12, 0.18); Gaussian distribution (left) and power-law (Cauchy) distribution (right).

Based on our results, one can introduce a rule for estimation of the company’s rating based on stock prices. The natural presumption is that company’s rating increases when return on assets increases and volatility decreases. Hence for estimation of the company’s rating one can use (very rough) index Rðt; f Þ ¼ f =Vðt; f Þ, where Vðt; f Þ ¼ EðV t jF t ¼ f Þ is calculated by formula (22). Figure 2 shows the plot of function Rðt; f Þ for three consequent time points for Gaussian (left) and power-law Cauchy (right) distributions, respectively. Parameters are as in Figure 1. We can see that in the Gaussian case the index does not rise monotonically with return. We can see that for strongly decaying initial distribution (Gaussian in our case), there exists a value of log return [corresponding to the maximum point at Figure 2 (left)], such that the rating of company is maximal. Further rise of return does not imply an increase of rating since the assets become too risky. For ‘fat-tail’ distributions this effect is feebly marked. 5.

Conclusion and further work

In this article, we studied a relation of SV with stock price data in the frame of the Heston model. This problem has been solved by calculation of the conditional expectation of SV given log-return of a stock price under the supplementary condition on initial distribution of log-return and SV. Namely, different cases of initial distributions of log-return have been studied: uniform, Gaussian and power-law ‘fat-tails’ distributions, intermediate between them. We revealed that the graph of the conditional expectation of SV considered as a function of the stock price log-return is convex downwards near a specific value of log-return for the Gaussian initial distribution and for certain distributions decreasing at infinity slower than the Gaussian distribution (for which we succeed to find the Fourier transform of the joint probability density of log-return and stochastic variance explicitly). For the Gaussian initial distribution of log-return this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down. Let us note that our formulae can be obtained in a different way, using the well-known expression for the joint characteristic function of the log-return and the stochastic variance in the Heston model [10,13] (in the correlated case). This expression was obtained exploiting the linearity of the coefficients in the respective PDE, in other words, the fact

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that the Heston model is affine [7]. Nevertheless, this way is not convenient for our purpose, since it requires an additional integration. Formulae for the conditional expectation of stochastic variance given log-return EðV t jF t ¼ f Þ are obtained in this study in the integral form, we compute the integrals numerically using standard algorithms and study asymptotics of the formulae for small time. The questions on analysis of the formulae for larger t and f and on the asymptotics of EðV t jF t ¼ f Þ as j f j ! 1 and t ! 1 are open. Moreover, the dependence of the stochastic variance on the properties of the initial distribution of returns has to be studied in general case, not only for separate examples, as it was done here. In this study, we concentrated on the function EðV t jF t ¼ f Þ and computed the conditional variance VarðV t jF t ¼ f Þ for the simplest case only. Nevertheless, the latter function can also be investigated by means of formula (7), though the computations are rather cumbersome. Furthermore, the conditional expectation of the log-return given the stochastic variance also is an interesting object of study. A similar problem was considered in Refs [15 – 17] in connection with a risk-sensitive portfolio selection problem. Namely, we studied a couple of Equations (1), where Ft describes the capital of portfolio, depending on the investment strategy h, and Vt is a market factor, for example, the interest rate. The problem consists in optimization of a functional EðF t jV t ¼ vÞ2 gVarðF t jV t ¼ vÞ, where g . 0 is a risk-aversion parameter, over the class of all admissible investment processes h. In Refs [15,16], the author deals with a Vasicek-type interest rate that obeys (25), whereas in Ref. [17] the CIR model of the interest rate was considered (see [4] for different models of evolution of interest rates). The latter model corresponds to (9). In the case of uniform distribution of the initial interest rate on the semi-axis v . 0, we succeed to obtain an explicit expression for EðF t jV t ¼ vÞ and VarðF t jV t ¼ vÞ, valid for all t $ 0. For other initial distributions of the initial interest rate (it is natural to consider the Gamma distribution [1]) seemingly it remains to be satisfied with an integral formula, moreover, there arises a restriction on the time t from above. Acknowledgements The authors thank heartily anonymous referees for their suggestions which helped to improve the paper significantly. This study was supported by the Ministry of Education of the Russian Federation, project 2.1.1/1399.

References [1] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, M. Abramowitz and I.A. Stegun, eds., Dover Publications, Inc., New York, 1992. [2] S. Albeverio and O. Rozanova, The non-viscous Burgers equation associated with random positions in coordinate space: A threshold for blow up behavior, Math. Models Methods Appl. Sci. 19 (2009), pp. 1 – 19. [3] S. Albeverio and O. Rozanova, Suppression of unbounded gradients in a SDE associated with the Burgers equation, Proc. Am. Math. Soc. 138 (2010), pp. 241 – 251. [4] A.J.G. Cairns, Interest Rate Models – An Introduction, Princeton University Press, Princeton, 2004. [5] A.J. Chorin and O.H. Hald, Stochastic Tools in Mathematics and Science, Springer, New York, 2006. [6] A.A. Dragulescu and V.M. Yakovenko, Probability distribution of returns in the Heston model with stochastic volatility, Quant. Finance 2 (2002), pp. 443 – 453. [7] D. Duffie, D. Filipovic´, and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Prob. 13 (2003), pp. 984 – 1053.

Stochastics: An International Journal of Probability and Stochastic Processes 927 [8] W. Feller, Two singular diffusion problems, Ann. Math. 54 (1951), pp. 173 – 182. [9] J.P. Fouque, G. Papanicolaou, and K.R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000. [10] J. Gatheral, The Volatility Surface, Wiley, Hoboken, NJ, 2006. [11] A. Gulisashvili and E.M. Stein, Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: The Hull-White model, C. R. Acad. Sci. Paris, Ser. I 343 (2006), pp. 519 – 523. [12] A. Gulisashvili and E.M. Stein, Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models, Math. Finance 30 (2010), pp. 447 – 477. [13] S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), pp. 327 – 343. [14] J. Hull and A. White, The pricing of options on asset with stochastic volatilities, J. Finance 42 (1987), pp. 281 – 300. [15] G.S. Kambarbaeva, Some explicit formulas for calculation of conditional mathematical expectations of random variables and their applications, Moscow Univ. Math. Bull. 65 (2010), pp. 186 –191. [16] G.S. Kambarbaeva, Composition of an efficient portfolio in the Bielecki and Pliska market model, Moscow Univ. Math. Bull. 66 (2011), pp. 197 – 204. [17] G.S. Kambarbaeva and O.S. Rozanova, On efficient portfolio depending on the Cox –Ingersoll – Ross interest rate, Moscow Univ. Math. Bull. (to appear). [18] S. Micciche`, G. Bonanno, F. Lillo, and R.N. Mantegna, Volatility in financial markets: Stochastic models and empirical results, Phys. A 314 (2002), pp. 756 – 761. [19] S. Mitra, A review of volatility and option pricing, Available at http://arxiv.org/pdf/0904.1392 [20] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer, Heidelberg, 2002. ¨ berhuber, and D. Kahaner, QUADPACK, [21] R. Piessens, E. de Doncker-Kapenga, C. U A Subroutine Package for Automatic Integration, Springer-Verlag, Berlin, 1983. [22] H. Risken, The Fokker – Planck Equation. Methods of Solution and Applications, 2nd ed., Springer, New York, 1989. [23] R. Scho¨bel and J. Zhu, Stochastic volatility with an Ornstein – Uhlenbeck process: An extension, Eur. Finance Rev. 4 (1999), pp. 23– 46. [24] L. Scott, Option pricing when the variance changes randomly: Theory, estimaton and an applications, J. Financial Quant. Anal. 22 (1987), pp. 419 –438. [25] E.M. Stein and J.C. Stein, Stock price distributions with stochastic volatility: An analytic approach, Rev. Financ. Stud. 4 (1991), pp. 727 –752.

Scandinavian Actuarial Journal, 2014 http://dx.doi.org/10.1080/03461238.2013.876927

On risk charges and shadow account options in pension funds PETER LØCHTE JØRGENSEN∗ † and NADINE GATZERT‡ †Department of Economics and Business, Aarhus University, Aarhus, Denmark ‡Department of Insurance Economics and Risk Management, Friedrich-Alexander-University Erlangen-Nürnberg (FAU), Nürnberg, Germany (Accepted December 2013) This paper studies the economic implications of regulatory systems which allow equityholders of pension companies to not only charge a specific premium to compensate them for their higher risk (compared to policyholders), but also to accumulate these risk charges in a so-called shadow account in years when they are not immediately payable due to e.g. poor investment results. When surpluses are subsequently reestablished, clearance of the shadow account balance takes priority over bonus/participation transfers to policyholders. We see such a regulatory accounting rule as a valuable option to equityholders and our paper develops a model in which the influence of risk charges and shadow account options on stakeholders’ value can be quantified and studied. Our numerical results show that the value of shadow account options can be significant and thus come at the risk of expropriating policyholder wealth. However, our analysis also shows that this risk can be remedied if proper attention is given to the specific contract design and to the fixing of fair contract parameters at the outset. Keywords: pension contracts; options; guarantees; valuation; risk management JEL classification: G12; G13; G23

1. Introduction When annual surpluses of life and pension (L&P) companies are distributed between the companies’ stakeholders, it is common that equityholders charge a risk premium in return for providing the capital to support the companies’ business and to compensate them for the risk they bear by ensuring policyholders’ claims. The procedure for determining the annual risk premium is often formulated as a simple mathematical rule. It can for example be a function of current surplus, the degree of leverage, and/or the value of liabilities. Whatever the case, the actual premium paid to equity will vary from year to year as the financial situation of the company varies stochastically over time. It may be zero in years when surplus is negative or simply insufficient to reward equity, and it may be correspondingly larger when ample surpluses are reestablished, so that over the long term, equity is fairly compensated for its risk. ∗ Corresponding

author. E-mail: [email protected] © 2013 Taylor & Francis

2

P.L. Jørgensen and N. Gatzert

The present paper is concerned with analyzing some important aspects of surplus distribution schemes in L&P companies stemming from the fact that in some countries, like for example Denmark, the regulatory system allows equity to determine its risk charge1 according to a scheme which yields a strictly positive result in all years. This will be the case, for example if the risk charge is specified as a flat percentage of the nominal value of liabilities. Under such a regime, the calculated risk charge may, however, not always be payable in the current year. In years when company surplus is insufficient to allow for immediate payment of equity’s (positive) risk charge, the regulatory framework then permits company management to park the year’s calculated risk charge in a so-called ‘shadow account’, the balance of which must be disclosed in a note in the annual financial report.2 The idea with the shadow account is that it can be cleared by payment of the amount ‘owed’ to equityholders when surpluses are reestablished at some later point in time. In a legal sense, the shadow account is thus not a liability (it has no claim in case of default), but economically it of course is, and the amounts involved can be significant. In Denmark, the risk charge is the main profit source of commercial L&P companies, but shadow accounts and risk charges are also in operation in mutual companies.3 A typical annual risk charge is in the order of 1% of liabilities, and shadow account balances in some companies exceed DKK 5bn (app. EUR 670mn). At the end of the year 2011, Danish life and pension companies’ shadow account balances totalled almost DKK 25bn (app. EUR 3.4bn) (Andersen & Dyrekilde 2012b). As pension systems and regulatory actions in Scandinavian countries are often considered role models for other countries,4 a detailed analysis of shadow account options is of high relevance for regulatory authorities when assessing whether surplus is fairly distributed and whether stakeholders are adequately compensated for the risk they bear. The inclusion of this option may also have severe implications regarding the attractiveness of private annuities which already exhibit an insufficient demand in some countries, cf. for example Mitchell et al. (1999) and Brown (2001). The purpose of this paper is to explain and analyze the consequences of a regulatory system that permits the operation of shadow accounts as introduced above. We show that permission to operate with a shadow account (or multiple shadow accounts in case of segregated pools of policyholders) is really an option to equityholders which makes equity more valuable and less risky than without it. The flip side of the coin is that shadow account options may seriously expropriate policyholder wealth. The remainder of the paper is organized as follows: In Section 2, we setup a simple and illustrative model to explain the nature of the shadow account option and to be able to analyze the effects of risk charges and shadow account options on equity and policyholder values. 1 We shall refer to equity’s risk premium as the ‘risk charge’ throughout this paper. An alternative term which is sometimes used is ‘risk allowance’. This term emphasizes the fact that the charge is ‘allowed’ by the regulator. 2 See the Appendix for an overview of the legal foundation for the risk charging and shadow account schemes of Danish L&P companies. 3 The operation of shadow accounts enables mutual companies to build up sufficient solvency buffers over time and they also serve a purpose in determining a fair distribution of surpluses to different policyholder risk groups (e.g. with different levels of guaranteed returns) who will each have their own designated shadow accounts. 4 See for example Mercer (2012) where in a global comparison of pension systems, Denmark comes out at the very top ‘achieving the first A-grade result in the history of this research’. Interestingly, and in close relation to the subject studied in this paper, the Mercer-report also concludes that ‘the overall index value (grade) for the Danish system could be increased by providing greater protection of members’ accrued benefits in the case of fraud, mismanagement or provider insolvency.’

Scandinavian Actuarial Journal

3

Section 3 briefly introduces and discusses additional assumptions necessary when using our model for valuation of the different components of the stakeholder claims including the shadow account option. Section 4 implements our model and presents a variety of illustrations and numerical results. By realistic parameterization of the model, we confirm that shadow account options are potentially very valuable to equityholders. Allowing equity to operate with a shadow account will therefore, ceteris paribus, transfer value from policyholders to equity. However, as our numerical analysis of fair contracts shows, the situation may be remedied with proper attention given to the design of the surplus distribution mechanism and in particular to the details of the risk charging scheme and the shadow account option design. Section 5 concludes.

2. Model The analysis of risk charges and shadow account options and their implications for valuation will be performed within a model of a pension company having issued a family of identical traditional guaranteed with-profits policies, i.e. policies with a guaranteed annual minimum return and with a right to receive bonus. To be able to focus clearly on the object of interest here – risk charges and shadow account options – we make several simplifying assumptions regarding for example the pension contracts, the regulatory environment, and the investment universe. The model and its assumptions are described below. We consider a pension company formed at time 0 by the infusion of capital by a group of pension savers and by a group of owners, i.e. equityholders. It is assumed that there are no further contributions from pension savers – i.e. we consider single premium contracts – and that there are no withdrawals (dividends) to equity prior to maturity and liquidation at time T . Considering only the pension savers’ accumulation phase – i.e. the period from contract initiation up until the beginning of the retirement and payout phase – we refrain from modeling mortality and explicit life insurance elements of the pension contract. The sum of the initial investments provided by the two investor groups forms the company’s initial asset base. In return for their investments, each investor group acquires a claim on the company which expires and pays off at time T , and which is to be described in further detail below. We shall henceforth refer to the pension savers and to the managers/owners as liabilityholders and equityholders and to their initial investments as L 0 and E 0 , respectively. The initial balance sheet looks as shown in Figure 1, where α = EA00 and λ = EL 00 = 1−α α are defined as the initial equity and leverage ratios, respectively.5 Upon formation of the pension fund at time 0, the assets will be invested in a well-defined (primarily wrt. volatility) reference portfolio of financial assets, and at the end of each year the (book) value of the entries on the liability side of the balance sheet will be updated according to well-specified rules that will depend on investment results as well as on specific contract details 5 In practice, equity ratios, E t , are controlled to satisfy regulators’ solvency capital requirements (SCRs), which may At in turn be affected by e.g. investment portfolio risk and by the types of policies sold by the pension company. SCRs are higher for companies which have issued policies with guaranteed minimum returns of the type we aim to model in this paper. In Denmark, actual equity ratios average around 10–15% with enormous variation across companies. We have set α = 20% in the base case in our later numerical examples.

4

Figure 1.

P.L. Jørgensen and N. Gatzert

Initial balance sheet.

and guarantees. We will refer to the liability entry in the balance sheet as the policyholders’ account since pension savers in with-profits pension funds actually have such an account where the balance gets updated annually. However, prior to time T where it is paid out, it cannot be withdrawn at face value. Hence, the account balance does not necessarily equal the market value of the policyholder’s claim prior to maturity (see also Guillen et al. (2006)). It is therefore emphasized that while a market value of the investment portfolio is easily obtained at all times (it is simply observed), the same does not hold for liabilities and equity. The market values of these claims must be determined by treating them as contingent claims and by pricing them via appropriate and consistent valuation methods. But before we can do this, the two claims and the rules that govern how investment surpluses (or deficits) are distributed between them period-by-period until maturity must be described. Since we are modeling traditional guaranteed with-profits policies, the mathematical description of liabilityholders’claim must reflect the fact that they have been promised a minimum return in each period. The discretely compounded constant guaranteed (annual) rate of return is denoted r G and it is assumed positive; i.e. r G ≥ 0. This claim must be honored before anything else, and we might say that in this sense policyholders have first priority on the company’s assets. This is similar to senior debt in a standard corporate capital structure. In addition to their guaranteed return, policyholders are entitled to receive a share of the company’s investment surplus when funds are adequate and the solvency situation allows this. The rate by which policyholders participate in the ‘upside’ is called the participation rate and it is denoted by δ, 0 ≤ δ ≤ 1. This right to participation is sometimes referred to – particularly by financial economists – as policyholders’ bonus option, see e.g. Briys & de Varenne (1997).6 It may be noted here that some newer actuarial literature essentially shares this view of policyholders’ right to receive bonus as an option. Norberg (2001) is a good example. Having defined technical surplus as the difference between the second-order retrospective reserve (based in part on experienced investment returns) and the first-order prospective reserve (based in part on the promised technical or guaranteed return), and having also noted that technical surplus belongs to the insured, he goes on to analyze various ways to calculate and distribute this surplus – if positive – as bonus. So although our terminologies and mathematical models differ, our approaches are in fact very closely related. 6 Policyholders’ right to share in pension insurance companies’ profits – i.e. to receive bonus – is typically a statutory right. In Denmark, for example, the Ministerial Order no. 358 on ‘The Principle of Contribution’ specifies rules for calculating and distributing the realized actuarial surplus of pension companies. In Germany, a similar Profit Sharing Act (MindZV) specifies, for example, that at least 90% (our δ) of investment surplus must be shared with policyholders, see e.g. Table 3 and the accompanying text in Maurer et al. (2013).

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We note that bonus which has been credited to policyholders’ account is guaranteed in the sense that such amounts are also entitled to receive a minimum return of r G in subsequent periods. This feature is sometimes referred to as a ratchet- or cliquet-style guarantee. The implications of such a ratcheting mechanism are studied in Grosen & Jørgensen (2000) and in many later papers. See also Jørgensen (2004). Equity is modeled as a residual claim – and as such with second priority status – as standard equity in usual corporate finance sense. However, here we shall explicitly model not only the feature concerning equity’s right to charge a periodic risk premium and thus to withhold a part of the ‘upside’ before bonus is distributed, but also its option to keep a shadow account where non-payable risk charges can be carried forward for later payment. The decisive feature here is that payment of equity’s risk charge and the clearance or reduction of the balance in the shadow account take priority over bonus payments to pension savers. This feature of equity’s risk charge can be seen as a parallel to the case of cumulative preference shares (see any corporate finance text such as e.g. Grinblatt & Titman (2002)) for which any unpaid preferred dividends from past periods must be paid in full before any dividends can be paid to common equity. In our model, equity’s risk charge in period t is calculated as a constant fraction θ ≥ 0 of liabilities in the beginning of the period, i.e. L t−1 . This corresponds to the standard practice in Danish life and pension companies (Pensionsmarkedsrådet 2004). Note that a risk charge scheme can be operated with or without the option to keep a shadow account. When we include the shadow account option, the time t balance in the shadow account is denoted as Dt . We stress that this is an off-balance sheet entry (liability) which does not affect the accounting identity At = L t + E t ,

(1)

which must hold at all updating times t ∈ [0, T ]∩N, where N refers to the set of natural numbers (including zero). It is one of the main points of this paper, however, to explain that shadow accounts should be thought of as economic/financial liabilities; although from an accountant’s point of view, they are not. The extended time t balance sheet in Figure 2, where we have added the shadow account, Dt , as a shaded entry (which does not enter in the sum of liabilities) is meant to serve as an illustration of this important insight. To keep the model simple, we will assume that a negative book value of equity is allowable before maturity. This corresponds to the assumption that regulatory authorities will not step in and force liquidation of an insolvent fund prior to maturity. It is furthermore assumed that

Figure 2.

Time t balance sheet and off-balance sheet shadow account entry (shaded).

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if equity is negative at the maturity date T , then equityholders will cover this loss with an infusion of additional capital so that liabilityholders’claim is in fact fully guaranteed at maturity.7 The assumption of no limited liability of equity – and thus essentially of default protection of policyholders – is more realistic than it may seem at first glance. In fact, it is a quite natural one when the insurer is a subsidiary of a larger, financially sound company – say a large bank – which in practice and for example for reasons of reputational protection would always stand behind its life and pension insurance arm. The assumption could also be seen as a proxy for a certain type of regulation. We shall return to this discussion later in the paper when our numerical results are presented. We now turn to describing the rules for the dynamic updating of the various accounts. We describe the general case in which both a non-trivial risk-charge scheme and a shadow account are in operation. Regimes without the shadow account option – or without the risk charge altogether – crystallize as special cases of the general case with appropriate parameters set equal to zero. The various cases will be further discussed and analyzed in the paper’s numerical section.

2.1. Periodic updating of account balances In period (year) t, the company’s investment return is given by At − At−1 , and as explained above, liabilities must be credited with a rate of return of at least r G ≥ 0 in each period (year). After this transfer of the guaranteed return to liabilities, the remaining surplus for year t is At − At−1 − r G L t−1 .

(2)

This is called the year’s realized result.8 We now turn to describing the rules for distributing the realized result (even if negative) among stakeholders. The state of the world at time t is divided into four cases according to the size of the realized result. We identify these situations as ‘Bad’, ‘Good’, ‘Better’, and ‘Best’, cf. Figure 3. 2.1.1. The Bad case A Bad year is characterized by At − At−1 − r G L t−1 < 0.

(3)

7 These assumptions are easily relaxed. One might equip equityholders with the put option to default at maturity (Briys & de Varenne 1997). Alternatively, one could impose a dynamic barrier on the asset value that would trigger premature liquidation should assets drop below a given, possibly time dependent, value as in e.g. Grosen & Jørgensen (2002). A third possibility would be to introduce a third-party guarantor of liabilities’ maturity claim as discussed in e.g. Gatzert & Kling (2007). Different assumptions regarding the default structure of the model would naturally affect fair values of the various balance sheet components, but they would not affect the qualitative results regarding the implications of risk charges and shadow account options focused on here. 8 The term ‘realized result’ is defined in Danish insurance legislation as the financial year’s surplus or deficit after policyholders’ accounts have been credited with the guaranteed return, and after deduction of insurance coverage expenses and costs as assumed in the company’s ‘technical basis’ (which must be filed with the Financial Supervisory Authority). The realized result is, in other words, the difference between the ‘actual’ and the ‘presumed’ development in the company’s accounts. In our simplified model without insurance risks and costs, this difference is given simply as in relation (2).

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Scandinavian Actuarial Journal “Bad”

“Good”

“Better”

“Best”

Realized result

0

Risk charge Figure 3.

Shadow account balance

Location of Bad, Good, Better, and Best case intervals on realized result value line.

So in Bad years, the company’s realized result is negative. This means that no risk charge can be taken by equityholders and that the book value of equity will decrease precisely by the amount on the left-hand side of (3) since liabilities’ guaranteed return must always be credited.9 Since no risk charge can be paid, the shadow account balance will increase by θ L t−1 in period t. In the Bad case, account balances are therefore updated as follows: L t = L t−1 (1 + r G ) E t = At − L t Dt = Dt−1 + θ L t−1 .

(4)

As can be seen from the last relation in (4), we have assumed – for simplicity – that the shadow account balance does not carry interest from time t − 1 to t.10  The union of the sets describing the three remaining cases (Good, Better, and Best) form the complement to the Bad case, i.e. (3), cf. again Figure 3. These cases are, in other words, all (partly) characterized by At − At−1 − r G L t−1 ≥ 0, (5) meaning of course that in all of the remaining cases the realized result is positive. The characterization is further refined as follows. 2.1.2. The Good case In the Good case, At − At−1 − r G L t−1 < θ L t−1 .

(6)

Combining this with condition (5), we have 0 ≤ At − At−1 − r G L t−1 < θ L t−1 .

(7)

The interpretation is straightforward: in the Good case, the realized result is positive (the investment return is large enough to cover liabilities guaranteed return) but not large enough to allow for full payment of equity’s risk charge. Consequently, the shadow account balance 9 It may be noted that E = A − L = A − A t t t t t−1 − (L t − L t−1 ) = At − At−1 − (L t−1 (1 + r G ) − L t−1 ) = At − At−1 − r G L t−1 . 10 In practice (in Denmark), the regulator allows for crediting of the shadow account balance with periodic interest as long as the particular scheme applied is disclosed.

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increases in this case and there is no bonus. Hence, account balances are updated as follows at time t: L t = L t−1 (1 + r G ) E t = At − L t   Dt = Dt−1 + θ L t−1 − (At − At−1 − r G L t−1 ) ,

(8)

At − At−1 − r G L t−1 ≥ θ L t−1 .

(9)

where the increase in the shadow account balance (the term in square brackets in (8)) is computed as the permissible risk charge minus the (smaller) realized result, which can actually be transferred to equityholders’ account. In both the Better and the Best cases, it holds that

This means that the realized result is large enough that the risk charge can be credited in full to equity’s account in these cases. What distinguishes the Better and Best cases is the status of the shadow account balance and thus the company’s ability to pay out bonus in the present year. 2.1.3. The Better case The Better case is characterized by At − At−1 − r G L t−1 ≤ θ L t−1 + Dt−1 .

(10)

Combining with condition (9), one obtains θ L t−1 ≤ At − At−1 − r G L t−1 ≤ θ L t−1 + Dt−1 ,

(11)

so the Better predicate covers years where the investment return is large enough to cover liabilities’ guaranteed return as well as equity’s risk charge. It may even be that the shadow account balance can be partly reduced (and even cleared), but in any case there will not be funds left for bonus distribution.11 In the Better case, account balances are updated as follows: L t = L t−1 (1 + r G ) E t = At − L t   Dt = Dt−1 − (At − At−1 − r G L t−1 − θ L t−1 ) .

(12)

In this case, the term in square brackets (in (12)) is the positive amount by which the shadow account balance can be reduced after full payment of the risk charge out of the realized return.  11 Note the following limiting cases: if the leftmost inequality in (11) is binding, i.e. if A − A t t−1 = r G L t−1 + θ L t−1 , then Dt = Dt−1 . The shadow account balance remains unchanged. If the rightmost inequality is binding, then At − At−1 − r G L t−1 − θ L t−1 = Dt−1 and Dt = Dt−1 − Dt−1 = 0, and the funds are just sufficient to fully clear the shadow account balance.

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2.1.4. The Best case The Best case is a situation characterized by At − At−1 − r G L t−1 > θ L t−1 + Dt−1 ,

(13)

which means that in the years where relation (13) holds, the investment return has been adequate to cover not only liabilities’guaranteed return, equity’s risk charge, and any balance in the shadow account. There will also be funds available for bonus distribution. As mentioned earlier, bonus is distributed with a share of δ > 0 to liabilityholders. Consequently, the remainder is deposited with equityholders’ account. Thus, in the Best case, accounts are updated as follows: L t = L t−1 (1 + r G ) + δ ( At − At−1 − r G L t−1 − θ L t−1 − Dt−1 ) E t = At − L t Dt = 0.

(14)

When looking at the increase in the (book) value of equity implied by the Best case system in (14), E t = At − L t   = At − At−1 − L t − L t−1   = At − At−1 − L t−1 (1 + r G ) + δ (At − At−1 − r G L t−1 − θ L t−1 − Dt−1 ) − L t−1 = θ L t−1 + Dt−1 + (1 − δ) (At − At−1 − r G L t−1 − θ L t−1 − Dt−1 ) ,

(15)

we can observe that the (accounting) return of equity in the Best case can be decomposed into a risk charge, a transfer equal to the full, previous balance in the shadow account, and a (1 − δ)-share of the surplus amount available for bonus.  At this point, note that one can simply set Dt ≡ 0, ∀t, to model a situation where risk charging by equityholders is allowed and practiced (θ > 0) but where the maintenance/keeping of a shadow account is not. In this case, there is no distinction between the Better and the Best case, cf. Figure 1. An even simpler model – corresponding to a situation in which risk charging is not allowed and where equity is solely compensated through the participation rate – is obtained by setting θ = 0. Along with the assumption that D0 = 0, this will ensure Dt = 0, ∀t, and it will correspond to the case where the Good, Better, and Best cases are combined. Returning to the general model, we finally observe that the period-by-period updating rules for liabilities, equity, and the shadow account that we have described in Equations (4), (8), (12), and (14) above, can be described in a more compact form which, to a certain extent, reflects the option elements embedded in the different claims. In particular, the development from time t − 1 to time t in the accounting (book) value of the liabilityholders’ claim is given by +  L t = L t−1 (1 + r G ) + δ At − At−1 − r G L t−1 − θ L t−1 − Dt−1 .

(16)

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Hence, for equity we have the relation +  E t = E t−1 + At − At−1 − r G L t−1 − δ At − At−1 − r G L t−1 − θ L t−1 − Dt−1 .

(17)

Finally, the development in the shadow account balance is governed by Dt = Dt−1 + θ L t−1 + +   − At − At−1 − r G L t−1 + At − At−1 − r G L t−1 − θ L t−1 − Dt−1 . (18) The ‘payoff functions’ in (16)–(18) are visualized in Figures 4–6, where the time t accounting value of liabilities (L t ), equity (E t ), and the shadow account balance (Dt ) conditional on (Dt−1 , L t−1 ) and on the fixed parameters are plotted as a function of the realized result in period t, i.e. At − At−1 − r G L t−1 . It is important to realize that Equations (16)–(18) and the accompanying figures merely represent a partial, one-period view on the development in the various accounts. It is not until multiperiod and cumulative pay-offs are studied that the operation of a shadow account becomes meaningful and that the consequences of risk charges and shadow account transfers become fully visible.

3. Valuation Having fully described the various claims and the mechanisms for determining their maturity pay-offs, we now focus on issues regarding valuation. Since final pay-offs to liabilities and to

Figure 4.

Development of liabilities from time t − 1 to time t.

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Figure 5.

Development of equity from time t − 1 to time t.

Figure 6.

Development of shadow account balance from time t − 1 to time t.

11

equity are fully determined by the account updating mechanisms, by the model parameters, and by the path of asset values from time 0 to time T , we assume that the investment portfolio is an asset that trades freely in a perfect and frictionless market.12 This means that we can price 12 To be more precise about the path-dependence, it is only the set of asset values sampled at the annual updating points that matter for final payoffs, not the entire path.

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contracts as replicable European-style financial contingent claims using standard risk neutral valuation techniques. Assuming a constant (and continuously compounded) riskless rate of interest, r f , the initial value of liabilities is given by    Q V0L A0 , D0 ; T, r f , r G , σ, α, θ, δ = e−r f T E0 L˜ T .

(19)

Similarly, the initial value of equity can be represented as    V0E A0 , D0 ; T, r f , r G , σ, α, θ, δ = e−r f T E0Q E˜ T .

(20)

In both of the above expressions, E0Q {·} refers to risk neutral- or Q-expectations conditional on time 0 information. In addition to the initial asset value, A0 , the initial balance in the shadow account, D0 , is specified as an argument of the valuation functions. The purpose of this is to emphasize the significance of this variable in the valuation problem(s). This significance is illustrated and quantified in further detail in the later numerical study. Before we can proceed with further analyses and evaluation of Equations (19) and (20), the stochastic dynamics of the investment portfolio needs to be defined. To this end, we assume that the asset value dynamics is governed by a geometric Brownian motion (GBM) as, for example, in Black & Scholes (1973) and many other studies related to ours (e.g. Briys & de Varenne (1997), Grosen & Jørgensen (2002), and Gatzert & Kling (2007)). This choice is not made to facilitate the derivation of closed-form solutions for claim values. As already noted, our claims payoffs are highly path-dependent. This will prevent the derivation of such closed-form solutions irrespective of the choice of asset dynamics. So we must resort to numerical methods such as Monte Carlo simulation in order to evaluate the central relations (19) and (20). We prefer the geometric Brownian motion to more complex dynamic models in order to keep matters reasonably simple. This allows us to focus on other more important details, and if deemed necessary, the assumption concerning the GBM is easily relaxed. The GBM process governing the dynamics in the asset value is given by d At = μAt dt + σ At d WtP ,

(21)

where μ denotes the (continuously compounded) expected return, σ is the constant asset return volatility, and W P is a standard Brownian motion defined on the filtered probability space (, F , (Ft ), P) on the finite time interval [0, T ]. The GBM process implies normal distributed log returns, see e.g. Bj˝ork (2009). For purposes of valuation – pertinent to the later Monte Carlo simulation work – the risk neutralized parallel to (21) is needed. It is given by (see again Bj˝ork (2009)) d At = r f At dt + σ At d WtQ ,

(22)

where W Q is a standard Brownian motion under the equivalent risk-neutral probability measure Q and r f is the constant riskless rate of interest.

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4. Numerical results and illustrations In this section, a range of numerical results are presented to illustrate and to further clarify and quantify various aspects of our model. First, single, simulated scenarios are presented to emphasize some essential implications of operating with risk charges and shadow accounts. We next study the design of fair contracts. By fair we mean that computed initial fair values of equity and liabilities should equal the amounts initially invested by these stakeholders. Finally, a sensitivity analysis is performed to illustrate the effects of changes in key parameters on the different value components – including the value of the shadow account option.

4.1. Single illustrative scenarios For a set of given parameters, Figure 7 illustrates the dynamic evolution of balance sheet entries for a single simulated scenario over a 20-year period. The point of reference is a simulated evolution of the market value of the underlying investment portfolio. The figure then contains plots of the evolution in the book values of equity and liabilities resulting from this particular asset value development under three different assumptions regarding the risk charge (RC) and shadow account (SA) regime: a regime without risk charging and shadow accounts, a regime where only a risk charge is imposed, and a regime where both risk charging and shadow account operation is in effect.13 Figure 7 illustrates how the imposal of a risk charge and the operation of a shadow account – ceteris paribus – benefit equityholders and hurt liabilityholders. This is seen by noting that the (book) value of equity is always higher when a risk charge is imposed than when not. Furthermore, the equity value is further increased if a shadow account also is in operation. Naturally, it is vice versa for liabilities. In the plotted scenario, the maturity value of equity is about 38 for a pure participating contract, it is 50 when a 1% risk charge is applied, and it is roughly 55 when a shadow account is also in operation. The corresponding maturity values of liabilities are 189, 177, and 172. In all three cases, equity and liability values add up to the market value of assets at maturity, which is about 227 in this scenario. The development in the balance of the shadow account – which is barely noticeable in Figure 7 – has been separated out and enlarged in Figure 8 for clarity. Note in this case how a positive shadow account balance is always brought back down to zero in the subsequent period. It also expires at zero. Not all cases are like this. It may take several periods to clear the shadow account, and the shadow account may expire with a positive balance. Figure 9 shows an alternative scenario for the shadow account development. It is emphasized that the plots in Figures 7–9 are merely randomly generated examples of the dynamics of the various balance sheet entries and the shadow account balance in our model.

13 In the current example, the parameter δ has been calibrated so that the contract, which includes both a risk charge and the operation of a shadow account, is initially fair, cf. the next subsection on fair contracts.

14

Figure 7.

P.L. Jørgensen and N. Gatzert

Dynamic evolution of balance sheet entries for a single simulated scenario.

Shadow account balance dynamics

0.0

0.5

1.0

1.5

Shadow account balance

0

5

10

15

20

Years T Figure 8.

Dynamic evolution of shadow account balance for single simulated scenario in Figure 7.

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4

Shadow account balance dynamics

0

1

2

3

Shadow account balance

0

5

10

15

20

Years T Figure 9.

Dynamic evolution of shadow account balance for an alternative single simulated scenario.

4.2. Fair contracts To some extent, the previous section has already illustrated how the values of the stakeholders’ ownership shares of the fund are affected by the specification of the risk charging and shadow account regime. In this section, we dig a bit deeper into the question of how parameter specification affects value components and explore and illustrate how parameters must be set in order to ensure that contracts are fair at initiation. By ‘fair’ we mean that parameters and contract characteristics are such that stakeholders’ initially invested amounts equal the computed arbitrage free initial value of their acquired contingent claim. In mathematical terms, this section will provide a host of examples of contract specifications and parameter combinations which ensure that   E 0 ≡ α A0 = V0E A0 , D0 ; T, r f , r G , σ, α, θ, δ , (23) and therefore also

  L 0 ≡ (1 − α)A0 = V0L A0 , D0 ; T, r f , r G , σ, α, θ, δ .

(24)

In Figure 10(a)–(f), we work from a base case where the fund operates with a shadow account and where A0 = 100, α = 0.20, D0 = 0, r f = 4.0%, r G = 0.0%, θ = 1.0%, and σ = 7.5%. With these parameters, the fair participation rate, δ, equals 0.688. Figure 10(a)–(f) is produced by varying the asset volatility and another key parameter, and by then solving (by iterated Monte Carlo simulation with 10 million paths) for the participation rate that ensures that the new parameter combination is fair to both sides of the contract. This procedure generates a family of fair contract curves which can be studied in the figures.

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Figure 10. Dependence of fair participation rates on key parameters. (a) Guaranteed rate (r G ) varied. (b) Risk charge (θ) varied. (c) Initial shadow account balance (D0 ) varied. (d) Initial leverage ratio (λ) varied. (e) Characteristics of shadow account/risk charge varied. (f) Time to maturity (T ) varied.

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The first thing to notice from these figures is the general negative relation between asset volatility and fair participation rates. This is as expected in the present setting without default during the contract term, since increasing asset volatility increases the value of the liabilityholders’ call option on the ‘upside’ of realized periodic returns, and the participation rate should therefore be lowered when volatility is increased – and vice versa – to reestablish a fair contract. A parallel view would be to think of the guarantee-issuing equity as having sold a put option on assets to liabilityholders. The value of such a put option also increases in volatility.14 It can also be noted that the negative relation between asset volatility and fair participation rates means that value is transferred from liabilityholders to equity if asset volatility is lowered without a corresponding increase in participation rates. This theoretical property of the model is quite consistent with empirical observations from recent years where managers of some pension funds with significant interest rate guarantees have lowered the level of risk in their investment portfolios – or have ‘threatened’ to do so unless liabilityholders would agree to renegotiate their interest rate guarantees or to give them up entirely.15 A final general observation from Figure 10(a)–(f) is that an asset volatility below a certain threshold will in some cases require participation rates above 100% in order for the contract to be fair. It is hard to imagine such a contract being effectuated in practice in the pension market. However, an interesting parallel worth mentioning is the retail market for structured investment products in which participation rates above 100% are quite standard, see e.g. Baubonis et al. (1993) and Chen & Wu (2007). Looking at the individual Figure 10(a)–(f), a further number of interesting observations can be made from studying the displacement of the fair (σ, δ)-curve as a third parameter is varied. Starting from the top-left Figure 10(a), it is seen that higher guaranteed rates and lower participation rates go hand-in-hand in fair contracts. Moreover, guaranteed rates above the riskless rate are not possible (if contracts are to be valued at par). When policyholders are guaranteed a return equal to the riskless rate they cannot also participate in the ‘upside’ and δ drops to zero as seen in the figure. The next Figure 10(b) (top-right) shows that as equity’s risk charge increases, policyholders participation rate must increase as well as in order for the contracts to remain fair. The middle-left Figure 10(c) compares situations in which the initial shadow account balances differ. In accordance with intuition, policyholders would, ceteris paribus, prefer to join a fund where policyholders are not already indebted to equity, i.e. they prefer an initial shadow account balance, which is as low as possible. Hence, the higher the initial shadow account balance, the higher the participation rate must be in order to fairly compensate policyholders at the outset. In addition, Figure 10(d) (middle-right) shows that policyholders should also require higher participation rates for higher initial leverage of the fund. Again this is as expected. The bottom-left Figure 10(e) illustrates fair participation and asset volatility pairs for different risk charge/shadow account regimes. Compared to a situation in which equity does not charge

14 Note that there is a natural connection here to the Put–Call parity, although the link is not simple since we are effectively dealing with a sequence on interrelated options on realized periodic returns. 15 Woolner (2010) describes how, in a controversial move, Danish pension company Sampension in 2010 redefined their guarantees from fixed to ‘intentional’. Sampension was subsequently sued by policyholders.

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Table 1. Equity value’s dependence on asset volatility (σ ), risk charge (θ ), and shadow account (SA) operation A0 = 100, α = 0.20, D0 = 0, T = 20, r f = 4%, r G = 0%, δ = 0.5918. Based on 107 simulations, δ calibrated so V0E = 20 in base case (circled).

for risk, policyholders would require higher participation rates when equity does impose a risk charge, and an even higher participation rate when a shadow account is also in operation. Finally, Figure 10(f) at the bottom right shows the effect of varying time to maturity. Time to maturity turns out to have negligible effect on the fair (σ, δ)-relationship when the initial shadow account balance is zero. The figure is therefore constructed with a positive initial shadow account balance (D0 = 25) where results are non-trivial. One can observe that when T is increased – and there is thus more time to earn an investment return and to clear the given shadow account balance – policyholders can accept a lower participation rate. This is also a quite intuitive result. 4.3. Contract valuation and sensitivity analysis Having considered in the previous section fair contract designs in some detail, we now move on to looking more directly at contract values and their sensitivities to parameter changes in our model setup. Since the market value of equity and liabilities always add up to the total value of assets (which is fixed to 100), a table of equity values also implies a table of liability values and vice versa. In Tables 1 and 2, we nevertheless take different perspectives – equity’s and liabilities’, respectively – in order to focus on changes to parameters that may seem more directly relevant to, or to some extent actually controlled by, one type of stakeholder. The point of departure of both tables is a particular set of parameters which lead to a fair contract and which are circled in the tables. From this ‘anchor point’, key parameters are then

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Table 2. Liability value’s dependence on riskless interest rate (r f ) and initial shadow account balance (D0 ). A0 = 100, α = 0.20, T = 20, θ = 1%, σ = 7.5%, δ = 0.6181. Results based on 107 simulations. δ calibrated so V0L = 80 in base case (circled).

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changed, and the resulting contract values are reported in the table. All other contracts in the tables are thus not strictly fair, but the exercise will give us a clear idea of which stakeholder(s) stands to gain or lose when parameters change, for example because of altered market conditions. Similarly, the tables will be informative about the strength of the incentive a stakeholder may have to attempt to manipulate a parameter (to the extent that this is possible). As before, all contract values in Tables 1 and 2 are obtained by Monte Carlo simulation. An average Monte Carlo error is given in the bottom line of the tables. Table 1 focuses on equity value from an initial point where asset volatility is 7.5% and where neither a risk charge nor a shadow account is in operation. With other parameters fixed as shown in the table’s header, such a contract is fair with δ = 0.5918. From the rest of the table, one can see the effects of changing asset volatility and of imposing a risk charge of increasing magnitude. In addition, the impact of introducing a shadow account is exhibited. To a certain extent, all of these variables are under equityholders’ control or influence. Table 1 reconfirms that equity value is decreasing in asset volatility. The sensitivity of equity value to asset value volatility reflects our assumption of no limited liability for equityholders. Given this assumption, equity may have a strong incentive to reduce risk in the fund’s investment portfolio. From the initial point, for example, equity value can be increased by approximately 33% by lowering volatility from 7.5% to 5.0%. This property of the model is consistent with the observed practice of many pension funds having switched their investments to less risky assets as interest rates have dropped further in recent years. The tendency is often stronger the stronger the pension funds’ exposure to interest rate guarantees, and we are aware of companies which used to have significant stock investments but which are now almost 100% invested in short-term, low-duration (Northern European) government bonds.16 Similarly, Table 1 shows that equity value is positively affected by imposing a risk charge and further so if also a shadow account is being operated. We see the value of the shadow account option as the difference between the ‘SA’ and ‘No SA’ values in the table. For example, by imposing a risk charge of 1%, which is in line with risk charges observed in practice, and by activating a shadow account, the market value of equity increases by approximately 30% (from 20.00 to 25.89) in our example. The shadow account option value is equal to about 1.75 in this case. In general, the shadow account option value increases in volatility. We finally note that the negative equity values in the top-right corner of Table 1 are not errors. They are a consequence of an increased volatility (and no risk charge or shadow account) in combination with our base assumption of no limited liability of equity. As was noted earlier in the paper, the assumption of no limited liability for equity is not as strange as it may seem. Life and pension companies are often subsidiaries of large financial conglomerates and/or banks that would only as an absolute last resort walk away from an insolvent L&P subsidiary and let it default. The assumption of no limited liability is therefore quite realistic, and in fact, one also regularly sees L&P companies trade at negative prices, i.e. other financial institutions or companies sometimes need to be paid to take over the L&P business of a bank, say, that does no longer want to embrace and support this type of business. This phenomenon is particularly pronounced for companies that are burdened by high-level guarantees. The assumption of default 16

This tendency is also a consequence of the advance of Solvency II-related risk-based capital requirements.

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protection of policyholders may alternatively be seen as a proxy for regulation designed to protect the pension benefits of policyholders in virtually all circumstances. This would then include situations where the regulator could force L&P company owners to supply additional equity capital when necessary. Having said that, an assumption of limited liability of equity may be more appropriate and correct in certain situations, and equityholders are of course in general not legally obliged to cover a possible default. We have therefore analyzed this case as well, and with such an assumption, negative equity values naturally cannot occur, but our qualitative results regarding risk charges and shadow account options are not materially affected.17 Turning to Table 2, where the perspective of liabilityholders is taken, the basis of comparisons is a case with a realistic risk charge of 1%, with a shadow account with zero initial balance in operation, and with fairly high riskless and guaranteed interest rates of 8% and 4%, respectively. Liabilities are fairly valued at 80.00 in this case with δ = 0.6181 and other parameters as given in the table header. The choice of these fairly high interest rate parameters for the base case is made to reflect a situation with contracts that were initiated (fairly, presumably) in a past where conditions, and interest rates in particular, were different (higher) than today. The first panel of the table then allows us to study what has happened to the value of the base contract as market riskless interest rates have dropped and as shadow account balances may have increased as a result of poor investment results. The effect of falling interest rates is of course significantly positive to liabilityholders. This is mainly because their guarantees have become more valuable, but we also see that the increase is partially reversed to the extent that shadow account balances are increased. The fact that positive shadow account balances hurt liabilityholder value is one that new pension savers should be particularly aware of – if it is not compensated for somehow – before they choose the company with which to trust their pension savings. To facilitate such comparisons by potential customers, in Denmark regulatory authorities specifically require pension companies to disclose their shadow account balances as well as their historically applied risk charging scheme. In the second panel of Table 2, the riskless rate has been fixed at 4%. We then vary D0 as before, as well as the guaranteed interest rate, r G , which is lowered from the original value of 4% (which is a rate applied in many actual contracts from some years back) in steps of 1% down to 0%. At first sight, it may seem unnatural to experiment with varying the guaranteed rate which is supposed to be fixed for the entire lifespan of a contract. However, in recent years, practice has seen some L&P companies lowering the guaranteed rates not only for new contracts but also for older, in-progress contracts. In some cases, the reduction of the guaranteed rates have been negotiated with policyholders, but in other cases it has simply been dictated.18 In any case, the effect of lowering r G is that policyholders’ value is expropriated, and with accompanying 17

Numerical results for the limited liability case are available from the authors upon request. In Denmark, L&P companies’ beginning practice of lowering of guaranteed rates was officially (and temporarily) sanctioned on 12 June 2012 when the Ministry of Business and Growth and the Danish Insurance Association signed an agreement that prevented L&P companies from paying dividends to equity and from crediting pensions savers’ accounts with returns exceeding 2%, irrespective of the level of their guaranteed rates, for the years 2012 and 2013. The agreement was part of a string of initiatives meant to ensure financial stability and to prepare the L&P sector for the upcoming Solvency II regulatory requirements. It has been criticized that the agreement did not prevent or regulate equity’s clearance of any positive shadow account balances during the period thus making the agreement a ‘Gift worth billions’ to pension fund owners (see e.g. Andersen & Dyrekilde (2012a)). A press release and the full text of the agreement are available at http://www.evm.dk. See also Footnote 15. 18

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increases in the shadow account balances, the effect to policyholder value can be detrimental as the table illustrates. Consider for example a contract as in the base case. If the riskless rate drops to 4%, then the immediate effect on the contract value is an increase from 80.00 to 107.88 (and equity becomes negative). But if the guaranteed rate is then lowered from 4% to 2% (as for example recently dictated by the Danish government, cf. Footnote 18), then the value of liabilities drops to 89.13. If, in addition, the shadow account balance has increased to 30, then all benefits from the significant fall in riskless interest rate is lost, and the contract value is back almost precisely at 80 where it started.

5. Conclusion This paper has analyzed risk charges and shadow account options in life and pension companies. We have explained that in combination with a simple proportional risk charging scheme, the permission to operate a shadow account is really an option to equityholders that can be very valuable. An important implication of this result is that if a shadow account option is granted to equity of an L&P company (e.g. by a country’s financial regulator) without a corresponding compensation to the company’s policyholders, then the wealth of the latter group can be seriously expropriated. An alternative – and perhaps more positive – way of stating this main conclusion of our paper would be to say that our research has shown that the presence of a shadow account option means that the fair risk premium that equity should require as compensation for the risk that it bears by providing the company’s equity buffer is lower than it would otherwise be. Regulators – and to a certain extent also policyholders – might see this as a positive thing, and it is certainly more comfortable to imagine this being the reason for the introduction of the shadow account option in the first place, rather than the fact that this instrument can be used to expropriate policyholders’ wealth. In any case, regulators should be aware of the potential impact of this option and they should work to ensure that the attractiveness of private pensions is still given for policyholders as well as for equityholders providing the capital to back the guarantees offered to the former group. This is also of high social relevance for most industrialized countries due to the prevailing problems encountered in public pension and social security systems. There are a number of directions in which to extend our work in future research. Firstly, it would be relevant and interesting to refine the default structure of our model, e.g. by allowing for premature default or restructuring, or by considering in more detail the alternative default assumptions that were only briefly discussed during our analysis. One could also extend the model with stochastic interest rates and (a) more advanced asset value process(es) – perhaps with a more realistic feedback mechanism from company solvency to the volatility of the investment portfolio. The inclusion of periodic premiums, surplus withdrawals (dividends) to equity, and/or heterogeneous policyholder risk groups and multiple shadow accounts are additional issues that could be analyzed and which could make the model setup more realistic. A final suggestion for future research would be to perform a study of how risk charges and shadow account options affect shortfall probabilities and the likelihood of default in the model.

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Acknowledgements The authors are grateful for the comments and useful suggestions received from an anonymous referee, Søren Andersen, Jan Bartholdy, Niels Bidilov, Claes Vestergaard, and from participants at the 39th Annual Seminar of the European Group of Insurance Economists (EGRIE) held in September 2012 in Palma de Mallorca, Spain. The authors gratefully acknowledge financial support by the German Research Foundation (DFG).

References Andersen, L. & Dyrekilde, B. (August 20, 2012a). Milliardgave til pensionskassers ejere (Gift worth billions to pension fund owners). Jyllands-Posten, Erhverv & Økonomi, p. 1. In Danish. Andersen, L. & Dyrekilde, B. (August 20, 2012b). Pensionskunder skylder ejerne milliarder (Pension savers owe billions to owners). Jyllands-Posten, Erhverv & Økonomi, p. 8. In Danish. Baubonis, C., Gastineau, G. L. & Purcell, D. (1993). The Banker’s guide to equity-linked certificates of deposit. Journal of Derivatives 1 (2), 87–95. Bj˝ork, T. (2009). Arbitrage theory in continuous time. 3rd ed. New York: Oxford University Press. Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81 (3), 637–654. Briys, E. & de Varenne, F. (1997). On the risk of life insurance liabilities: debunking some common pitfalls. Journal of Risk and Insurance 64 (4), 673–694. Brown, J. R. (2001). Private pensions, mortality risk and the decision to annuitize. Journal of Public Economics 82 (1), 29–62. Chen, K. & Wu, L. (2007). An anatomy of bullish underlying linked securities. Global Finance Journal 18 (1), 34–46. Gatzert, N. & Kling, A. (2007). Analysis of participating life insurance contracts: a unification approach. Journal of Risk and Insurance 74 (3), 547–570. Grinblatt, M. & Titman, S. (2002). Financial markets and corporate strategy. 2nd International ed. New York: McGrawHill. Grosen, A. & Jørgensen, P. L. (2000). Fair valuation of life insurance liabilities: the impact of interest rate guarantees, surrender options, and bonus policies. Insurance: Mathematics and Economics 26 (1), 37–57. Grosen, A. & Jørgensen, P. L. (2002). Life insurance liabilities at market value: an analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework. Journal of Risk and Insurance 69 (1), 63–91. Guillen, M., Jørgensen, P. L. & Nielsen, J. P. (2006). Return smoothing mechanisms in life and pension insurance: path-dependent contingent claims. Insurance: Mathematics and Economics 38 (2), 229–252. Jørgensen, P. L. (2004). On accounting standards and fair valuation of life insurance and pension liabilities. Scandinavian Actuarial Journal 104 (5), 372–394. Maurer, R., Rogalla, R. & Siegelin, I. (2013). Participating payout life annuities: lessons from Germany. ASTIN Bulletin 43, 159–187. Mercer (2012). Melbourne Mercer global pension index. Melbourne: Australian Centre for Financial Studies. Mitchell, O. S., Poterba, J. M., Warshawsky, M. & Brown, J. R. (1999). New evidence on the money’s worth of individual annuities. American Economic Review 89 (5), 1299–1318. Norberg, R. (2001). On bonus and bonus prognoses in life insurance. Scandinavian Actuarial Journal 2, 126–147. Pensionsmarkedsrådet (2004). Pensionsmarkedsrådets redegørelse for driftsherretillæg [The pension market council’s explanation of risk charges]. Paper dated July 2, 2004, Pensionsmarkedsrådet. Woolner, A. (2010). Sampension moves to intentional guarantees, risk.net. Published July 9, 2010 at http://www.risk.net/ insurance-risk.

Appendix In brief, the legal foundation of risk charges and shadow account operation by Danish L&P companies is the following. The Danish Financial Business Act (‘Lov om finansiel virksomhed’) requires L&P companies to file their ‘technical basis’ with the Danish Financial Supervisory

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Authority (DFSA). The technical basis should explain, among many other things, the company’s rules for calculating and distributing the realized result (see also footnote 8) between company stakeholders. These rules must be ‘precise, clear, and fair’. The quite general guidelines of the Financial Business Act are clarified in the Ministerial Order no. 358 on the Principle of Contribution (‘Bekendtgørelse om kontributionsprincippet’, see also footnote 6). The order specifies that equity’s total return must be decomposed into its share of investment asset returns and a risk charge, where the latter must be justified by the risk that equity assumes by ensuring policyholder claims. The risk charging scheme must be disclosed via the DFSA. The same order states that if equity in a previous year has not received its calculated risk charge in full, then the lacking amount can be charged in later years’ positive realized results. Finally, the DFSA’s ‘Guide to the Ministerial Order on the Principle of Contribution’ (‘Vejledning om bekendtgørelse om kontributionsprincippet’) specifically refers to the construct where equity can park its unpaid risk charge receivable as a ‘shadow account’. The above-mentioned legal documents are in Danish. They are available for example from http://www.retsinformation.dk.

The European Journal of Finance Vol. 19, No. 2, February 2013, 145–164

On risk management determinants: what really matters? Georges Dionnea∗ and Thouraya Trikib a HEC

Montréal, Montreal, QC, Canada; b African Development Bank, Tunis, Tunisia

We develop a theoretical model in which debt and hedging decisions are made simultaneously, and test its predictions empirically. To address inefficiencies in current estimation methods for simultaneous equations with censored dependent variables, we build an original estimation technique based on the minimum distance estimator. Consistent with predictions drawn from our theoretical model, we show that more hedging does not always lead to a higher debt capacity. We also find that financial distress costs, information asymmetry, the presence of financial slack, corporate governance, and managerial risk aversion are important determinants of corporate hedging. Overall, our evidence shows that modeling hedging and leverage as simultaneous decisions makes a difference in analyzing corporate hedging determinants.

Keywords: risk management determinants; corporate hedging; debt structure; managerial risk aversion; simultaneous Tobit equations; panel data JEL Classification: C33; C34; C51; D80; G10

1. Introduction The financial literature puts forward several arguments to explain why and how corporations hedge (or should hedge) against risks they face.1 Yet, empirical tests offer conflicting evidence on whether hedging increases firm value and managerial utility. This reflects, among others, differences in samples and methodologies used to implement the tests.2 Failure to capture the simultaneity between hedging activities and other corporate policies could have also led to erroneous conclusions in past empirical tests. Rogers (2002), for instance, shows that when the simultaneous nature of the risk management and CEO risk-taking incentives are ignored, the relationship between both decisions seems weak at best, while simultaneous regressions show a much stronger relation. Roger’s (2002) conclusion was more recently supported by empirical findings in Chen, Jin, and Wen (2008). The objective of this paper is to offer a regression methodology that can improve the econometric modeling of corporate hedging given its potential interaction with other financial policies within the firm.3 More specifically, we extend the minimum distance estimation (MDE) technique developed by Lee (1995), for cross-sectional samples, to the case of unbalanced panel datasets. Besides the proposed new estimation methodology, the paper contributes to the risk management literature by offering a theoretical model that allows for the firm to make simultaneous hedging and leverage decisions. In contrast with the existing models, we show that, under a standard debt contract, hedging can lead to a lower debt capacity.

∗ Corresponding

author. Email: [email protected]

ISSN 1351-847X print/ISSN 1466-4364 online © 2013 Taylor & Francis http://dx.doi.org/10.1080/1351847X.2012.664156 http://www.tandfonline.com

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We rely on a sample of firms from the North American gold mining industry to conduct our tests. These companies share a common view on gold price fluctuations. Gold is a highly liquid and volatile commodity and firms can use a wide range of instruments to hedge against its price fluctuations. This fosters variations in hedging policies, making it fit for our tests. Furthermore, we have detailed information on firms’ exposure and hedging activities by instrument type. This allowed us to implement the delta percentage, first proposed by Tufano (1996) and more recently used by Dionne and Garand (2003) as a measure of corporate hedging.4 Unlike other measures of corporate hedging available in the literature, the delta percentage captures information on the level of hedging, the types of instruments used to hedge, as well as hedging activities by non-derivatives.5 Our key empirical finding is that hedging does not increase debt capacity once the endogenous nature of the debt is modeled. This finding stands in stark contrasts with our results drawn from single equation models. Overall, our findings highlight that modeling hedging and leverage as simultaneous decisions makes a difference in analyzing risk management determinants, at least for the gold mining industry. While we acknowledge that our results might be industry-specific, we believe that our work will allow further efficient tests of corporate hedging. Our results also show that firms hedge because of managerial risk aversion and to reduce costs arising from information asymmetry and financial distress. The proportion of unrelated directors sitting on the board and the availability of attractive acquisition opportunities also have positive effects on the hedging ratio, while liquidity cushions seem to reduce the magnitude of hedging. The remainder of the paper is divided into six sections. Section 2 describes our theoretical model and hypotheses. Section 3 summarizes the different variables we use in empirical tests. Sections 4 and 5, respectively, describe our sample and the methodology we develop to estimate our system of simultaneous equations. Section 6 presents a discussion of our empirical findings, while Section 7 concludes the paper.

2. The model We model the hedging and leverage decisions as simultaneous by extending the general framework of costly state verification developed by Townsend (1979), Gale and Hellwig (1985), and Caillaud et al. (2000). Unlike the model of Froot, Scharfstein, and Stein (1993) (FSS hereafter), our model does not focus on how the investment policy is financed but rather on the simultaneous decisionmaking of the hedging and leverage policies for a given investment decision. This supposes that the firm seeks to maximize its revenues by choosing optimal hedging and leverage ratios, and finances its investments with available funds. We focus on linear hedging strategies because they are more popular in hedging commodity risk. Consider a mining firm that produces gold and holds a second asset or commodity whose price cannot be hedged. The firm’s total revenue w is equal to: w = w0 [h + (1 − h)x] + y,

(1)

where w0 describes the firm’s revenues from gold in certainty, x the random gold return, h the hedging ratio, and y the random revenue generated by the second asset in place.6 Consistent with FSS (1993), we assume that x is distributed normally with a mean of 1 and a standard deviation of σ and we set y equal to αxw0 , where α is a measure of the correlation between both revenues. The firm must simultaneously choose its hedging ratio h and the face value of its debt F in order

The European Journal of Finance to maximize the following program:  +∞ [w0 [h + (1 − h)x] + αxw0 − F]g(x) dx max F,h

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(2)

xF

under the bank’s financial constraint:  xF [w0 [h + (1 − h)x] + αxw0 − c]g(x) dx + F[1 − G(xF )] ≥ D(1 + r),

(3)

−∞

where D is the amount given by the bank to the firm at the inception of the debt contract; c are the audit costs paid by the bank in default states; r the interest rate during the period; g(x) and G(x) are, respectively, the density and cumulative distribution function of x; xF is defined such that w0 [h + (1 − h)xF ] + αxF w0 − F = 0 and corresponds to the minimal value of x that allows the firm to avoid default. The maximization problem in Equations (2) and (3) yields the following first-order conditions with λ corresponding to the Lagrange multiplier for Equation (3): λ=

1 − G(xF ) 1 − G(xF ) − cg(xF )1/(w0 (1 − h + α))

w0 σ 2 λ= . w0 σ 2 − c(F − w0 (1 + α))/(w0 (1 − h + α)2 )

(4)

The first equation in (4) looks familiar and corresponds to the condition for a standard debt contract. Hence, λ will be greater than 1 (a standard result for a debt contract) as long as 1 − h + α > 0 and F − w0 (1 + α) > 0. In the following, we limit our analysis to the standard debt contract by assuming that these two conditions are satisfied. Solving for F ∗ and h∗ yields (see the appendix for details):    1 1 − G(xf ) F ∗ − w0 (1 + α) ∗ (5a) h =1+α− 2 σ g(xf ) w0   g(xf ) ∗ ∗ 2 F = w0 1 + α + σ (1 − h + α) (5b) 1 − G(xf ) According to Equation (5a), the firm’s optimal hedging ratio is increasing in σ 2 and in the hazard rate g(x)/1 − G(x). This is consistent with Leland’s (1998) findings that hedging magnitude and benefits increases with default costs. Equation (5a) also shows that when the firm deals with one source of uncertainty by selling only gold (α = 0), the optimal hedging ratio h∗ will be lower than 1 for a given F ∗ . This result contrasts with FSS’s (1993) conclusion that the optimal hedging ratio is equal to 1 when the firm faces a single source of risk. Differences in results are driven by the fact that FSS’s (1993) model assumes that the firm hedges only its internal revenues and therefore captures all the benefits of hedging, whereas in our model, the firm hedges its total revenues and therefore a fraction of the hedging benefits will be captured by creditors. Thus, in our model, firms have a lower incentive to hedge their risks fully.7 Equation (5a) shows that the optimal hedging ratio is lower when α < 0, meaning that firms generating negatively correlated revenues benefit from natural hedging that decreases their need for hedging. Equation (5a) also shows that when both revenues are positively correlated (α > 0), the optimal hedging ratio is higher than in the previous two cases and can even be greater than 1.

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Similarly, Equation (5b) shows that F ∗ , the face value of the standard debt contract, is increasing in σ 2 and in the hazard rate. Also, for a fixed value of the hazard rate, F ∗ is a decreasing function of h∗ . This means that if the default intensity were independent from the level of hedging and debt, firms that hedge more would be able to reduce the face value of their debt. This supports the argument that hedging could increase the debt capacity. However, the default intensity (the hazard rate) is usually not constant and we can prove that it is increasing in x when the latter is normally distributed.8 Therefore, the relation between the hedging ratio and the face value of debt will always not be positive. The derivative of F ∗ with regard to h∗ is equal to:

g(xF ) d g(x ) ∂F ∗ 1 − G(xF ) F + (1 − h∗ + α) = −σ 2 ∗ ∂h 1 − G(xF ) dxF      0



dxF dh∗ 

(6)

The first term in Equation (6) shows that an increase in the hedging ratio reduces the firm’s riskiness, which allows it to lower the face value of its debt (the firm reimburses less per dollar borrowed) and consequently increases its debt capacity. The second term describes an indirect effect of hedging on the firm’s debt face value. Because the firm is less risky, it could contract more debt and ends up with higher default intensity and consequently a higher face value. This will yield an indirect second positive effect between hedging and debt, making the relation between both variables unclear. Our findings contrast with Leland (1998), who shows that more hedging increases leverage. Yet, differences could reflect the fact that Leland’s (1998) results are based on an ex-post choice of hedging while ours are based on simultaneous decision-making regarding hedging and leverage. Numerical simulations run in Leland (1998) show that ex-post optimal strategies lead to lower leverage than ex-ante ones. Thus, it is possible that agency costs resulting from the ex-post choice of hedging make the firm reluctant to contract more debt while it reduces its default intensity by hedging. This makes the second effect in Equation (6) negligible. To conclude, our model allows us to formulate two hypotheses:

H1: The firm’s hedging ratio is an increasing function of its default intensity. Because financially distressed firms support higher distress costs we should observe a positive relation between h (the hedging ratio) and the firm’s financial distress costs. H2: Higher hedging ratios will not lead to increased debt capacity, unless the increase in the debt capacity dominates the increase in the default intensity.

3. Regression specification 3.1

Measures of corporate hedging and leverage

As stated earlier, we test our hypotheses on a sample of gold mining firms. The hedging equation uses the delta percentage as a dependent variable. The delta percentage measures, for a given quarter, the fraction of the planned gold production that is being hedged over the next three years. The dependent variable in the debt equation is leverage measured as the book value of long-term debt divided by the firm’s market value.

The European Journal of Finance 3.2

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Determinants of corporate hedging

3.2.1 Information asymmetry Stulz (1990) shows that corporate hedging reduces one of the costs associated with managerial discretion in the presence of information asymmetry while DeMarzo and Duffie (1995) conclude that corporate hedging increases firm’s value when the information asymmetry concerns the source and magnitude of risks rather than managers’ competence. Similarly, Breeden and Viswanathan (1998) find that hedging can reduce the noise in the learning process concerning managers’ capacities when there is information asymmetry about management competence. DeMarzo and Duffie (1991) also conclude that hedging can be profitable for shareholders faced with information asymmetry about the dividend stream. As in Graham and Rogers (2002), we measure information asymmetry by the percentage of shares held by institutions. Institutional shareholders typically have privileged access to information and facilitate its processing on financial markets. Therefore, we expect a negative coefficient for this variable.

3.2.2 Taxes According to Smith and Stulz (1985), in the presence of a convex tax function, hedging reduces the variability of the firm’s pre-tax value and its tax liability because it locks taxable earnings in a predefined level. This conclusion is supported by empirical findings reported by Nance, Smith, and Smithson (1993), among others. We measure the tax function’s convexity using a modified version of the simulation approach proposed by Graham and Smith (1999). Unlike Graham and Smith, we apply to each firm the tax code of its home country instead of the US code and we repeat our simulations 1000 times instead of 50.9 We calculate the tax savings resulting from a 5% reduction in the volatility to be consistent with the empirical findings reported by Guay (1999).10 As in Graham and Rogers (2002), we scale tax savings by the firm’s sales in the regression analysis and expect this variable to have a positive coefficient.

3.2.3 Financial distress costs Smith and Stulz (1985) show that hedging increases shareholders’ wealth because it decreases the expected value of direct bankruptcy costs and the loss of the debt tax shield. As in Tufano (1996), we measure financial distress costs with two variables: cash cost and leverage. Cash cost measures the operating costs of producing one ounce of gold, excluding all non-cash items such as depreciation, amortization and other financial costs. It captures the operating efficiency of a gold mining firm. Leverage is measured as the book value of the long-term debt divided by the firm’s market value. We expect a positive relation between the delta percentage and both variables measuring financial distress costs.11

3.2.4 Firm size If hedging costs are proportional to the firm’s size as stated by Smith and Stulz (1985), small firms should hedge more. Large firms may hedge less because they are better diversified geographically and by lines of business. Conversely, large firms might hedge more if hedging costs are fixed, especially when these costs are substantial. We use the natural logarithm of the sales revenues to control for firm size and we do not offer any expectations about the sign of this variable.

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3.2.5 Investment opportunities FSS (1993) conclude that firms with attractive investment opportunities are more likely to engage in corporate hedging to ensure the availability of internally generated funds when external financing is costly. Thus, hedging can reduce the underinvestment problem. Morellec and Smith (2002) show that hedging has two opposite effects on the manager’s risk shifting incentives: (i) over the short term, hedging decreases the firm’s free cash flow level and therefore constrains the investment policy, and (ii) over the long run, hedging decreases financial distress costs and improves credit risk, which leads to an increase in investment levels. This second effect prevails when the number of investment opportunities is important. Thus, corporate hedging should be positively associated with the number of growth opportunities. More recently, Lin, Phillips, and Smith (2008) also show that firms with greater investment opportunities hedge more to reduce the probability of financial distress and increase their activities in risky investments. Nance, Smith, and Smithson (1993), Géczy, Minton, and Schrand (1997), Gay and Nam (1998), and Knopf, Nam, and Thornton (2002) report empirical results that support these conclusions. Gold mining firms often expand either internally by exploring new mines or externally by acquiring existing mines. Therefore, we use two measures of the firm’s investment opportunities, namely exploration expenditures and acquisition expenditures, both scaled by the firm’s market value. We expect both variables to have positive coefficients. 3.2.6 Managerial risk aversion According to Smith and Stulz (1985), managers should hedge less when their expected utility is a convex function of the firm’s value. Similarly, compensation packages that create a concave relation between managers’ expected utility and the firm’s value should encourage the latter to hedge more. Hence, managers with large option (shares) holdings should seek more risk (hedging) than those with small or non-existent option (shares) holdings. Tufano (1996), Rogers (2002), and Chen, Jin, and Wen (2008) find support for these conclusions. Nevertheless, Carpenter (2000) shows that stock option-based compensation creates two opposing effects on managerial wealth. First, managers’ wealth increases as stock return volatility and option payoffs increase. This first effect should motivate managers to hedge less. Second, payoffs from options become less important as the stock price decreases. This should cause risk-averse managers to increase their hedging activities to avoid a reduction in the share price. If the second effect prevails, managers will hedge more when they are paid with stock options. This conclusion is supported by findings of Knopf, Nam, and Thornton (2002) and explains the positive relation between option holdings and hedging reported by Géczy, Minton, and Schrand (1997) and Gay and Nam (1998). We measure managerial risk aversion with two variables: the value of the common shares held by directors and officers at the quarter end, and the value of options held by directors and officers.12,13 We expect a positive coefficient for the value of shares and offer no expectations for the sign of the options value coefficient. 3.2.7 Other determinants 3.2.7.1 Board composition: Several explanations provided for corporate hedging are based on agency theory. Given that the mandate of the board of directors is to mitigate problems arising from the agency relations within the firm, it is likely that its composition affects the hedging policy. Unrelated directors and the separation between the CEO and the chairman of the board positions are expected to limit managerial discretion and consequently lead to a hedging ratio that increases shareholders’ wealth. For instance, Borokhovich et al. (2004) report a positive relation between the number of outside directors sitting on the board and the level of interest rate hedging,

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while Adams, Lin, and Zou (2011) show that firms subject to tax monitoring are more likely to buy insurance for managerial self-interest. We use two variables to control for the composition of the board: the share of independent directors sitting on the board and a dummy variable that equals 1 if the CEO holds also the chairman position.14 As long as corporate hedging increases shareholders’ (managers’) wealth, these two variables should have a positive (negative) and a negative (positive) coefficient, respectively. 3.2.7.2 Financial slack: Derivatives and gold loans are not the only tools firms can use to manage gold price risk. For instance, gold mining firms can decide to absorb losses caused by adverse movements in the gold price, especially when the hedging costs are comparable or lower than expected losses. Hence, the existence of financial slack should be negatively related to the level of hedging. Nance, Smith, and Smithson (1993) and Tufano (1996) find support for this hypothesis. Yet, one could argue that this negative relation will disappear when the debt decision is endogenously determined and the financial slack is funded by debt.15 We use the quick ratio to measure the firm’s financial slack. The quick ratio is defined as the value of the cash on hand, short-term investments and clients’ accounts divided by the short-term liabilities. We offer no expectations about the sign of this variable. Finally, we include a dummy variable equal to 1 if the firm is US, 0 otherwise, to control for the firm’s home country. 3.3 Determinants of corporate leverage We include the delta percentage as an explanatory variable. A positive and significant coefficient will support the conclusion of Graham and Rogers (2002) and Belghitar, Clark, and Judge (2007) that hedging increases debt capacity. Nevertheless, our theoretical model suggests that hedging does not lead to a higher debt capacity if the second effect (an increase in the default intensity) prevails or is significant enough to neutralize the first one (a decrease in the firm’s risk). Therefore, we offer no directional expectations about the sign of the delta percentage coefficient. The other explanatory variables are standard in the literature (e.g. Titman and Wessels (1988)). We control for the firm’s collateral value, the non-debt tax shield, the tax advantage from debt financing, the firm’s growth opportunities, the firm’s uniqueness, the firm’s size, the firm’s profitability and operational risk, and finally the firm’s home country. Table 1 summarizes the variables used in the regression analysis and the predicted signs for their coefficients. 4. Sample construction Our initial data consist of quarterly detailed information describing the hedging activities of 48 North American gold mining firms over the period 1991–1999. These data were graciously provided by Ted Reeve, a Canadian analyst covering the North American gold mining industry.16 We used this information to calculate a measure of corporate hedging introduced by Tufano (1996) called the delta percentage. The delta percentage measures, for a given quarter, the fraction of the planned gold production that is being hedged over the next three years through derivatives and gold loans.17 For each firm-quarter observation, we collected additional data from COMPUSTAT Quarterly on the firm’s market value, leverage, liquidity, acquisition expenses, operating income, selling and general expenses, depreciation and amortization, as well as on the book value of its property, plant

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Table 1. Summary of the variables used in the analysis. Predicted sign for the variable What we want to measure

How we measure it

The risk management activity Delta % The delta of the risk-management portfolio held by the firm divided by its expected gold production The financial distress costs Leverage The book value of the firm’s long-term debt divided by its market value Cash cost ($US/oz) The operating cost of producing one ounce of gold, excluding all non-cash items such as depreciation, amortization, and other financial costs Information asymmetry Institutional shareholding The percentage of shares held by institutions The tax advantage of hedging Tax save Tax savings resulting from a 5% reduction in the volatility of the taxable income. This variable is constructed using a modified version of Graham and Smith’s (1999) approach Size ln (sales) The natural logarithm of the firm’s sales revenues Investment opportunities Exploration The firm’s exploration expenditures scaled by its market value Acquisition The firm’s acquisitions expenditures scaled by its market value Managerial risk aversion D&O CS value The number of common shares held by D&O multiplied by their market price D&O value of options The value of options held by D&O Composition of the board % of unrelated The number of unrelated directors divided by the board’s size. A director is defined as unrelated if he is independent of the firm’s management and free from any interest and any business or relationship that could be perceived to affect his ability to act as a director with a view to the best interests of the firm, other than interests arising from shareholdings. A director who is a former employee of the firm is defined as related

RM equation

Debt equation ?

+ +

−

+

?

+

+

+

+

+

+ ? + (−) if hedging is in the interest of shareholders (managers)

(Continued)

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Table 1. Continued. Predicted sign for the variable What we want to measure Dummy COB

Liquidity Quick ratio

The firm’s collateral value BV of pp&eq

How we measure it A dummy equals 1 if the CEO is also the chairman of the board

The value of firm’s cash on hand, short-term investments. and client’s accounts divided by its short-term liabilities

RM equation − (+) if hedging is in the interest of shareholders (managers) ?

+

The firm’s book value of property, plant and equipment divided by its book value of total assets

The non-debt tax shield Dep & Amt

Depreciation and amortization divided by the book value of total assets The tax advantage for financing with debt MTR The firm’s marginal tax rate defined as the additional taxes paid on an additional dollar of income The firm’s uniqueness Sgl&Adm The firm’s selling, general, and administrative expenses divided by the net sales Profitability Operating income The firm’s operating income scaled by its sales Operational risk Volatility of % change in The volatility of the % change in the OI firm’s operating income Nationality Dummy US A dummy equals 1 if the firm is US, 0 if it is Canadian

Debt equation

−

+

−

−

−

?

?

Notes: This table reports a description of the variables used in the multivariate analysis as well as the predicted sign for their coefficients. Some variables are scaled by the firm’s market value calculated as the number of common shares multiplied by their unit market price plus the number of preferred shares multiplied by their value at par plus the book value of debt. RM, risk management; D&O, directors and officers; CS value, common shares value; pp&eq, property, plant and equipment; Dep&Amt, depreciation and amortization; OI, operating income; Sgl&Adm, selling, general and administrative expenses.

and equipment, and sales. Data describing operating costs and exploration expenditures were handcollected from quarterly reports, while directors and officers’ shareholdings and option holdings, the percentage of shares owned by institutions, the board size and composition were hand-collected from proxy statements and annual reports. To maximize our sample size, we complement our

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G. Dionne and T. Triki Table 2. Descriptive statistics for the delta percentage. Delta % (1993–1999)

Number of observations

Exactly 0 88 0–0.1 125 0.1–0.2 89 0.2–0.3 59 0.3–0.4 32 0.4–0.5 23 0.5–0.6 29 0.6–0.7 16 0.7–0.8 12 0.8–0.9 6 Over 0.9 29 Number, 508; mean, 0.2451; median, 0.1381; standard deviation, 0.2808 Notes: The delta % is the fraction of the gold production that is hedged over the next three years. It is our measure of the firm’s risk management activity. This table reports the descriptive statistics of the delta percentage.

Table 3. Distribution of the delta percentage over the sampling period (1993–1999). Year Delta % 1993 1994 1995 1996 1997 1998 1999

Number of observations

Mean

Median

Standard deviation

16 50 52 55 108 123 105

0.0233 0.1992 0.1632 0.1886 0.2374 0.3068 0.3058

0.0000 0.0777 0.0754 0.1076 0.1692 0.2309 0.1854

0.0448 0.3003 0.2618 0.2446 0.2342 0.2864 0.3209

Note: This table reports the distribution of the delta percentage over the years as well as the descriptive statistics for each year.

dataset with information collected from internet research and additional documents provided by sample firms that we managed to contact. Our final sample consists of 485 quarter-company observations over the period 1993–1999 concerning 36 North American gold mining companies: 25 Canadian and 11 US. As Table 2 shows, most firms composing our sample have delta percentages ranging between 0% and 50%, which indicates that these firms do not fully hedge their production. This contrasts with FSS’s (1993) full hedging expectation. Table 3 also suggests that corporate hedging becomes more popular as we approach the end of the 1990s. This is probably caused by the growing popularity of the derivatives’ market during this period. Table 4 provides summary statistics for our variables and shows that our sample firms have an average debt ratio of 11.9% and relatively small exploration opportunities compared with the firm’s market value. Managers in these firms hold on average USD 16.9 million and USD 2.1 million in the form of the firm’s shares and stock options, respectively. Also, more than two-third of the board members are on average outside directors while institutional shareholders hold on average 17.7% of the firm’s shares.

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Table 4. Descriptive statistics for the independent variables.

Variable Cash cost Tax save Leverage ln (sales) Acquisition Exploration Quick ratio Institutional shareholding D&O CS value D&O value of options % of unrelated Operating income Sgl&Adm Volatility of % change in OI Dep & Amt BV of pp&eq MTR

N

Mean

Median

Standard deviation

516 494 506 513 517 517 517 517 517 485 517 513 513 485 516 516 497

247 0.1381 0.1186 3.3220 0.0119 0.0037 3.1937 0.1766 16.9145 2.1 0.7018 −0.0761 0.1483 12.9680 0.0181 0.6187 0.1578

239 0.0371 0.0842 3.1247 0.0000 0.0022 2.2022 0.0000 2.3021 0.039 0.7143 0.1797 0.1105 2.1128 0.0152 0.6463 0.1625

61.7266 0.2822 0.1271 1.3674 0.0774 0.0088 3.0946 0.2536 46.8042 6.98 0.1580 1.1046 0.1854 125.3759 0.0176 0.1713 0.1248

5. Methodology Empirical papers commonly use Tobit specifications to model hedging and leverage decisions given significant proportions of the observations that are censored at zero. While estimation techniques are widely available to implement such tests for single equation models and simultaneous equations models with cross-sectional data, to date, there is no widely accepted methodology in financial economics that provides efficient estimation of censored dependent variables that are determined simultaneously for panel datasets. This raises a problem for researchers working with panel datasets who seek to better understand the simultaneity of hedging and leverage decisions. In order to address this issue, we develop an estimation technique that could be applied to a model of two simultaneous equations with two censored dependent variables using a panel dataset. We decided to rely on the MDE technique developed by Amemiya (1978) because it provides unbiased and consistent estimates of the system structural form (SF) coefficients even when the error terms in the equations are correlated. Lee (1995) applies the MDE to a system of three simultaneous equations with a censored, a dichotomous, and a regular dependent variable, respectively. However, his work applies only to cross-sectional data. To address this problem, we extend Lee’s (1995) methodology to panel data by (i) deriving the system reduced form (RF) and mathematical relations between all relevant parameters; (ii) generating the log-likelihood function corresponding to the model; and (iii) introducing numerical techniques to compute derivatives of the log-likelihood with regard to η, θ , and σ for the variance–covariance matrix. Given that our log-likelihood function is more complicated, we were not able to use closed solutions for the derivatives as in Lee (1995). We introduce numerical techniques to do so (see Dionne and Triki, 2004, for details). The system SF can be written as:  ∗ ∗ ′ = α12 y2it + X1it β1 + u1i + e1it y1it (7) ∗ ∗ ′ y2it = α21 y1it + X2it β2 + u2i + e2it

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∗ ∗ where y1it and y2it are the, respective, hedging and leverage ratios targeted by the firm; X1it and X2it are (k1 × 1) and (k2 × 1) vectors of exogenous variables (including a constant term) that putatively affect the hedging and leverage policies, respectively; u and e correspond, respectively, to the random firm effect and error term; and α’s and β are the parameters to be estimated. Only ∗ ∗ maximum (y1it , 0), maximum (y2it , 0), X1it , and X2it are observed. We assume that (u1i , u2i ) and (e1it , e2it ) are jointly normally distributed with a 0 mean and that α12 × α21 = 1. The system RF can be derived as:  ∗ = Xit′ η1 + l1i + γ1it y1it (8) ∗ y2it = Xit′ η2 + l2i + γ2it , ′ ′ ]; η1 , X2it where Xit′ is a vector including all exogenous variables in the system, such that Xit′ = [X1it and η2 are vectors including RF parameters; l1t and l2t are the RF random firm effects; and γ and 1it

β1 /

β1 α21 /

γ2it are the error terms. Let = 1 − α12 × α21 . We can show that η1 = β2 α12 / , η2 = β2 / , and ⎧ 1 α12 ⎪ l1i = u1i + l1i ∼ N(0, θ12 ) u2i ; ⎪ ⎪ ⎪



⎪ ⎪ ⎪ ⎪ 1 α ⎪ ⎪l2i = 21 u1i + u2i ; l2i ∼ N(0, θ22 ) ⎨



⎪ ⎪γ = 1 e + α12 e ; λ ∼ N(0, σ 2 ) ⎪ ⎪ 1it 1it 2it 1it 1 ⎪



⎪ ⎪ ⎪ ⎪ ⎪ 1 α21 ⎩ e1it + e2it ; λ2it ∼ N(0, σ22 ) γ2it =



The first step in the MDE procedure consists of estimating the RF parameters. In our case, each equation in the RF corresponds to a random effect Tobit model which we estimate by the maximum-likelihood method. This step provides us with estimates of etas (η), thetas (θ ), and sigmas (σ ). Next, the relationships between the RF and the SF parameters are used to formulate the following restrictions:  η1 = α11 η2 + J1 β1 (9) η2 = α21 η1 + J2 β3 , ′ ′ . and Xit′ J2 = X2it where J1 and J2 are the exclusion matrices constructed such as: Xit′ J1 = X1it These restrictions are used to recover consistent but inefficient estimates of the SF parameters. To do so, we replace η1 and η2 by ηˆ 1 and ηˆ 2 obtained from step 1, add an error term ωk (k = 1, 2) to each equation in (9) and estimate using ordinary least-squares (OLS). The last step of the procedure consists in calculating a variance–covariance matrix based on the effective scores of ETAs from each RF equation, and using it as a weighting matrix to compute efficient estimates of the SF parameters. Lee (1995) defines the effective score of ETA as the residual from the regression of the score of ETA on the score of the error term standard deviation. In our case, the score of the error term standard deviation is a matrix with two components: the first component corresponds to the score of θ (the standard deviation of the random firm effect) and the second component corresponds to the score of σ (the standard deviation of the error term). Getting an effective score requires the computation of the log-likelihood derivatives with regard to η, θ, and σ , which is very challenging. To overcome this problem, we first generate the log-likelihood function corresponding to a random effect Tobit model and evaluate the derivatives numerically using the derivative definition. Numerical integration is done using the Gauss–Hermite quadrature

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method. Because our dataset corresponds to an unbalanced panel, we compute the derivatives firm by firm. The effective scores of ETAs are then multiplied by the inverted information matrix and used along with the inefficient estimates of the SF parameters obtained from the previous step, as inputs to calculate the variance–covariance matrix. Finally, we estimate the system using a procedure analogous to a SURE method while using the Moore–Penrose regularization technique to address problems arising from the singularity of the variance–covariance matrix.18

6. Empirical results To asses the importance of appropriate econometric modeling, we report in this section, the results for single equation models (both for leverage and hedging) and for simultaneous equations. We use Tobit specifications to estimate single equation models to account for the censoring of the dependent variables. Greene (2004) shows that the incidental parameter problem that affects fixed effects specifications does not lead to biased estimates of the slope in the case of a Tobit specification, but does cause a downward bias in estimated standard deviations. This might lead to erroneous conclusions about the statistical significance of variables included in the regressions. Therefore, we use random effect Tobit specifications to estimate these models. 6.1

Results with single equation models

Table 5 summarizes our results drawn from single equation models. The first two columns correspond to our findings for the hedging policy. Consistent with H1, the two measures of financial distress costs have positive and significant coefficients at the 1% level, suggesting that firms hedge to reduce financial distress costs. Our results also show that larger firms are more likely to hedge which is consistent with conclusions of Haushalter (2000). This probably reflects the high costs of corporate hedging. We also provide support for the assumption that firms with important liquidity hedge less. Similarly, our two measures of managerial risk aversion have significant coefficients at the 1% level and support Smith and Stulz’s (1985) conclusion that paying managers with shares will motivate them to hedge more, whereas paying them with options will give them an incentive to seek more risk. The institutional shareholding variable also has a negative and significant coefficient at the 1% level, suggesting that reducing information asymmetry costs is another motive for corporate hedging. Results in Table 5 show that CEOs who are also chairmen of the board seek more hedging, which suggests that hedging is mainly motivated by managerial risk aversion. The last two columns of Table 5 summarize our results for the leverage equation. The delta percentage has a positive and significant coefficient at the 1% level, suggesting that more hedging leads to higher debt capacity. This finding contrasts with H2. We also find that higher non-debt tax shields make debt less-attractive while unique firms are more likely to contract debt. Further, our results show that firms with important exploration opportunities are able to contract more debt while those reporting higher profitability levels contract less debt, reflecting their lower needs for external funding. Overall, empirical evidence reported in Table 5 suggests that hedging in the gold mining industry is motivated by managerial risk aversion, and firm value maximization through a reduction of financial distress costs and costs related to information asymmetry. Hedging is also positively related to the firm’s size, confirming the argument of scale economies. Finally, our results with

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G. Dionne and T. Triki Table 5. Results with single equation models. RM equation Coefficient Constant Tax save Leverage Delta % Cash cost ln (sales) Acquisition Exploration Quick ratio Institutional shareholding D&O CS Value D&O value of options % of unrelated Dummy COB Operating income Sgl&Adm Dep & Amt BV of pp&eq MTR Volatility of % change in OI Dummy US Number of observations Uncensored observations P-value

P-value

−0.310 0.078 0.855

0.021∗∗

0.001 0.055 0.099 −0.001 −0.010 −0.348 0.002 −0.010 0.152 0.083

0.000∗∗∗ 0.015∗∗ 0.341 0.576 0.024∗∗ 0.000∗∗∗ 0.001∗∗∗ 0.000∗∗∗ 0.187 0.012∗∗

−0.025 485 404 0.000

Leverage equation Coefficient 0.010

0.842

0.119

0.000∗∗∗

0.016 0.003 0.001

0.115 0.927 0.009∗∗∗

−0.007 0.086 −0.380 0.006 0.007 0.0001 −0.042 485 401 0.000

0.020∗∗ 0.035∗∗ 0.082∗ 0.860 0.839 0.253 0.191

0.173 0.000∗∗∗

0.761

P-value

Notes: This table reports results for single equation models estimated using random effect specification. The dependent variable in the risk-management equation is the delta percentage defined as the fraction of the gold production that is hedged for the next three years. The delta % is measured at the quarter end. The dependent variable in the leverage equation is debt ratio defined as the value of long-term debt divided by the firm’s value. RM, risk management. ∗ Statistical significance at 90%. ∗∗ Statistical significance at 95%. ∗∗∗ Statistical significance at 99%.

the single equation models show that the relationship between hedging and debt goes in both directions. 6.2

Results with an endogenous debt decision

Table 6 summarizes our results for the simultaneous equations system. The first two columns describe our findings for the hedging policy. Leverage maintains a positive and significant coefficient at the 5% level, suggesting that firms hedge to reduce financial distress costs arising from higher debt levels. This result is consistent with Lin, Phillips, and Smith (2008) findings. Cash cost loses its statistical significance, suggesting that only costs arising from financial distress rather than operational inefficiencies affect corporate hedging. The managerial risk aversion argument is also supported because the coefficients of the two variables remain significant at the 1% level. We also find that liquidity remains negatively related to the hedging ratio when debt is set endogenous. Other results are also consistent with conclusions drawn from the single equation model:

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Table 6. Results for the simultaneous equations system. RM equation Coefficient Constant Tax save Leverage Delta % Cash cost ln (sales) Acquisition Exploration Quick ratio Institutional shareholding D&O CS value D&O value of options % of unrelated Dummy COB Operating income Sgl&Adm Dep & Amt BV of pp&eq MTR Volatility of % change in OI Dummy US Number of observations Uncensored observations

Debt equation

P-value

Coefficient

−0.255 0.119 1.711

0.000∗∗∗

−0.042

0.332

−0.061

0.138

−0.0002 0.016 1.027 −0.002 −0.013 −0.370 0.002 −0.007 0.146 0.069

0.414 0.395 0.006∗∗∗ 0.142 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗

0.014 −0.652 0.003

0.020∗∗ 0.094∗ 0.000∗∗∗

−0.001 −0.080 −0.473 0.057 −0.126 −0.001 0.009 485 401

0.826 0.001∗∗∗ 0.058∗ 0.017∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.564

0.022 485 404

P-value

0.122 0.024∗∗

0.555

Notes: This table reports results for the simultaneous equations system using the minimum distance method. The first equation in the system models the risk management (RM) decision in the firm, and the second equation models the debt decision (Debt). The dependent variables in both the equations are, respectively, the delta % and the firm’s leverage. ∗ Statistical significance at 90%. ∗∗ Statistical significance at 95%. ∗∗∗ Statistical significance at 99%.

firms hedge to reduce their information asymmetry costs, and CEOs who are also chairmen of the board are more attracted to corporate hedging. This last result supports the idea that risk management benefits managers. One of the variables controlling the firm’s investment opportunities, acquisition expenditures, becomes significant at the 1% level, suggesting that firms that expand through active acquisitions might hedge more to secure sufficient resources to complete acquisitions during periods of low gold prices. This is not surprising given that gold mining firms are more likely to get good deals on their acquisitions when the gold price is low. As in Borokhovich et al. (2004), we find that unrelated directors encourage corporate hedging. This result, combined with the positive coefficient we reported for CEOs who are also chairmen, suggests that corporate hedging benefits both the firm and managers. Additionally, when we set the debt decision endogenous, the variable controlling for the firm’s size loses its explanatory power. The last two columns of Table 6 report the results for the leverage equation. Interestingly, the coefficient of the delta percentage is no longer significant, suggesting that firms do not necessarily hedge to increase their debt capacity, as reported by Graham and Rogers (2002). This finding contrasts with our conclusions drawn from the single equation model and illustrates the importance

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of modeling leverage as endogenous. It also provides support for H2. Obviously, our results could be driven by industry specificities related to the type of risk hedged or to gold mining firms’ difficulties to raise debt funding during the period under study. Indeed, Belghitar, Clark, and Judge (2007) show that the effect of hedging on debt capacity depends on the type of risk hedged. For instance, they show that the debt capacity benefits of interest rate hedging are six times larger than those generated by foreign exchange only hedging. The superior benefits of interest rate hedging on debt capacity are intuitive and arise from the cost-reducing effect of hedging on synthetic debt issues and the greater flexibility it offers to alter debt characteristics. The benefits of foreign currency hedging could reflect their potential to reduce risks arising from foreign debt financing. Therefore, Graham and Rogers’ (2002) and Borokhovich’s et al.’s (2004) conclusion on a positive relation between corporate hedging and debt could result from their focus on hedging activities of interest and foreign exchange risks, which potentially affects the characteristics of the firm’s debt contract. Thus, we fail to find such relation between hedging and debt because commodity price hedging may not directly affect debt contract characteristics. Alternatively, our results could be driven by gold mining firms’ difficulties to raise debt financing. Table 4 shows that an average firm in our sample has a leverage ratio of 11.86%, while 50% of our firms have leverage ratios that do not exceed 9%.19 Adam (2009) notes that low leverage reported by gold mining firms, over the period covered by our sample, reflects these firms’ difficulties to attract debt financing. According to Adam (2009), average profit margin over cash production costs in the North American gold mining industry during the 1990s stood at 47%. Such slim margins make only firms with stable cash flows able to contract more debt. Thus, it is possible that gold mining firms that were able to contract more debt because of stable cash flows resulting from active hedging used this benefit excessively to increase leverage. This ultimately translated into higher default intensity and reduced their capacity to increase leverage further. Results in Table 6 show that leverage is negatively related to the firm’s uniqueness, supporting the hypothesis that unique firms are more difficult to evaluate and consequently are less able to attract debt financing. Results also show that the existence of non-debt tax shields reduces the attractiveness of leverage. Interestingly, when we model hedging and leverage simultaneously, the firm’s profitability is no longer a significant determinant of hedging while the firm’s size and collateral value become positively related to the debt ratio. Furthermore, firms with higher levels of operational risk and those having more attractive acquisition opportunities report lower leverage. Overall, our empirical findings suggest that once we control the endogenous relation between risk management and debt decisions, firms hedge to reduce their financial distress and information asymmetry costs, and to increase their manager’s utility. The size argument loses its explanatory power while the share of unrelated directors and acquisition expenditures become significant determinants of the hedging ratio. We show that the hedging ratio no longer affects leverage, which contrasts with our results drawn from single equation models. Nevertheless, our results could be driven by gold mining firms’ difficulties to raise debt funding or to the type of risk under study. Regardless, the results illustrate the importance of appropriate econometric modeling to avoid erroneous conclusions. 7. Conclusion Theoretical models by Stulz (1996), Leland (1998), and Lin, Phillips, and Smith (2008), along with empirical work by Graham and Rogers (2002), Borokhovich et al. (2004), and Belghitar, Clark, and Judge (2007) show the need to model leverage as endogenous when studying corporate

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hedging determinants. This paper builds on this literature. We first develop a theoretical model where the debt and the hedging decisions are set simultaneously and use it as a background for our empirical tests. Our theoretical model suggests that, under a standard debt contract, hedging has two opposite effects on leverage and does not always lead to a higher debt capacity. We run empirical tests to investigate corporate hedging determinants using a newly developed methodology that allows us to account for the simultaneity of leverage and hedging decisions while using a panel dataset and controlling for censored dependent variables. Failure to account for such simultaneity could have led to erroneous conclusions in previous research. We construct a database that contains detailed quarterly information on risk-management operations, as well as financial and managerial characteristics for a sample of North American gold mining firms over a seven-year period. Our results confirm Tufano’s (1996) conclusion that managerial risk aversion is an important determinant of the hedging ratio. Our evidence also shows that financial distress costs, information asymmetry costs, and the presence of liquidity cushions are important determinants of corporate hedging. Similarly, unrelated directors and CEOs who also chair the board of directors seem to encourage corporate hedging. Our results suggest that while firms hedge to reduce their financial distress costs, more hedging does not increase debt capacity. This conclusion contrasts with results drawn from single equation models and evidence reported in the study of Graham and Rogers (2002). While our findings are likely to be industry-driven, they underscore the importance of properly modeling leverage and hedging decisions as simultaneous in empirical tests of corporate hedging. Acknowledgements This research was financed by SSHRC Canada, the Canada Research Chair in Risk Management, and IFM.2 We thank Ronald Leung, Ousman Gagigo, and an anonymous referee for their comments, and Karima Ouederni for excellent research assistance.

Notes 1. See, among others, Stulz (1984), Smith and Stulz (1985), Stulz (1990, 1996); DeMarzo and Duffie (1991); Froot, Scharfstein, and Stein (1993); Morellec and Smith (2002); Breeden and Viswanathan (1998); Holmström and Tirole (2000); Carpenter (2000); and Hoyt and Liebenberg (2011). 2. Although differences in results could reflect dissimilarities in samples, conflicting results were also reported for studies focusing on a single industry. For instance, Tufano (1996) shows that North American gold mining firms hedge only because of managerial risk aversion while Dionne and Garand (2003) find support for value-maximizing theories. 3. Stulz (1996), Leland (1998), and Lin, Phillips, and Smith (2008) show, for example, that hedging can increase debt capacity while empirical results reported in Graham and Rogers (2002), Borokhovich et al. (2004), and more recently Belghitar, Clark, and Judge (2007) support this argument. Based on this evidence, the leverage decision should be modeled as endogenous when studying the determinants of corporate hedging. 4. The delta percentage is defined as the delta of the risk-management portfolio held by the firm divided by its expected production. 5. For further discussion related to the use of the gross and notional values as measures of corporate hedging, refer to Triki (2006). 6. For a gold mining firm, w0 corresponds to the current forward price of gold multiplied by the total production of the firm, while x corresponds to the ratio of the revenues obtained if the production is sold later on the spot market divided by the revenues obtained if the production is sold at the current forward price. 7. Leland (1998) contrasts the case where the firm chooses its hedging and debt policies simultaneously with the case where the hedging strategy is defined ex-post given the amount of debt in place. He uses the difference in maximal firm values between both cases as a measure of agency costs and shows that while hedging is always beneficial to

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8. 9.

10. 11.

12.

13.

14. 15. 16. 17. 18. 19.

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the firm, the inability to pre-commit ex-ante to hedging reduces hedging benefits by almost one-third. Thus, based on Leland’s (1998) findings, introducing agency costs with an ex-post choice of hedging strategy in our model is likely to lead to lower optimal hedging ratio. The increasing relation between x and the hazard rate could be verified for other distributions, e.g. exponential and uniform. Using the drift and volatility estimates from the first step in Graham and Smith’s (1999) procedure, we generate for each US firm a normal variable ε with 18 realizations (15 years to account for carry forwards and 3 years to account for carry backs). In the United States, since late 1998, net operating losses can be carried back 2 years and forward 20 years. However, because our sample ranges mainly between 1992 and 1999, we use the old legislation that allows firms to carry back losses for 3 years and forward for 15 years. For Canadian firms, we generate a normal variable with only 10 realizations because the Canadian legislation allows firms to carry back net operating losses for 3 years and forward only for 7 years. Based on the existing methodology, it is impossible to construct the tax save variable on a quarterly basis. Therefore, we calculate it on an annual basis and assume that it is constant for the four quarters of the year. The two measures we use for financial distress costs reflect the likelihood of encountering distress rather than distress costs. This should not be a problem given that our sample includes firms operating in a single industry (gold mining) and within a homogenous region (North America). Thus, both variables remain acceptable measures given that our sample firms are likely to face similar distress costs if distress happens. Given that information on option holdings is disclosed on an annual basis, we assume that the number of options held by directors and officers is constant over the fiscal year. This hypothesis is acceptable because firms usually wait for fiscal year end performance to determine the number of options that will be granted to directors and officers. We did not use the sensitivities of D&O option portfolio to stock return (Delta) and stock return volatility (Vega) as proxies for managerial risk aversion because of data limitation. Moreover, as stated in Rajgopal and Shevlin (2002), the partial derivatives used to calculate the sensitivities likely overstate the real values of the ESO risk incentive (Vega) and the ESO wealth effect (Delta). Although our variables admittedly have their own limitations, we think they represent acceptable proxies for managerial risk aversion. Refer to Table 3 for further details about how we define independent directors. Holmström and Tirole (2000) show that the link between hedging and liquidity is unclear when leverage is endogenously determined and hedging is costly. During the 1990s, Ted Reeve published quarterly reports containing detailed three-year information on hedging activities for North American gold mining firms. Mr Reeve stopped publishing these reports in 1999. For further details about the methodology used to calculate the delta percentage, please refer to Tufano (1996). The Moore–Penrose inverse of a matrix (pseudo inverse) is calculated using singular value decomposition. For further details about regularization techniques, refer to Moore (1920), Penrose (1955), and Yuan and Chan (2008). Such ratios are low when compared with other industries. For instance, Carter, Rogers, and Simkins (2006) report debt ratios ranging between 25.7% and 31.4% over the period 1993–1999 for the US airline industry.

References Adam, T. 2009. Capital expenditures, financial constraints, and the use of options. Journal of Financial Economics 92: 238–51. Adams, M., C. Lin, and H. Zou. 2011. Chief executive officer incentives, monitoring, and corporate risk management: Evidence from insurance use. The Journal of Risk and Insurance 78, no. 3: 551–82. Belghitar, Y., E. Clark, and A. Judge. 2007. The value effects of foreign currency and interest rate hedging: The UK evidence. Working Paper, Middlesex University, UK. Borokhovich, K., K. Brunarski, C. Crutchley, and B. Simkins. 2004. Board composition and corporate use of interest rate derivatives. The Journal of Financial Research 27: 199–216. Breeden, D., and S. Viswanathan. 1998. Why do firms hedge? An asymmetric information model. Working Paper, Duke University, NC. Caillaud, B., G. Dionne, and B. Jullien. 2000. Corporate insurance with optimal financial contracting. Economic Theory 16: 77–105. Carpenter, J. 2000. Does option compensation increase managerial risk appetite? The Journal of Finance 55: 2311–31. Carter, D., D. Rogers, and B.J. Simkins. 2006. Does hedging affect firm value? Evidence from the US airline industry. Financial Management, 35: 53–86. Chen, C., W. Jin, and M.M. Wen. 2008. Executive compensation, hedging, and firm value. Working Paper, California State University, USA.

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Cross, J. 2000. Gold derivatives: The market view. London, England: The World Gold Council Publications. DeMarzo, P., and D. Duffie. 1991. Corporate financial hedging with proprietary information. Journal of Economic Theory 53: 261–86. DeMarzo, P., and D. Duffie. 1995. Corporate incentives for hedging and hedge accounting. The Review of Financial Studies 8: 743–71. Dionne, G., and M. Garand. 2003. Risk management determinants affecting firms’ values in the gold mining industry: New empirical evidence. Economics Letters 79: 43–52. Dionne, G., and T. Triki. 2004. On risk management determinants: What really matters? http://papers.ssrn.com/sol3/papers. cfm?abstract_id=558761 (accessed February 22, 2012). Froot, K., D. Scharfstein, and J. Stein. 1993. Risk management: Coordinating corporate investment and financing policies. The Journal of Finance 48: 1629–58. Gay, G., and J. Nam. 1998. The underinvestment problem and corporate derivatives use. Financial Management 27: 53–69. Géczy, C., B. Minton, and C. Schrand 1997. Why firms use currency derivatives. The Journal of Finance 52: 1323–54. Graham, J., and D. Rogers. 2002. Do firms hedge in response to tax incentives? The Journal of Finance 57: 815–39. Graham, J., and C. Smith. 1999. Tax incentives to hedge. The Journal of Finance 54: 2241–62. Greene, W. 2004. The behaviour of the maximum likelihood estimator of limited dependent variable models in the presence of fixed effects. Econometrics Journal 7: 98–119. Guay, W. 1999. The sensitivity of CEO wealth to equity risk: An analysis of the magnitude and determinants. Journal of Financial Economics 53: 43–72. Haushalter, D. 2000. Financing policy, basis risk, and corporate hedging: Evidence from oil and gas producers. The Journal of Finance 55: 107–52. Holmström, B., and J. Tirole. 2000. Liquidity and risk management. Journal of Money, Credit and Banking 32: 295–319. Hoyt, R.E., and A.P. Liebenberg. 2011. The value of enterprise risk management. Journal of Risk and Insurance 78: 795–822. Knopf, J., J. Nam, and J. Thornton. 2002. The volatility and price sensitivities of managerial stock option portfolios and corporate hedging. The Journal of Finance 57: 801–13. Lee, M.J. 1995. Semi-parametric estimation of simultaneous equations with limited dependent variables: A case study of female labour supply. Journal of Applied Econometrics 10: 187–200. Leland, H. 1998. Agency costs, risk management, and capital structure. The Journal of Finance 53: 1213–44. Lin, C.M., R.D. Phillips, and S.D. Smith. 2008. Hedging, financing, and investment decisions: Theory and empirical tests. Journal of Banking and Finance 32: 1566–82. Moore, E.H. 1920. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society 26: 394–5. Morellec, E., and C. Smith Jr. 2002. Investment policy, financial policies, and the control of agency conflicts. Working Paper No. FR 02-16, The Bradley Policy Research Center, Rochester, NY. Nance, D., C. Smith Jr., and C. Smithson. 1993. On the determinants of corporate hedging. The Journal of Finance 48: 267–84. Penrose, R.A. 1955. A generalized inverse for matrices. Proceedings of Cambridge Philosophical Society 51: 406–13. Rajgopal, S., and T. Shevlin. 2002. Empirical evidence on the relation between stock option compensation and risk taking. Journal of Accounting and Economics 33: 145–71. Rogers, D. 2002. Does executive portfolio structure affect risk management? CEO risk-taking incentives and corporate derivatives usage. Journal of Banking and Finance 26: 271–95. Smith, C. and R. Stulz. 1985. The determinants of firms’ hedging policies. Journal of Financial and Quantitative Analysis 20: 391–405. Stulz, R. 1984. Optimal hedging policies. Journal of Financial and Quantitative Analysis 19: 127–40. Stulz, R. 1990. Managerial discretion and optimal financing policies. Journal of Financial Economics 26: 3–28. Stulz, R. 1996. Rethinking risk management. Journal of Applied Corporate Finance 9: 8–24. Titman, S., and R. Wessels. 1988. The determinants of capital structure choice. The Journal of Finance 43: 1–19. Triki, T. 2006. Research on corporate hedging theories: A critical review of the evidence to date. ICFAI Journal of Financial Economics IV: 14–40. Tufano, P. 1996. Who manages risk? An empirical examination of risk management practices in the gold mining industry. The Journal of Finance 51: 1097–37. Yuan, K.H., and W. Chan. 2008. Structural equation modeling with near singular covariance matrices. Computational Statistics and Data Analysis 52: 4842–58.

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Appendix. Derivative of the debt face value expression with regard to the hedging ratio The problem for the firm is to choose the level of hedging h and the face value of its debt F simultaneously in order to maximize the following program:  +∞ [w0 [h + (1 − h)x] + αxw0 − F]g(x) dx L= xf

+λ



xf −∞

 [w0 [h + (1 − h)x] + αxw0 − c]g(x) dx + F[1 − G(xF )] − D(1 + r) ,

where xF = (F − w0 h)/w0 (1 − h + α) and it corresponds to the minimal value of x that allows firms to avoid bankruptcy. Because xF is a function of both F and h, we need to compute its derivative with regard to these two variables. We can show that: 1 dxF = , dF w0 (1 − h + α) dxF F − w0 (1 + α) . = dh w0 (1 − h + α)2 The first-order conditions with regard to F and h are, respectively:   ⎧ dxF ⎪ ⎪ + [1 − G(xF )] = 0 ⎪[G(xF ) − 1] + λ −cg(xF ) ⎨ dF   ⎪ dxF ⎪ ⎪ ⎩−w0 σ 2 g(xF ) + λ −cg(xF ) + w0 σ 2 g(xF ) = 0 dh ⎧ 1 − G(xF ) ⎪ ⎪λ = ⎪ ⎨ 1 − G(xF ) − cg(xF )(1/w0 (1 − h + α)) ⎪ ⎪ ⎪ ⎩λ =

w0

σ2

(A.1)

w0 σ 2 − c(F − w0 (1 + α))/(w0 (1 − h + α)2 )

λ will be greater than 1 as long as 1 − h + α > 0 and F − w0 (1 + α) > 0. Equating both to solve for F ∗ and h∗ gives us: 1 − G(xF ) w0 σ 2 = , ∗ 2 ∗ 1 − G(xF ) − cg(xF )1/(w0 (1 − h + α)) w0 σ − c(F − w0 (1 + α))/(w0 (1 − h∗ + α)2 ) from which we obtain: 1 h∗ = 1 − 2 σ  



1 − G(xf ) g(xf ) 



h∗ for α=0

F ∗ = w0 1 + α + σ 2



⎡

⎢ F ∗ − w0 1 ⎢ +α ⎢1 + 2 ⎣ w0 σ  

g(xf ) (1 − h∗ + α) 1 − G(xf )





⎤

 1 − G(xf ) ⎥ ⎥ ⎥ ⎦ g(xf )  

(A.2)

>0

(A.3)

To isolate the relation between F ∗ and h∗ , we derive Equation (A.3) with regard to h∗ ∂F ∗ g(xF ) d(g(xF )/(1 − G(xF ))) dxF = −σ 2 + (1 − h∗ + α) ∂h∗ 1 − G(xF ) dx dh∗  F      0

The first term in Equation (A.4) indicates that an increase in h reduces the firm’s risk and gives it access to a higher debt capacity by reducing the face value of debt. The second term describes an indirect effect via the hazard rate. Under the normal assumption for x, the default intensity is an increasing function of xF , which is also an increasing function of h under the standard debt contract assumption. Therefore, increasing the hedging activities will increase both the debt capacity and the default intensity. This will lead to a first negative effect and a second positive effect between the firm’s face value and its hedging activities.

International Journal of Computer Mathematics, 2014 http://dx.doi.org/10.1080/00207160.2014.898065

REVIEW

On the understanding of profiles by means of post-processing techniques: an application to financial assets Karina Giberta,b∗ and Dante Contic a Statistics and Operation Research, Universitat Politècnica de Catalunya-Barcelona Tech, Dep. EIO. Ed C5. Campus Nord, C. Jordi Girona 1-3, Barcelona 08034, Barcelona, Spain; b Knowledge Engineering and Machine Learning Group, UPC-Barcelona Tech, Spain; c Department of Operations Research, Universidad de Los Andes, Mérida, Venezuela

(Received 11 October 2013; revised version received 30 January 2014; accepted 19 February 2014) In last years, mining financial data has taken remarkable importance to complement classical techniques. Knowledge Discovery in Databases provides a framework to support analysis and decision-making regarding complex phenomena. Here, clustering is used to mine financial patterns from Venezuelan Stock Exchange assets (Bolsa de Valores de Caracas), and two major indexes related to that market: Dow Jones (USA) and BOVESPA (Brazil). Also, from a practical point of view, understanding clusters is crucial to support further decision-making. Only few works addressed bridging the existing gap between the raw data mining (DM) results and effective decision-making. Traffic lights panel (TLP) is proposed as a postprocessing tool for this purpose. Comparison with other popular DM techniques in financial data, like association rules mining, is discussed. The information learned with the TLP improves quality of predictive modelling when the knowledge discovered in the TLP is used over a multiplicative model including interactions. Keywords: clustering; Knowledge Discovery in Databases; data mining; patterns interpretation; postprocessing; traffic lights panel; financial assets; Venezuela Stock Exchange; association rules; frequent itemsets; general linear model 2010 AMS Subject Classifications: 62H30; 68T10; 68U35; 97R40; 91G80

1. Introduction Nowadays with an exponential increase in data in many fields, tools for an efficient management of these data to obtain meaningful information became critically necessary. Knowledge Discovery in Databases (KDD), often known as data mining (DM), is oriented to manage large and complex data sets and to discover useful and non-trivial information from them [27]. Temporal data sets, specifically time-series have a potential related to the time parameter involved and they have been widely studied for long. Temporal data analysis [3], patterns extraction [18], profiles description, rules association and mining of frequent sequences [14] or temporal relational learning [30] are topics of research within this type of data in the real world [29]. Analysing financial time-series is particularly difficult, as financial data have high volatility and non-stationarity [7]. Classical Statistical modelling approaches such as ARIMA, regression and related methods show poor performance as most of the technical assumptions to apply these ∗ Corresponding

author. Email: [email protected]

© 2014 Taylor & Francis

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methods do not hold in real data [28]. Also, the traditional approaches show technical limitations to handle large amounts of data as the ones available in current finance [32]. In fact, the financial crisis in 2007 enacted an active discussion about the gap between classical models’ requirements and the nature of real financial data and the consequences of the negligent use of financial models. The Financial-Modellers-Manifesto [15] provides deep thoughts on the risks of misuses of classical financial models, already stated by the authors some years before the crisis. New approaches seem to be required to model financial data and the application of DM methods to financial time-series is becoming a popular alternative. In fact, mining financial time-series has shown successful results to describe the underlying processes involved with financial data by supporting tasks such as forecasting, patterns extraction and finding relationships between assets and markets [44]. In financial mining, association rules, clustering and temporal mining are mainly used [26]. These techniques provide a significant support to the financial analyst in terms of understanding, usefulness and prediction [33,46]. This research is in collaboration with the Bolsa de Valores de Caracas in Venezuela and it is aimed to evaluate the benefits of KDD approaches for a better understanding of Venezuelan stock market. Venezuelan stock market is particularly difficult to model because of its special behaviour along the years in the Chavez’s era, who deployed specific government policies, control of foreign exchange rate and regulation in economy. However, the case of Venezuela is very interesting to model for some reasons. Even being a close and small stock market and despite of Chavez’s regulations in the country, the Venezuelan Stock Exchange in Caracas, the IGBC, has shown the best performance all over the world during these years [9,11,40,41]. Many investors believe that this behaviour, called strange rally, is related on the one hand to the Chavez’s policies themselves: local investors use IGBC as a way to obtain profitability and benefits in front of an increasing inflation, excess of liquidity and the inability to access to foreign currencies; in addition, treasure bonds and PDVSA (the Venezuela’s oil company) are present in the market by using banking assets as financial intermediaries. On the other hand, major changes are expected from Venezuela’s economy in the next years which will bring a new but unpredictable overview. Under this scenario, providing tools to support local investors to understand their market and to detect potential profiles and relationships among the assets in both pre and post Chavez’s era is of great interest for the near future. However, few works can be found in the literature regarding the Venezuela stock market, as it is discussed in Section 2. Also, a recent work [6] verifies the difficulties of getting reliable models for Venezuela compared with the modelling of other South American Countries. Under these premises our main goal is to improve predictive modelling by using KDD methods to identify multivariate relationships among assets quoting in IGBC and the IGBC itself, as well as to understand the relationship between IGBC and the two more influent indexes in the geographical zone, i.e. Dow Jones (New York, USA) and BOVESPA (Sao Paulo, Brazil). Association rules mining and clustering seem to be the most suitable approaches for this purpose. Thus, in this work, suitability of both approaches is assessed with data from the Venezuelan Stock market and results are compared. This is the first time that these kinds of techniques are applied to Venezuelan data, but the methodology proposed in this paper can also be extrapolated to other international markets. The results of this step will be later used to improve classical predictive modelling of IGBC. Furthermore, this work is not only providing an application of classical DM techniques to a new domain. It is also making a contribution towards the conception of an integral KDD process, aligning with Fayyad’s conceptions. In the seminal paper [16] Fayyad declares: ‘Non trivial identifying of valid, novel, potentially useful, ultimately understandable patterns in data’ and ‘Most previous work on KDD has focused on [...] DM step. However, the other steps are of considerable importance for the successful application of KDD in practice.’ Indeed, much of the research in KDD is focused on the improvement of the modelling step itself, providing new and more powerful DM methods for more and more complex data, but few works are oriented to

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Figure 1. Interpretation support tools bridging the gap between DM and decision support.

guarantee understandability of DM results. In this work, the models obtained with data mining procedures are sometime difficult to understand by end-users. Understanding models is key to using them for decision-making. Indeed, the main potential of DM results is that they become useful to support decision-making and better help the understanding of the underlying system. However, from a practical point of view, there is an important gap between the raw DM results and effective decision-making [23]. The key is in the interpretation process required, which stands in the middle of these two steps (Figure 1) [13,21]. Post-processing steps are required in both association rules and clustering to obtain the valid, novel and understandable knowledge claimed by Fayyad. In the paper, the Traffic Lights Panel (TLP) [8] is used as a post-processing for the clustering results to achieve understandability. In this work the analysis of financial data by using clustering followed by a post-processing based on TLP is compared with the more popular alternative of mining financial data by using association rules mining and related techniques, like finding frequent itemsets. The paper is structured as follows: Section 1 gives the introduction with the motivation of this research; Section 2 is devoted to the state of the art. In Section 3, the Application domain is presented together with previous work performed with the target data regarding clustering analysis followed by post-processing by means of TLP. In Section 4 association rules’ mining is performed. In Section 5 frequent itemsets analysis is performed. In Section 6 the results are compared with clustering results and a classifier is provided to identify different typical situations of the Venezuelan stock market. Section 7 shows how the induced classifier can be used to improve a classical predictive modelling for IGBC. Finally, in Section 8, discussion, conclusions and future work are provided.

2. Related work Most of the previous works about KDD and DM in financial databases focus on mining financial data by using association rules. Association rules is a technique that refers to the widely mentioned problem of the market basket introduced by Agrawal during the early 1990s [1] and is applied to a wide range of different time-series data sets including financial data sets [14]. Financial timeseries are converted to transactional databases by following a preprocessing step which performs different techniques: descriptive analysis, characterization, discretization/categorization of these data sets to finally get frequent itemsets suitable to apply association rules mining. Different researches describe this association approach: In [36,43] DM is used to find temporal rules in stock-exchange time-series on the Singapore and Taiwan markets. In [31] the NASDAQ and the NYSE in the USA are analysed. The Thai market is focused in Mongkolnavin and Tirapat [37] and

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Conti et al. [10] does a similar research for Madrid Stock Exchange (IGBM). However, there is no research regarding the Venezuelan Stock Exchange (IGBC) by using this association approach. Only two references could be obtained from IGBC and refer to mathematical programming with nonlinear models [9] and artificial neural networks [11]. In both cases, data are used to evaluate investment portfolios supported by financial engineering methods and diversification theory. However, an interesting approach is to identify first typical punctual patterns that can be economically interpreted and afterwards find temporal relationships between those patterns. Different references are found in the literature using time-series clustering under raw-data-based approaches, feature-based approaches and model-based approaches. In [34] a detailed survey on this field is found. Also, in [17] a more recent review in time-series mining (clustering) is shown. This approach has been successfully applied in complex domains such as patient’s follow-up [39] or environmental systems [24]. However, in financial fields there are just a few references regarding clustering. In [6] clustering is used to model profiles in the financial data of several South American countries; the analysis includes an instantaneous picture of the stock market situation in several countries, but does not include the temporal component. Additionally, most of the works pay little attention on the process required to transform the raw results of both clustering or association rules mining into useful information for the decision support. The work of Cervello [6] introduces some visualization of the local distributions of the clusters. Authors of [13,23] stress the importance of devoting proper post-processing tools to guarantee the effective knowledge transfer to the end-user. But few references address these issues. In fact, for the specific case of clustering techniques, some references address the problem from the structural point of view: in [45] fractal theory is used to post-process clusters and detect artificial divisions that can be merged; in [42] a method based on silhouettes is proposed to analyse the structure of the clusters; in [38] automatic interpretation of Self Organizing Maps is done by approaching the contributions of the features to the classes. However, this does not guarantee interpretability of clusters. Only few references tackle the problem of finding the meaning of the clusters independent of the underlying clustering method. In [2] an interesting procedure in the field of image clustering is introduced: independent experts are asked to label the classes and common hyperonims of labels are then found; experts use the whole set of pictures for every cluster, which is not viable with non-graphical data. In previous works, class panel graph (CPG) is introduced as a tool to identify relevant variables in the classes based on conditional distributions. In [12,19] explanations based on concept induction are proposed, with some limitations on scalability. In [22], TLP is proposed as visual symbolic abstraction of CPG that supports conceptualization by experts. In [35] heat maps are used inside every clustering to find local relationships among variables in a health application. In [21] it is successfully applied to environmental data, in [20] to medical data and in [8] to financial data. TLP appears as a promising post-processing tool to support understanding of clustering results.

3. Application domain and previous work This work analyses a database composed of some financial assets from Venezuela Stock Exchange (Bolsa de Valores de Caracas) and the two major indexes related to this market (Dow Jones–USA and BOVESPA–Brazil). Table 1 shows the names, description and economical sector of the considered assets. Data are collected from January 1998 to April 2002 and March 2003 to April 2008, with a total of 364 records, those weeks when the target assets are quoted in the stock market. No data are available between April 2002 and March 2003, since Venezuelan stock market was closed for political reasons related with the coup d’état against Chavez and oil strike. Weekly

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Table 1. List of variables. Variable name CANTV EDC MZV BPV IGBC IBOVESPA DJIA

Description

Economical sector

Compania Nacional Teléfonos Electricidad de Caracas Banco Mercantil Banco Provincial BBVA Índice General Bolsa de Caracas Índice Bolsa de Valores São Paulo Dow Jones Industrial Average

Telecommunications Electrical supply/energy Banking and insurance Banking and insurance Caracas Index Stock Exchange (Venezuela) Sao Paulo Index Stock Exchange (Brazil) Dow Jones Index NY Stock Exchange (USA)

Figure 2. Traffic light panel (colour online only).

variations in both the price of stock assets and the indexes (IGBC, DJIA and BOVESPA) are considered. Xk being one of the considered variables (asset or index), and xkt the price of the asset or index k at the end of week t, the weekly variation of Xk , namely Yk is computed as ykt =

xkt − xk(t−1) × 100. xk(t−1)

In [8] the data were clustered using a hierarchical method under Ward’s method, and normalized Euclidean distance, as all variables are numerical. Five classes were considered according to Calinski-Harabaz’s criterion [5]. The TLP was used as a post-processing tool to support the classes understanding. The resulting TLP is shown in Figure 2 and it was used to build the classes description: C359: All variables denote an uptrend (favourable market situation). This is one of the most common patterns in finances: if most assets are up, then Index trend will be up as well. In this case, CANTV, EDC, MVZB and BVP are green and so is also IGBC (Venezuelan Index). IBOVESPA (Brazil) and DJIA (USA) reflect the same trend, owing to positive correlation often present in these two major indexes. C356: In this case, the four assets (CANTV, EDC, MVZB and BVP) have a neutral variation and it seems that the IGBC Index follows the same trend. Indexes IBOVESPA and DJIA

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have the same trend which is common because of high financial correlation between New York Market and Sao Paulo Market, same as C359. In fact, it is usual that when DJIA closes high, most market in strong economies like Brazil close also high. In this cluster, the IGBC Index moves in a different way than the other major indexes such as IBOVESPA and DJIA, which could be denoted as an important and non-trivial characteristic. C347: This is an important pattern. It denotes that low changes in CANTV, EDC, MVZB and a negative trend in BVP do not affect very much the Venezuelan Index (IGBC) which appears in yellow. This has two possible explanations: first, there are other assets in the market that have not been included in the study, and second, BVP participation in the market is not so strong to affect by itself the global index IGBC. Second, BVP (with major stakeholder Spaniard BBVA Bank) does not negotiate big amounts of action due to Venezuelan policies regarding banking sector, more even if investors are related to foreign ones. Again, IBOVESPA, and DJIA follow the financial correlation explained in C356 and C359, while IGBC shows another trend. C357: This situation is similar to C356, except to IBOVESPA (down) and DJIA (neutral). This cluster represents the situations in which little or null changes in New York might affect the Brazil market. However, this effect is not necessarily observed in Venezuela. Again it is shown that the Venezuelan Market moves in a different way than Brazil and New York. C346: Here, we have another important pattern. When major assets in Venezuela market closed down (red colour), IGBC also closed down, regardless of the trends in New York and Brazil. This clustering represents five patterns that are associated with the major particularities of the Venezuelan market. It has the two main facts: (1) The Venezuelan market is a small and closed market and has its own particular trends, not necessarily moving according to what happened in other bigger markets. Government policies and the presence of public funds in the companies which operate in the market give a special denotation to this stock market exchange. (2) The clustering also shows the current connection that exists between two bigger markets (Brazil and New York) and the possible disconnection with Venezuelan trends. In fact, BOVESPA and DJIA show the same colours in TLP (same trends) in all classes except one, while IGBC show coincidence with them only in the situation that everything is up in both the inner and foreign markets.

4. Understanding by means of association rules Association rule algorithms require qualitative data. For this reason, the data set was discretized by a group of three experts in financial analysis (two with the liberal approach and one with the conservative approach, just to avoid bias to a particular financial view). They were asked to discretize the original variables in a set of qualitative levels proposed by consensus. Five categories were provided for every variable (high down changes, low down changes, no major changes, low up changes and high up changes). Experts decided to take into account the variable’s distribution to range those categories. Thus, the quartiles were proposed as cutpoints between high and low changes and a fix neighbourhood around 0 was considered as no relevant changes from an investor point of view. Q1k and Q3k being the first and third quartiles of each variable Xk . Table 2 shows the cut-offs induced in the variables by this criterion. For each variable, a discretization in five

7

International Journal of Computer Mathematics Table 2. Discretization of variables. Label

Definition

CANTV

EDC

MVZB

BVP

IGBC

BOVESPA

DJIA

High down < Q1k < −4.54 < −4.65 < −2.79 < −2.5 < −2.22 < −2.05 < −1.3 change Low down [Q1k , −0.5) [−4.54, 0.5) [−4.65, 0.5) [−2.79, 0.5) [−2.5, 0.5) [−2.22, 0.5) [−2.05, 0.5) [−1.3, 0.5) change No major [−0.5, 0.5] [−0.5, 0.5] change Low up change (0.5, Q3k ] (0.5, 4.11] (0.5, 3.4] (0.5, 2.63] (0.5, 2.22] (0.5, 2.79] (0.5, 3.93] (0.5, 1.83] High up > Q3k > 4.11 > 3.4 > 2.63 > 2.22 > 2.79 > 3.93 > 1.83 change

Table 3. Rules with higher confidence or higher support, sorted by lift. Rule.id 1 3 2 7 6 55 56 94 95 105 106 143

Rules {ANTECEDENT} => {CONSEQUENT } {CANTV.PR = DA-CANTV EDC.PR = DA-EDC

IBOVESPA=DAIBOVESPA} => {IGBC = DA-IGBC} {CANTV.PR = DA-CANTV MVZB.PR = DA-MVZB BVP.PR = DA-BVP} => {IGBC=DA-IGBC} {CANTV.PR = DA-CANTV EDC.PR = DA-EDC MVZB.PR = DAMVZB} => {IGBC=DA-IGBC} {EDC.PR = CA-EDC MVZB.PR = CA-MVZB DJIA = CB-DJIA} =>{IGBC=CA-IGBC} {CANTV.PR=CA-CANTV EDC.PR=CA-EDC MVZB.PR=CA-MVZB} => {IGBC=CA-IGBC} {CANTV.PR=DA-CANTV} => {IGBC=DA-IGBC} {IGBC=DA-IGBC} => {CANTV.PR=DA-CANTV} {IGBC=CA-IGBC} => {EDC.PR=CA-EDC} {EDC.PR=CA-EDC} => {IGBC=CA-IGBC} {IGBC=CA-IGBC} => {CANTV.PR=CA-CANTV} {CANTV.PR=CA-CANTV} => {IGBC=CA-IGBC} {IBOVESPA=DA-IBOVESPA} => {DJIA=DA-DJIA}

Support Confidence Lift 0.05

1.00

3.82

0.05

1.00

3.82

0.07

1.00

3.82

0.05

1.00

3.56

0.08

1.00

3.56

0.18 0.18 0.19 0.19 0.18 0.18 0.18

0.70 0.68 0.66 0.66 0.65 0.64 0.68

2.67 2.67 2.35 2.35 2.30 2.30 1.52

qualitative levels is derived, according to the following scheme: Assign ‘High down change’ if Xk < Q1k Assign ‘Low down change’ if Xk ∈ [Q1k , 0.5) Assign ‘no major change’ if Xk ∈ [−0.5, 0.5] Assign ‘low up change’ if Xk ∈ (0.5, Q3k ] Assign ‘High up change’ if Xk > Q3k The data were discretized according to this criterion and the a priori [1] algorithm was used to find association rules with minimum support 5% and a minimum confidence of 0.6. No limits were imposed to the length of both antecedent and consequent of the rules. A set of 144 rules is obtained. Table 3 shows the subset of rules with higher confidence and higher support sorted by lift. All resulting rules have simple consequent and none exceeds four items in the antecedent. Most of them refer patterns about high changes, either up or down. Only 20 rules have a support greater than 10%. No rules show supports in between (13%, 17%). Only seven rules have support greater than 17%, and none shows support over 19%. All rules have confidences between 0.6 and 0.7. All rules have simple antecedent, and three of them (r56, r95, r106, see Table 3) are

8

K. Gibert and D. Conti Table 4. Shorter frequent itemsets sorted by support. 159 160 9 10 150 116

{EDC.PR = CA-EDC {CANTV.PR = CA-CANTV {MVZB.PR = DB-MVZB} {BVP.PR = SC-BVP} {IBOVESPA = DA-IBOVESPA {CANTV.PR = DA-CANTV

IGBC = CA-IGBC} IGBC = CA-IGBC} DJIA = DA-DJIA} IGBC = DA-IGBC}

0.185 0.183 0.180 0.180 0.180 0.177

the symmetric of other rules (r54, r94, r105), just changing order of causality, but providing non new information. Lifts are high, between 1.5 and 2.67. Rules basically link the same trend (either increasing or decreasing) between IGBC and CANTV or EDC, eventually including other assets such as MVZB or BVP. Only one rule links high down change between international indexes BOVESPA and DJIA. Only five rules show no predictive errors (confidence = 1), all of them with high lifts (over 3.5), but quite low support (between 5% and 8%); in this case, consequent is always IGBC, with either high down change (three rules) or high up (two rules). Linking IGBC with CANTV, EDC, MVZB or BCP is nothing new, as these are the assets used to determine IGBC by construction. BOVESPA appears involved in down changes of IGBC, while DJIA seems relevant when IGBC goes up. The rules involving international indexes could indicate that the Venezuelan market moves according to international markets, but analysts know that this happens with very low frequency (as support indicates). In summary, the rules seem to say that when several Venezuelan assets go down, IGBC also, and the same for uptrends.

5. Understanding by means of frequent item sets Often the rules show combinatorial configurations of a single group of items. In fact, given a frequent item set (i1, i2 … ik), k different rules with simple consequent can be derived from it, as is the case of rules r55 and r56 or r94 and r95 (Table 3). This is just proposing different causality patterns, but not introducing new information about the multivariate association between items. For this reason, the frequent itemsets are obtained using the ECLAT algorithm [4] with a minimum support of 5%. In this case, 277 itemsets are found. From those, 30 singletons were obtained, with support never greater than 44%. Table 4 shows the most frequent itemsets, with support greater than 17%. From those, only four are singletons (in yellow, colour online; printed version in grey). These four itemsets are producing the five rules with multiple antecedents referred in Table 3. In the provided solution, only 14 itemsets involve 4 items and only 2 of those show support greater than 7% (Table 5). In Table 5, the itemsets that produced the best rules shown in Table 3 are marked in yellow (colour online; printer version in lighter grey) and the consequent is marked in green (colour online; printed version in darker grey).

6. Inducing frequent itemsets from clusters The results show that the TLP can also be used to induce general patterns from the data and the size of the class can be interpreted as the support of the pattern as shown in Table 6. In this case, the colours of the cells in Figure 2 induce three qualitative values for the variables (green: high up change (CA); yellow: stability (NC); and red: high down change (DA)). These qualitative levels must be associated with intervals of original values of the variables, in order to make the profile explicit, and to allow evaluation over new data. Thus, the corresponding cut-offs are defined by inspecting the conditional distributions of the variables against classes to

9

International Journal of Computer Mathematics Table 5. Longer frequent itemsets sorted by support. 263

{CANTV.PR = CA-CANTV

265 {EDC.PR = CA-EDC 254 {CANTV.PR = DA-CANTV 256

{CANTV.PR = DA-CANTV

261 258 262 255

{CANTV.PR = CA-CANTV {CANTV.PR = DA-CANTV {CANTV.PR = CA-CANTV {CANTV.PR = DA-CANTV

266 {EDC.PR = CA-EDC 253 {CANTV.PR = DA

EDC.PR = CA-EDC

MVZB.PR = CA-MVZB MVZB.PR = CA-MVZB BVP.PR = CA-BVP EDC.PR = DA-EDC MVZB.PR = DA-MVZB IGBC = DA-IGBC IBOVESPA = DA-IBOVESPA EDC.PR = CA-EDC BVP.PR = CA-BVP MVZB.PR = DA-MVZB IGBC = DA-IGBC EDC.PR = CA-EDC IGBC = CA-IGBC MVZB.PR = DA-MVZB IGBC = DA-IGBC MVZB.PR = CA-MVZB EDC.PR = DA-EDC

IGBC = CA-IGBC IGBC = DA-IGBC

257 {CANTV.PR = DA-CANTV MVZB.PR = DA-MVZB BVP.PR = DA-BVP 259 {CANTV.PR = CA-CANTV EDC.PR = CA-EDC IGBC = CA-IGBC 264

{CANTV.PR = CA-CANTV

EDC.PR = CA-EDC

IGBC = CA-IGBC

267

{CANTV.PR = CA-CANTV

MVZB.PR = CA-MVZB

BVP.PR = CA-BVP

IGBC = CA-IGBC}

0.084

IGBC = CA-IGBC} IGBC = DA-IGBC}

0.076 0.065

DJIA = DA-DJIA}

0.065

IGBC = CA-IGBC} DJIA = DA-DJIA} DJIA = CB-DJIA} IBOVESPA = DA-IBOVESPA} DJIA = CB-DJIA} IBOVESPA = DA-IBOVESPA} IGBC = DA-IGBC} IBOVESPA = CA-IBOVESPA} IBOVESPA = CB-IBOVESPA} IGBC = CA-IGBC}

0.065 0.060 0.057 0.054 0.054 0.0518 0.0518 0.0518 0.0518 0.0518

Table 6. Support of the pattern from TLP. C357: C356: C359: C346: C347:

{CANTV = NC,EDC = NC,MVZB = NC,BVP = NC,IGBC = NC,BOVESPA = DA,DJIA = NC} 22.74% {CANTV = NC,EDC = NC,MVZB = NC,BVP = NC,IGBC = NC,BOVESPA = CA,DJIA = CA} 33.79% {CANTV = CA,EDC = CA,MVZB = CA,BVP = CA,IGBC = CA,BOVESPA = CA,DJIA = CA} 9.06% {CANTV = DA,EDC = DA,MVZB = DA,BVP = DA,IGBC = DA,BOVESPA = NC,DJIA = NC} 4.39% {CANTV = NC,EDC = NC,MVZB = NC,BVP = DA,IGBC = NC,BOVESPA = CA,DJIA = CA} 28.02%

Figure 3. Criteria to deduce the cutpoints from the clustering results.

minimize overlapping areas. For two qualitative levels, given a set of classes C ∈ P, P a partition of the original set of individuals, and a variable X, the cutpoint for X, namely p, is computed in the following way (Figure 3):  min{P(X|C ∩ [p, ∞)), P(X|C ∩ (−∞, p])} being r = ∩C∈P (range(X|C)). p = argminp∈r C∈P

When finding cutpoints for k qualitative levels, classes are previously ordered from left to right according to X values and divided into k packs. Cutpoints are searched by applying the same procedure locally to the subset of classes overpassing the frontiers of the packs.

10

K. Gibert and D. Conti Table 7. Variable’s cut-offs resulting from the TLP. Label

CANTV

EDC

MVZB

BVP

IGBC

BOVESPA

DJIA

High down change No major change High up change

< −4.54

< −4.65

< −2.79

< −2.5

< −2.22

< −2.05

< −1.3

> 4.11

> 3.4

> 2.63

> 2.22

> 2.79

> 3.93

> 1.83

In this case, the meaning of the qualitative levels is shown in Table 7. This permits to obtain a classifier for the situation registered in the Venezuelan stock market, which, after eliminating redundant conditions, is the following: C346: All assets down and indexes stable (CANTV < −4.54). C347: Quasi-Internal stability (BVP down) and international uptrends (CANTV∈[−4.54, 4,11]& BVP < −2.5). C357: Internal stability and Brazil down, NY stable (CANTV∈[−4.54, 4,11] & BOVESPA < −2.05). C356: Internal stability, indexes uptrend (CANTV ∈ [−4.54, 4.11]&BVP ∈ [−2.5, 2.22]& BOVESPA > 3.93). C359: All uptrend (CANTV > 4.11).

7. Improving classical modelling with the pattern understanding From a predictive point of view, it is known that IGBC is computed by means of a model that combines daily average quotations of the national assets. The matrix plot has shown no relevant nonlinearities between IGBC and the other variables. Thus, a regression model has been adjusted between IGBC and the national assets: IGBC = 0.303 + 0.241 CANTV + 0.258 EDC + 0.148 MVZB + 0.184 BVP, R2 = 83%. The residuals indicate a valid model, all the assets are significant and the goodness-of-fit is quite high, as expected by construction. However, when the international context is added to the model, it seems to explain the scenario a little bit better and the new model shows R2 = 83.8%. Model is significant and the goodness-of-fit improves. One could feel comfortable with this model. However, looking at the TLP and the classifier induced, it can be seen that the market can fit in different situations, behave under different patterns depending on the week, and TLP also elicits that the relationship between the assets and the indexes with IGBC might depend on the class itself. Thus, adding the class and the interactions between regressors and classes in a multiplicative model, the predictive power improves to R2 = 86%: IGBC = 0.249 − 6.502 C346 − 1.378 C347 + 0.199 CANTV∗ C346 + 0.238 CANTV∗ C356 + 0.248 CANTV∗ C357 + 0.384 CANTV∗ C359 + 0.092EDC∗ C347 + 0.242 EDC∗ C356 + 0.238 EDC∗ C357 + 0.260 EDC∗ C359 + 0.311 MVZB∗ C347 + 0.124 MVZB∗ C356 + 0.118 MVZB∗ C357 + 0.040 MVZB∗ C359 + 0.236 BVP∗ C356 + 0.150 BVP∗ C359 + 0.349 IBOVESPA∗ C346 + 0.500 IBOVESPA∗ C359−0.315 DJIA∗ C347− 0.889 DJIA∗ C359, R2 = 86%

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International Journal of Computer Mathematics

Splitting this model in independent equations for every class, interpretation becomes easier. IGBC346 = −6.253 + 0.199 CANTV + 0.349 IBOVESPA, IGBC347 = −1.129 + 0.092 EDC + 0.311 MVZB − 0.315 DJIA, IGBC356 = 0.249 + 0.238 CANTV + 0.242 EDC + 0.124 MVZB + 0.236 BVP, IGBC357 = 0.249 + 0.248 CANTV + 0.238 EDC + 0.118 MVZB, IGBC359 = 0.249 + 0.384 CANTV + 0.260 EDC + 0.040 MVZB + 0.150 BVP + 0.500 IBOVESPA − 0.889 DJIA, R2 = 86%. For class C346 high negative values of IGBC are obtained and the index is determined by CANTV, the most important blue chip in Caracas Stock Exchange and IBOVESPA, both negative in this class, even if IBOVESPA is near −0, but DJIA does not influence the Venezuelan market in this situation. Class C347 neutral values of ICGB are determined by EDC and MVZB blue chips in a certain period and DJIA is going in the contrary sense, remarking again a different behaviour in Venezuela than the one in the international context. Class 356 shows internal stability and uptrend in international markets. The equation reflects that all assets influence the IGBC in this scenario, but international indexes do not. Class 357 shows stability both in and out of Venezuela (except for Brazil which shows downtrend).Only blue chips seem to be required to determine IGBC when Brazil is down. C359 is the optimistic scenario in which all is uptrend. And it is the only situation in which all variables appear together in the equation. The negative coefficient of DJIA indicates that even in this optimistic situation, NY works with lower values than Venezuela. So, the situation in Brazil seems to influence IGBC only when Venezuelan market is non stable. The situation in NY only influences the Venezuelan market when it has higher values, and Venezuelan market is always behaving below it. Considering internal stability (C356, C357) international context do not affect, and all assets and indexes appear to be important when IGBC is downtrend.

8. Discussion, conclusions and future work Financial data from the Venezuelan stock exchange are mined under three approaches: the one proposed by the authors consisting clustering followed by a post-processing based on TLP and frequent itemsets induction; the second one is association rule mining and, as a third approach, frequent itemsets mining, both frequently used in financial DM. The patterns learned will be later used to improve predictive modelling. A main advantage of the proposed methodology is that it can be used with the original data, without requiring a previous discretization of data. Both association rules and frequent item sets require discretization and the results will always depend on the selected cutpoints. Using association rules and frequent itemsets, the discretization is provided by a group of experts on the basis of the variables’ distributions. However, finding optimal discretization for numerical variables is still an open problem and the impact of discretization over results’ biases is well known [25]. In fact, in the presented application, even if the experts recommended five qualitative levels on the variables (from high up change to high down change), only extreme values appear in the rules and this might suggest that another discretization would better model

12

K. Gibert and D. Conti

the situation. Instead, post-processing the clusters with TLP provides a quick, visual and easy way to determine the appropriate variables’ cut-off a posteriori, as a result of the process and not as a previous constraint that might bias results, thus giving a better characterization. Also, the TLP supports the process of understanding the underlying multivariate relationships among variables which seem to provide more perspective on what is happening when using association rules or frequent itemsets. In fact, the TLP shows one general pattern per class involving the whole set of variables and giving a quite easy to understand overview. In this particular application, it permits to identify situations in which the Venezuelan stock market does not correlate with international markets and some particular assets which might not affect the Venezuelan index (IGBC) in certain situations. On the contrary, both the association rules and the frequent itemsets return an enormous set of rules/itemsets involving no more than four items at a time, which give a fragmented view of the reality and also require some pruning as a post-processing to become understandable for the end-user. The association rules with higher support are much focused, involving small number of items (mainly two items), retrieving well-known trends by financial experts (no new knowledge). Even though, they show quite high risk of error as confidence is not so high. On the contrary, longest rules (never with more than four items), look more valuable to the financial experts, as they provide some interesting information, and have high confidence. However, there is a few number of long rules and refer to quite unusual patterns showing really low support. It seems that association rules show a gap between quantity of information and quality, giving rules either much focused or very rare. Also, many of the rules become redundant by deriving combinations from the same itemset, as already discussed in the previous section. That is why the pure frequent itemsets were analysed. From seven variables only four appear in the longest itemsets, with supports between 5% and 8%, which are not very high. The itemsets with higher support contain only two items and have supports between 17% and 18% which are not very high. In this work, the symbolic patterns shown in the TLP have been associated with frequent itemsets in a direct way by using a conjunctive form over the variables and by identifying the ranges for the qualitative levels of the variables associated with the colours from TLP. The itemsets induced on the basis of the clustering plus the TLP provide a reduced number of more global patterns, involving the whole set of variables, with higher support than those obtained by classical frequent itemsets algorithms such as ECLAT, giving more perspective and coverage. Both frequent and more focused patterns are included in the same picture (linked to bigger and smaller classes from the clustering) and new and valuable information for investors appears, like the idea that BPV might be unlinked with the general trend in the Venezuelan market. It is interesting to see that almost all patterns provided by both association rules and frequent itemsets are subsumed by those provided in the TLP, as shown in Table 8. This seems to indicate that clustering + TLP gives a more global insight to the data structure. For the particular application presented here, the profiles obtained by the proposed approach (clustering + TLP + itemsets induction) represent a financial scenario that matches with experts’ feedback and represents very well what happens in the stock market in Venezuela. Thus, it can be said that the TLP contributed to the understanding of DM results for the particular case of clustering, making clusters easily interpretable and enabling experts validation. This provided a picture of the typical situations that can be found in the Venezuelan stock market and establishes a formal basis to support further analysis of temporal transits between the identified typical situations. In fact using a colour for any pattern (or class) and having a clear conceptualization, from a financial point of view, of classes’ meanings, the temporal patterns can be associated with colour sequences along time and might provide high granularity patterns quite useful for the decision-making support. A formalization of the process to induce itemsets from the TLP is in progress.

International Journal of Computer Mathematics

13

Also, it has been shown that building a predictive model for IGBC, based on both assets and indexes, significantly improves when using the information provided by the TLP in a multiplicative regression model , increasing the original goodness-of-quality of the regression from 83% to 86% (both models with all significant coefficients and valid residuals). The predictive equations found, provide a quantitative approach of the qualitative and intuitive view given provided by the TLP. Thus, for high-level decisions where distinguishing between up or down trends is enough, the TLP provides a sufficient qualitative insight to the decision-maker, whereas the regression equations add a detailed quantitative prediction for the value of the ICGB, taking into account the profiles discovered by the TLP, to be used in more sensitive decisions. The classifier induced from the TLP is the link that permits the decision-maker to select which of the local predictive equations might be used in a real situation. Acknowledgements We want to thank Albany Sanchez and Maria Elena Naranjo (Universidad de Los Andes, Venezuela) for their valuable collaboration in exploratory analysis of financial data set used in this work, as well as their ETL process (Extract, Transform and Load) in data, prior association rules mining and their meetings and reports with Venezuelan financial analysts that were extremely useful in the discussion and conclusions of this research.

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[22] K. Gibert, A. García-Rudolph, and G. Rodríguez-Silva, The role of KDD support-interpretation tools in the conceptualization of medical profiles, Acta Informatica Medica 16(4) (2008), pp. 178–182. [23] K. Gibert, G. Rodríguez-Silva, and R. Annicchiarico, Post-processing: Bridging the gap between modelling and effective decision-support. The profile assessment grid in human behaviour, Math. Comput. Model. 57(7–8) (2013), pp. 1633–1639. [24] K. Gibert, G. Rodríguez-Silva, and I. Rodríguez-Roda, Knowledge discovery with clustering based on rules by states: A water treatment application, Environ. Model. Softw. 25 (2010), pp. 712–723. [25] K. Gibert, Z. Sonicki, and J.C. Martín, Impact of data encoding and thyroids dysfunctions, Stud. Health Technol. Inf. 90 (2002), pp. 494–498. [26] E. Hajizadeh, H. Ardakani, and S. Jamal, Application of data mining techniques in stock markets: A survey, J. Econ. Int. Financ. 2(7) (2010), pp. 109–118. [27] J. Han, M. Kamber, and J. Pei, Data Mining: Concepts and Techniques, 3rd ed., Morgan Kauffman, Waltham, MA, 2011. [28] A. Hirotugo and G. Kitagawa, The Practice of Time Series Analysis, Springer-Verlag, New York, 1998. [29] B. Kovalerchuk and E. Vityaev, Data mining for financial applications, in The Data Mining and Knowledge Discovery Handbook, 2nd ed., 2005, O. Maimon and L. Rokach, eds., Springer, Heidelberg, 2010, pp. 1203–1224. [30] B. Kovalerchuk and E. Vityaev, Data Mining in Finance: Advances in Relational and Hybrid Methods, Kluwer Academic Publishers, New York, 2000. [31] M. Last, Knowledge discovery in time series databases, IEEE Trans. Syst. Man Cybern. 31(1) (2001), pp. 160–169. [32] M. Last, Data mining in time series databases, Series in Machine Perception and Artificial Intelligence, Vol. 57, World Scientific Publishing Co. Pvt., Ltd, London, 2004. [33] S. Laxman, A survey of temporal data mining, Sadhana 31(2) (2006), pp. 173–198. [34] T.W. Liao, Clustering of time series-data: A survey, Pattern Recognit. 38(11) (2005), pp. 1857–1878. [35] J.C. Lindsey, D.L. Jacobson, H. Li, E.A. Houseman, G.M. Aldrovandi, and K. Mulligan, Using cluster heat maps to investigate relationships between body composition and laboratory measurements in HIV-infected & HIV-uninfected children & young adults, J. Acquir. Immune Defic. Syndr. 59(3) (1999), pp. 325–338. [36] H. Lu, J. Han, and L. Feng, Stock movement prediction and n-dimensional inter-transaction association rules, Proc. of the ACM-SIGMOD workshop on research issues on DM & KDD, Seattle, WA, 1998, pp. 1–12. [37] J. Mongkolnavin and S. Tirapat, Marking the close analysis in Thai Bond Market surveillance using association rules, Expert Syst. Appl. 36 (2009), pp. 8523–8527. [38] M. Siponen, J. Vesanto, S. Olli, and P. Vasara, An approach to automated interpretation of SOM, Advances in Self-Organizing Maps, London, UK, 2001. [39] C.A. Sugar, R. Sturm, T.T. Lee, C.D. Sherbourne, R.A. Olshen, K.B. Wells, and L.A. Lenert, Empirically defined health states for depression from the SF-12, Health Serv. Res. 33 (1998), pp. 911–928. [40] The Business Week, 2012.Available at http://www.businessweek.com/articles/2012-10-14/venezuelas-strange-stockmarket-rally. [41] The Guardian. Available at http://www.theguardian.com/business/2012/dec/31/venezuela-best-stock-market-2012. [42] T. Trauwaert, P. Rouseew, and L. Kauffman, Some silhouette-based graphics for clustering interpretation, Belgian J. Oper. Res. Stat. Comput. Sci. 29(3) (1998), pp. 35–55. [43] A. Tung, H. Lu, J. Han, and L. Feng, Breaking the barrier of transactions: Mining inter-transaction association rules, Proc. of the fifth ACM SIGKDD, San Diego, CA, USA, 1999, pp. 297–301. [44] A. Weigend, Data mining in finance: Report from the post-NNCM-96 Wksh. on teaching computer intensive methods for financial modelling and data analysis, Proc. 4th NNCM-96, Pasadena, CA, USA, 1997, pp. 399–411. [45] G.H. Yan, Z.H. Li, and J.W. Dang, Finding natural cluster hierarchies based on MultiFractal, J. Softw. 19(6) (2008), pp. 1283–1300. [46] Q. Zhao, Sequential Pattern Mining, Nanyang Technological University, Singapore, 2003.

Scandinavian Actuarial Journal, 2014 Vol. 2014, No. 7, 583–601, http://dx.doi.org/10.1080/03461238.2012.750621

Original Article

Optimal investment of an insurer with regime-switching and risk constraint JINGZHEN LIU†‡, KA-FAI CEDRIC YIU∗ ‡ and TAK KUEN SIU§¶ †School of Insurance, Central University Of Finance and Economics, Beijing 100081, P.R. China ‡Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, P.R. China §Cass Business School, City University London, London, UK ¶Faculty of Business and Economics, Department of Applied Finance and Actuarial Studies, Macquarie University, Sydney, Australia (Accepted November 2012) We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regimeswitching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly. Keywords: optimal investment; entropy risk; risk constraint; regime-switching; model uncertainty; stochastic differential game

1. Introduction Portfolio allocation is one of the key problems in the interplay between financial mathematics and insurance mathematics. Due to the rapid convergence of insurance and financial markets, many insurance companies are actively involving in investment activities in capital markets. An important issue in optimal investment problem for an insurer is model uncertainty which is attributed to uncertainty in the ‘true’ data generating processes for financial prices and insurance liabilities. Indeed, model uncertainty is an important issue in all modeling exercises in finance and insurance, (see, e.g. Derman 1996 and Cont 2006). Mataramvura & Oksendal (2008) investigated a risk-minimizing portfolio selection problem, where an investor is to select

∗ Corresponding

author. E-mail: [email protected] © 2013 Taylor & Francis

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an optimal portfolio strategy so as to minimize a convex risk measure of terminal wealth in a jump-diffusion market. Note that model uncertainty is incorporated in the penalty function of the convex risk measure. It seems that Zhang & Siu (2009) were the first who attempted to incorporate model uncertainty in an optimal investment–reinsurance model for an insurer, where the insurer select an optimal investment-reinsurance strategy so as to maximize the expected utility in the ‘worst-case’ scenario. Elliott & Siu (2010) studied the optimal investment of an insurer in the presence of both the regime-switching effect and model uncertainty. To incorporate model risk, Elliott and Siu considered a robust approach where a family of probability measures, or scenarios, was used in the formulation of the optimal investment problem of an insurer. The goal of the insurer was then to maximize the expected utility in the ‘worst-case’ scenario. For an overview on robust representations of preferences, please refer Follmer & Schied (2002). In the past few decades or so, the insurance sector has grown significantly making it the secondlargest in the European financial services industry. Consequently, any negative disturbances in the insurance industry can have adverse impacts on the entire financial system and the whole economy. In response to this, regulators in the European Union region introduced the Solvency II, where some requirements are imposed for insurance companies to minimize the risk of insolvency. In quite a substantial amount of the existing literature, much emphasis has been put on investigating the ultimate, or long-run, risk of insurance companies. However, as pointed out in Bingham (2000), a significant loss from the risky investment within a short-term horizon could be more serious than what will happen in the long term. Inspired by this fact, we will consider the situation in the presence of risk constraint. In order to monitor the financial strength of an insurer, assume that the regulator will regularly evaluate the potential current risk of the company. For risk evaluation, the regulator will look at the risk based on the current portfolio holdings of the insurer and potential future fluctuation in the risky asset. Traditionally, volatility is used as a measure of risk in finance while ruin probability is used as a risk measure in insurance. Value at Risk (VaR) has emerged as a popular risk measure in both the finance and insurance industries. It has become a benchmark for risk measurement in both industries. However, as pointed out by Artzner et al. (1999), VaR does not, in general, satisfy the subadditivity property. In other words, the merge of two risky positions can increase risk, which is counter-intuitive. Artzner et al. (1999) introduced an axiomatic approach for constructing risk measures and the concept of coherent risk measures. A risk measure satisfying a set of four desirable properties, namely, translation invariance, positive homogeneity, monotonicity and subadditivity, is said to be coherent. Though theoretically sound, the notion of coherent risk measures cannot incorporate the impact of liquidity risk of large financial positions in risk measurement. To articulate the problem, Follmer & Schied (2002) and Frittelli & Gianin (2002) introduced the class of convex risk measures which can incorporate the nonlinearity attributed to the liquidity risk of large financial positions. They relaxed the subadditive and positive homogeneous properties of coherent risk measures and replaced them with the convexity property. They extended the notion of coherent risk measures by the class of convex risk measures and obtained a representation for convex risk measures. Here, we adopt a particular form of a convex risk measure in a short time horizon as a risk constraint. In particular, the convex risk measure with the penalty function being a relative entropy is adopted here in the short time horizon. Consequently, the convex risk measure used here may be related to the

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entropic risk measure which is an important example of convex risk measures and corresponds to an exponential utility function (see, e.g. Frittelli & Gianin 2002, Barrieu & El-Karoui 2005; Barrieu & El-Karou in press). Dynamic VaR risk constraint, first proposed in Yiu (2004), has been considered in the literature for financial portfolio optimization under a Markovian regime-switching environment in Yiu et al. (2010) or with Poisson jumps in Liu et al. (2011). When taking risk constraint into consideration for an insurer, Liu et al. (2012) considered the problem of minimizing the ruin probability with the classic diffusion model. In this paper, we study the optimal investment problem for an insurer, whose objective is to maximize the expected utility of the terminal wealth in the worst-case scenario with the risk constraint. To incorporate model uncertainty, here, we use a set of probability models surrounding an approximating model, where the set of probability models is characterized by a family of probability measures, which are absolutely continuous with respect to a real-world probability measure, introduced by a Girsanov-type transformation for jump-diffusion processes. Moreover, we consider a general model for the aggregate insurance claims, namely, a Markov regime-switching random measure. This model is flexible enough to incorporate the regime-switching effect in modeling both the mean rate of claims arrivals and the distribution of claim sizes. To model the price process of a risky financial asset, we adopt a Markov, regime-switching, geometric Brownian motion, where key model parameters such as the appreciation rate and the volatility of the risky asset are modulated by a continuous-time, finite-state, observable Markov chain. Some works using this price process model include Zariphopoulou (1992), Zhou & Yin (2003), Sass & Haussmann (2004), Yin & Zhou (2004), Zhang & Yin (2004), Elliott & Siu (2010) and Elliott et al. (2010). Here, we interpret the states of the observable Markov chain as proxies of different levels of some observable economic indictors, such as Gross Domestic Product and Retail Price Index. Indeed, the states of the observable Markov chain may also be interpreted as the credit ratings of a country, (or a corporation). In our modeling framework, we incorporate four important sources of risk in our model, namely, financial risk due to price fluctuations, insurance risk due to uncertainty about insurance claims, economic risk (or regime-switching risk) that is attributed to structural changes in economic conditions, and model risk due to uncertainty about the ‘true’ model. We formulate the optimal investment problem of the insurer with risk constraint as a constrained Markov, regimeswitching, zero-sum, stochastic differential game with two players, namely, the insurer and the nature (i.e. the market). In such a game, the insurer chooses an optimal investment strategy so as to maximize the expected utility of the terminal wealth in the worst-case probability scenario subject to a risk constraint. The market can be viewed as a ‘fictitious’ player in the sense that it responds antagonistically to the choice of the insurer by choosing the worst-case probability scenario which minimizes the expected utility. This is a standard assumption in the robust approach to model uncertainty used in economics. One popular approach to discuss stochastic differential games is based on solving the Hamilton–Jacobi–Bellman–Issacs (HJBI) equations, (see, e.g. Evans & Souganidis 1984, Mataramvura & Oksendal 2008, Zhang & Siu 2009, Elliott & Siu 2010). One of the key assumptions of the HJBI dynamic programming approach is that the value function of the game problem should be C 2 function and it is attained at a saddle point(N ash equilibrium) so as to obtain a classical solution to the problem. However,

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it appears that in the situation with constraint, such a C 2 solution to the HJBI equation or the N ash equilibrium cannot be found and it could be difficult to verify the underlying assumptions even though it holds. For the Markov, regime-switching, stochastic differential game, the problem becomes more involved due to the system of nonlinear differential equations. In this work, we consider the class of exponential utility functions as which plays an important role for pricing insurance contracts. Here, we transform the original game control problem with uncertainties in financial prices, insurance risk, and market uncertainty, (i.e. the Markov chain), into one with only market uncertainty being included and the problem is simplified greatly. Then we derive a closed-form optimal strategy of the transformed control problem. The paper is organized as follows. The next section first describes the model dynamics in the financial and insurance markets. Then we describe the worst-case scenario risk and use it as a dynamic risk constraint in the optimal investment problem, where the expected utility of the terminal wealth in the worst-case probability scenario is maximized under the dynamic risk constraint. We then formulate the problem as a zero-sum stochastic differential game between the insurer and the market. We consider an important case based on an exponential utility and derive a closed-form solution to the game problem in Section 3. The final section gives concluding remarks.

2. Model dynamics, optimal investment and stochastic differential game We start with a complete probability space (, F , P ), where P represents a reference probability measure from which a family of probability measures absolutely continuous with respect to P are generated. We suppose that the probability space (, F , P ) is rich enough to incorporate all sources of uncertainties, such as financial and insurance risks, in our models. We call such a probability space a reference model, or an approximating model. Suppose T is a time parameter set [0, T ], where T < ∞. To describe economic risk attributed to transitions in economic, or environmental, conditions, we consider a continuous-time, N -state, observable Markov chain Z := {Z(t)|t ∈ T } with state space Z := {z1 , z2 , · · · , z N } ⊂ ℜ N . These states are interpreted as proxies of different levels of observable, (macro)-economic, conditions. For mathematical convenience, we follow the convention in Elliott et al. (1994) and identify the state space of the chain Z with a finite set of standard unit vectors E := {e1 , e2 , . . . , e N } ∈ ℜ N , where the j th -component of ei is the Kronecker delta δi j for each i, j = 1, 2, · · · , N . This is called the canonical representation of the state space of the chain Z. Let Q be the constant rate matrix, or the generator, [qi j ]i, j=1,2,··· ,N of the chain under P so that the probability law of the chain Z under P is characterized by Q. With the canonical representation of the state space of the chain Z, Elliott et al. (1994) derived the following semimartingale dynamics for the chain Z: Z(t) = Z(0) +



0

t

QZ(u)du + M(t) ,

t ∈T.

(2.1)

Here {M(t)|t ∈ T } is an ℜ N -valued, (FZ , P )-martingale; FZ := {F Z (t)|t ∈ T } is the rightcontinuous, P -completed natural filtration generated by the Markov chain Z.

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In what follows, we shall specify how the state of the economy described by the chain Z influences the price process of a risky financial asset and the insurance surplus process.

2.1. The model 2.1.1. The surplus of an insurer We model the risk process of the insurer by a Markov, regime-switching, random measure which is flexible enough to allow the Markov chain influencing the distribution of claim sizes and the claims arrival rate. Let C := {C(t)|t ∈ T } be the aggregate claim process, where C(t) is the aggregate claim up to time t. We suppose that C is a real-valued, Markov, regime-switching pure jump process on (, F , P ). Then for each t ∈ T , C(t) =



C(0) = 0 ,

C(u) ,

0 0. Let R := {R(t)|t ∈ T } denote the surplus process of the insurer. Then R(t) := x0 + = x0 +



t

0 N  i=1

p(s)ds − C(t) pi



0

t

Z(s), ei  ds −

 t 0

0

∞

cγ (dc, ds), t ∈ T .

(2.9)

2.1.2. The investment process We assume that the insurer is allowed to invest in a risky financial asset S, say a stock. Suppose that the price process {S(t)|t ∈ T } of the risky financial asset S is governed by the following Markov, regime-switching, geometric Brownian motion:   d S(t) = S(t) μ(t)dt + σ (t)dW (t) , S(0) = s. (2.10) Here W := {W (t)|t ∈ T } is a standard Brownian motion on (, F , P ) with respect to FW := {F W (t)|t ∈ T }, the P -augmentation of the natural filtration generated by the standard Brownian motion W , μ(t) and σ (t) are the appreciation rate and the volatility of the risky financial asset S at time t, respectively. We assume that μ(t) and σ (t) are modulated by the chain Z as: μ(t) := µ, Z(t) , σ (t) := σ , Z(t) ,

(2.11)

where µ := (μ1 , μ2 , . . . , μ N )′ ∈ ℜ N and σ := (σ1 , σ2 , . . . , σ N )′ ∈ ℜ N with μi > 0 and σi > 0, for each i = 1, 2, . . . , N .

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For each t ∈ T , let π(t) be the amount of money allocated to the risky asset S at time t. Suppose that {X (t)|t ∈ T } is the surplus process of the insurer investing in the financial market with initial wealth X (0) = x0 > 0. Then, the surplus process of the insurer after taking into investment is governed by the following Markov, regime-switching, jump-diffusion process: ⎧  ∞ ⎨ d X (t) = ( p(t) + π(t)μ(t))dt + π(t)σ (t)dW (t) − cγ (dc, dt), 0 ⎩ X (0) = x 0 .

(2.12)

Let G (t) := F Z (t) ∨ F W (t) ∨ F C (t), the minimal σ -field containing F Z (t), F W (t) and Write, then, G := {G (t)|t ∈ T }. A portfolio process π := {π(t)|t ∈ T } is said to be admissible if it satisfies the following three conditions:

F C (t).

(1) (2)

π(·) is a G-adapted, measurable, process; for each t ∈ T ,  t 0

2

| p(u) + π(u)μ(u)| + (π(u)σ (u)) +

P − a.s.;

(3)



∞ 0

2



c νZ(u−) (dc, du) du < ∞ , (2.13)

the stochastic differential equation governing the surplus process X has a unique, strong solution.

Write for the space of all admissible portfolio processes without any constraint. 2.1.3. Model risk by change of measures In this section, we incorporate model uncertainty, or model risk, by a family of probability measures introduced via Girsanov’s transformation of jump-diffusion processes. Let θ := {θ (t)|t ∈ T } be a G-progressively measurable process characterizing a probability measure in the family. Suppose θ (·) satisfies the following conditions: (1) (2)

for each t ∈ T , θ (t) ≤ 1, P -a.s.; T 2 0 θ (t)dt < ∞, P -a.s.

Write for the space of all such processes θ (·). Then, the family of probability measures for capturing model risk is parameterized, or indexed, by . Note that is the space of admissible strategies adopted by the market without constraint. For each θ (·) ∈ , we define a G-adapted process θ := { θ (t)|t ∈ T } by putting:   t  t ∞  1 t θ (u)2 du + ln(1 − θ (u))γ˜ (dc, du) θ (u)dW (u) −

θ (t) = exp − 2 0 0 0   t  ∞0 (ln(1 − θ (u)) + θ (u))νZ(u−) (dc, du) . (2.14) + 0

0

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Or equivalently,

d θ (t) = θ (t − )(−θ (t)dW (t) −

θ (0) = 1, P − a.s.

∞ 0

θ (t)γ˜ (dc, dt)),

(2.15)

Then, for each θ (·) ∈ , θ is a (G, P )-(local)-martingale. For each θ (·) ∈ , a real world probability measure P θ is defined as:  d P θ  := θ (T ). d P G (T )

(2.16)

Suppose, the insurer is averse to model uncertainty. He/she may consider a family of probability measures which are introduced by perturbing the approximating model described by the reference probability P . Here we suppose that the family of probability measures is given by {P θ |θ (·) ∈ }. The following lemma is a well-known result which can be obtained from applying the Girsanov theorem. We state the result without giving the proof. Lem m a 2.1 For each θ ∈ , under P θ , W θ (t) := W (t) −



t

θ (u)du , 0

t ∈T ,

(2.17)

is a standard Brownian motion and the random measure γ θ has the following compensator: θ νZ(u−) (dc, du) :=

N  (1 − θ (u−)) Z(u−), ei  λi Fi (dc)du.

(2.18)

i=1

Then under P θ ,

∞ d X (t) = [ p(t) + π(t)(μ(t) − σ (t)θ(t))]dt + π(t)σ (t)dW θ (t) − 0 cγ θ (dc, dt), X (0) = x 0 .

(2.19)

2.2. The model uncertainty constraint and risk constraint In this work, the risk is evaluated according to the current information and the model parameters at time t. As the insurer is averse to model uncertainty, he/she will take into account the maximal risk in the family of probability measures {Qθ |θ (·) ∈ † } to be defined later in this section and among all regimes of the Markov chain. 2.2.1. The model uncertainty constraint For each t ∈ T , let H := F Z (t) ∨ F W (t). Write H := {H(t)|t ∈ T }. Suppose † is the space of all processes θ := {θ (t)|t ∈ T } such that (1) (2)

θ (·) is H-progressively measurable; T 2 0 |θ (t)| dt < ∞;

591

Optimal investment of an insurer

(3)

θ (·) satisfies the Novikov condition, (i.e., E[exp( 12

T 0

|θ (t)|2 dt)] < ∞).

Consider, for each θ (·) ∈ † , the following H-adapted process θ1 := { θ1 (t)|t ∈ T } on (, F , P):   t   1 t 2 θ θ (u)dW (u) − θ (u)du .

1 (t) := exp − 2 0 0 Then d θ1 (t) = −θ (t) θ1 (t)dW (t). Consequently, θ1 is a (H, P )-(local)-martingale. Since θ satisfies the Novikov condition, θ1 is a (H, P )-martingale. For each θ (·) ∈ † , we define the new probability measure Qθ which is absolutely continuous with respect to P on H(T ) by putting:  d Qθ  := θ1 (T ). d P H(T )

We now define the relative entropy between Qθ and P in the short duration [t, t +t) as follows:

Kt,t+t (Qθ , P ) := E

θ1 (t + t)

θ1 (t)

ln



θ1 (t + t)

θ1 (t)

 |H(t) .

Here E[·] is expectation under P . By Bayes’ rule,

Kt,t+t (Qθ , P ) =

E[ θ1 (t + t) ln(

θ1 (t+t) )|H(t)]

θ1 (t)

E[ θ1 (t)|H(t)]



θ1 (t + t) H (t) = Eθ ln |

θ (t) 

 t+t  1 1 2 = Eθ θ (u)du |H(t) . 2 t



For the evaluation in the short duration [t, t + t), we use the current regime and model (i.e. Z(u) = Z(t) and θ (u) = θ (t), for all u ∈ [t, t + t)). Consequently, Kt,t+t (Qθ , P ) can be approximated as follows: Kt,t+t (Qθ , P ) ≈ t

N  i=1

Z(t−), ei 



For each i = 1, 2, · · · , N , write t,t+t (Qθ , P , ei ) := K

1 2 θ t. 2 i

 1 2 θi . 2

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Constraint 2.1 We impose the following constraint for the aversion to the model uncertainty: t,t+t (Qθ , P , ei ) ≤ R1,t . max K i

For each i = 1, 2, · · · , N , let θ(t,i,−) and θ(t,i,+) be the negative and positive roots, respectively, of the following nonlinear equation in θ : t,t+t (Qθ , P , ei ) = R1,t . K

Let

K t := [θ(t,−) , θ(t,+) ] =



i=1,2,··· ,N

[θ(t,i,−) , min(1, θ(t,i,+) )].

†

Define the subspace 1 of † as follows: †

1 := {θ (·) ∈ † |θ (t) ∈ K t , ∀t ∈ T }.  of constrained admissible strategies adopted by the market is given by: Then the space  := † ∩ . 1

2.2.2. The risk constraint on investment

Suppose that the insurer evaluate the risk based on the current portfolio holdings and potential future fluctuations in the risky asset. In order to evaluate the potential investment risk at time t, we evaluate the change in the surplus of the insurer in the horizon [t, t + t) under P , which can be approximated by X (t) ≈ μ(t)π(t)t + σ (t)π(t)(W (t + t) − W (t)). Consequently, under Qθ , X (t) ≈ (μ(t) − θ (t)σ (t))π(t)t + σ (t)π(t)(W θ (t + t) − W θ (t)).  , let For each i = 1, 2, · · · , N and each θ ∈

Lθt (X (t), ei ) := Eθ [−X (t)|H(t), X(t) = ei ]

and Eθ is expectation under Qθ . We define a convex risk measure as

t,t+t (Qθ , P , ei ) . ρ(X (t)|G (t)) = max max Lθt (X (t), ei ) − K i

 θ ∈

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Under the time horizon, the risk measure can be approximated by

1 ρ(X (t)|G (t)) ≈ max max t −(μi − θ (t)σi )π(t) − θ 2 (t) i θ (t)∈ (t) 2 ˜ = max h i (π(t)),

i

(2.20)

where

h i (π(t)) :=

⎧ 1 2 ⎪ ⎪ ⎨ t[−(μi − θ(t,+) σi )π(t) − 2 θ(t,+) ], if π(t) ≥ π(t))2

θ(t,−) σi

θ(t,+) σi ,

t[ (σi 2 − μi π(t)], ≤ π(t) ≤ ⎪ ⎪ θ 1 2 ⎩ t[−(μ − θ θ ], if π(t) ≤ (t,−) σ )π(t) − i (t,−) i 2 (t,−) σi .

θ(t,+) σi ,

Constraint 2.2 Supposes that, for each t ∈ T , the risk level described by the convex risk measure is constrained below R2,t , i.e., (2.21)

max h i (π) < R2,t . i

For each i = 1, 2, · · · , N , let r(t,i,±) :=

when when

 μi ± μi2 +2R2,t

. Obviously, h i (π(t)) decreases with π(t)

σi2 θ(t,−) π ≤ σi . Here, we assume that μi − θ(t,+) σi < 0, θ π(t) ≥ (t,+) σi . Write, for each t ∈ T and i = 1, 2, · · · ,

π(t,i,−) := and π(t,i,+) :=

⎧ ⎨ r(t,i,−) , ⎩

−

R2,t 1 2 t + 2 θt μi −θ σi

⎧ ⎨ r(t,i,+) , ⎩

−

R2,t 1 2 t + 2 θt μi −θ σi

ift[ ,

N,

−

μi θ(t,−) ] σi

≥ R2,t ,

(2.22)

−

μi θ(t,+) ] σi

≥ R2,t ,

(2.23)

otherwise,

ift[ ,

2 θ(t,−) 2

then h i (π(t)) increases with π(t)

2 θ(t,+) 2

otherwise,

then the constraint (2.21) leads to: π(t) ∈ [π(t,−) , π(t,+) ] :=



i=1,··· ,N [π(t,i,−) , π(t,i,+) ].

2.3. Optimal investment problem as a stochastic differential game  := {π ∈ |πt ∈ [π(t,−) , π(t,+) ]}. Denote the expected utility on the Problem 2.1 Let

terminal wealth J (t, x, π(·), θ(·)) = Eθ [U (X π (T ))|X π (t) = x].

(2.24)

˜ so as Then the optimal investment problem of the insurer is to select an investment process in

to maximize the following minimal, (or ‘worst-case’), expected utility on the terminal wealth: sup inf J (t, x, π(·), θ(·)).

  θ∈ π ∈

(2.25)

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This is a two-player, zero-sum, stochastic differential game between the insurer and the market in the sense that the insurer tries to maximize the value of J (t, x, π(·), θ(·)) and the market tries to minimize this value. We call V+ (t, x) = sup inf J (t, x, π(·), θ(·)) the upper value of the game and

  θ ∈ π ∈

V− (t, x) = inf sup J (t, x, π(·), θ(·)) the lower value of the game. If

 π ∈

θ ∈ 

V+ (t, x) = V− (t, x), it is called the value of the game. If the value of the game exists, a popular approach, in the literature, is applying the dynamic programming principle, then the problem is reduced to find a C 2 solution to the resulted HJBI equations and the C 2 solution is attained at a Markov control point. However, when there is a constraint on the strategy, it is often difficult to find the C 2 solution due to the constraint on the strategy. In the insurance literature, the class of exponential utility functions play an important role for pricing insurance contracts. In this work by an technique in stochastic analysis, we derive an optimal solution of the investment problem of the insurer when the insurer has an exponential utility of the following form: U (x) = − exp(−αx) ,

(2.26)

where α is a positive constant.

3. The optimal strategy ×  , we define the following quantities: For each t ∈ T and (π(·), θ(·)) ∈



 T − α X (t) + − α( p(s) + π(s)(μ(s) − σ (s)θ(s))) t    T  ∞ 1 (1 − θ (s))λ(s) (eαc − 1)F(dc, ds) , + α 2 π 2 (s)σ 2 (s) ds + 2 t 0    T  T 1 π,θ 2 2 2 θ X 2,t (T ) := exp − α π (s)σ (s)ds + απ(s)σ (s)dW (s) × 2 t t   T ∞   ∞ αc θ exp − (e − 1)F(dc, ds) . αcγ (dc, ds) − (1 − θ (s))λ(s) π,θ (T ) := − exp X 1,t

t

0

0

(3.1)

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Let Eθt be the conditional expectation under Qθ given G (t). Then Eθt [U (X π (T )]   T θ [ p(s) + π(s)(μ(s) − σ (s)θ(s))]ds = −Et exp − α X (t) − α t   T  T ∞ −α cγ θ (dc, ds) π(s)σ (s)dW θ (s) − α 0 t t   T θ = −Et exp − α X (t) − α [ p(s) + π(s)(μ(s) − σ (s)θ(s))]ds t   T  ∞  T 1 + α2 π(s)2 σ 2 (s)ds + (1 − θ (s))λ(s) (eαc − 1)F(dc, ds) 2 t t 0   T  T ∞  1 2 T 2 2 θ × exp − α π (s)σ (s)ds − α π(s)σ (s)dW (s) − αcγ θ (dc, ds) 2 t t 0 t   T  ∞ − (1 − θ (s))λ(s) (eαc − 1)F(dc, ds) 0

t

= Eθt [X 1,t (T )X 2,t (T )].

(3.2)

For each t ∈ T , let G † (t) := F Z (T ) ∨ F W (t) ∨ F C (t). Write G† := {G † (t)|t ∈ T }. Then for each fixed t ∈ T , it is not difficult to see that {X 2,t (u)|u ≥ t} is a (G† , Qθ )-martingale. Consequently, Eθt [X 2,t (T )|G † (t)] = 1 ,

P -a.s.

,  ), Now for any (π(·), θ(·)) ∈ (

π,θ π,θ Eθt [X 1,t (T )X 2,t (T )]

π,θ π,θ (T )X 2,t (T )|G † (t)]} = Eθt {E θ [X 1,t

π,θ π,θ (T )Eθ [X 2,t (T )|G † (t)]} = Eθ {X 1,t π,θ (T )]. = Eθt [X 1,t

(3.3)

Consequently, Problem 2.1 can be simplified into the following problem: Problem 3.1 Find π,θ V (θ(·), π(·), t, x) := sup inf Eθt [X 1,t (T )].   θ∈ π ∈

(3.4)

It is important to note that the only source of uncertainty involved in Problem 3.1 is from Markov chain Z.

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Write (Z) for the path space generated by the sample paths of the Markov chain Z. For each ×  , we define: ωz ∈ (Z), t ∈ T , (π, θ) ∈

H (t, ωz , π(t, ωz ), θ(t, ωz ))

1 := α[ p(t, ωz ) + π(t, ωz )(μ(t, ωz ) − σ (t, ωz )θ(t, ωz ))] − π(t, ωz )2 σ 2 (t, ωz )α 2 2  ∞ −(1 − θ (t, ωz ))λ(t, ωz ) (eαc − 1)F(dc, dt). 0

 (t) and  (t) be the coordinate spaces of

 and  at time t, respectively. For each t ∈ T , let

 (t) ×  (t) such Now if, for each ωz ∈ (Z) and t ∈ T , we can find (π ∗ (t, ωz ), θ ∗ (t, ωz )) ∈

that H (t, ωz , π ∗ (t, ωz ), θ ∗ (t, ωz )) = then ∗

sup

inf

(t)  (t) θ (t,ωz )∈ π(t,ωz )∈

∗

π ,θ Eθt [X 1,t (T )] = sup

H (t, ωz , π(t, ωz ), θ(t, ωz )), (3.5)

π,θ inf Eθt [X 1,t (T )].

  θ (·)∈ π(·)∈

In other words, to solve the stochastic differential game, we can solve the corresponding pathwise minimax problem with respect to the path space of the Markov chain (Z). The pathwise minimax problem is presented as follows: Problem 3.2 If, for each t ∈ T and ωz ∈ (Z), we can find (π ∗ (t, ωz ), θ ∗ (t, ωz )) such that H (t, ωz , π ∗ (t, ωz ), θ ∗ (t, ωz )) =

sup

inf

(t)  (t) θ (t,ωz )∈ π(t,ωz )∈

H (t, ωz , π(t, ωz ), θ(t, ωz )), (3.6)

and (π ∗ (·), θ ∗ (·)) ∈ × , then (π ∗ (·), θ ∗ (·)) solves the original Problem 2.1. 3.1. The Nash equilibrium For (π(·), θ(·)) ∈ × , let

  1 (θ(t), π(t)) = α p(t) + π(t)(μ(t) − σ (t)θ(t)) − α π 2 (t)σ 2 (t) 2  ∞ αc −(1 − θ (t))λ(t) (e − 1)F(dc, dt). 0

(3.7)

and i (θ(t), π(t)) = (θ(t), π(t)|Z (t) = i)   1 2 2 = α p(t) + π(t)(μi − σi θ (t)) − α π (t)σi 2  ∞ (eαc − 1)F(dc, dt). −(1 − θ (t))λ(t) 0

for each i = 1, 2, . . . , N .

(3.8)

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Then Problem 3.2 is equivalent to the following set of N minimax problems:

inf

sup

(t)  (t) θ (t)∈ π(t)∈

i (θ(t), π(t)) ,

i = 1, 2, · · · , N .

(3.9)

for any time t ∈ T . For each i = 1, 2, · · · , N , a pair (πi∗ (t), θi∗ (t)) is said to achieve an N ash equilibrium of th i problem in (3.9), or equivalently, a saddle point of i th problem in (3.9) if it satisfies the following Issac’s condition:  (t) ×  (t). i (θi∗ (t), π(t)) ≤ i (θi∗ (t), πi∗ (t)) ≤ i (θ(t), πi∗ (t)), ∀(θ(t), π(t)) ∈

(3.10)

We have the following lemma which gives an N ash equilibrium of i th problem in (3.9). From the saddle-point theorem, the problem is equivalent to the following minmax problem:

inf

sup

˜ θ (t)∈ (t) ˜ π(t)∈ (t)

i (θ(t), π(t)),

(3.11)

t ∈T.

In the following, we solve (3.9) by considering (3.11). Firstly, we present the results without constraint, namely, when (π(·), θ(·)) ∈ ( , ). (0,∗)

(0,∗)

Lem m a 3.1 For each i = 1, 2, · · · , N and t ∈ T , let (πi (t), θi (t)) denote the Nash equilibrium of the game when there are no constraints on π(·) and θ (·) and Z (t) = ei . Then (0,∗) (0,∗) (t), θi (t)) (πi

=



(0,∗)

Proof According to (3.8), (πi conditions:

λi

∞ 0

 (eαz − 1)F(dz) μi − πi(0,∗) (t)ασi2 , . σi α σi

(0,∗)

(t), θi

(t)) can be obtained from the following first-order

−μi + σi2 π(t)α + θ (t)σi = 0, ∞ σi π(t)α + λi 0 (1 − eαz )F(dz) = 0.

(3.12)

 The following lemma will be needed in the proof of Theorem 3.1.

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i Lem m a 3.2 Denote , for each i = 1, 2, · · · , N , π¯ i (θ) := − θ σi −μ 2 . Then

ασi

inf

sup

θ (t)∈ (t) π(t)∈ (t)

=

i (θ(t), π(t))

1 α[ p(t) + π¯ i (θ(t))(μ(t) − σ (t)θ(t)) − α π¯ i2 (θ(t))σ 2 (t)] θ (t)∈ (t) 2  ∞ αz (e − 1)F(dz, dt) +(1 − θ (t))λ inf

0

= =

inf

gi0 (θ(t))

θ (t)∈ (t) gi0 (θi(0,∗) (t)),

(3.13)

where gi0 (θ)

2  θ σi − μi 1 θ σi − μi := α[ p(t) − (μi − σi θ ) − α − σi2 ] 2 ασi2 ασi2  ∞ (eαc − 1)F(dc, dt). +(1 − θi )λ 0

Note that gi0 (θ) is increasing with θ when θ (t) > θi This property will be used later in Theorem 3.1. We now state the main result of this paper:

(0,∗)

and decreasing with θ when θ (t)
π(t,+) , then (θi∗ (t), πi∗ (t)) = (θ(t,−) , πˇ (θ(t,−) )),

(2)

(0,∗)

if πi

(t) < π(t,−) , then (θi∗ (t), πi∗ (t)) = (θ(t,+) , πˇ (θ(t,+) ),

(3)

(0,∗)

if πi

(3.15)

(t) ∈ [π(t,−) , π(t,+) ] and

(3.16)

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(a) (b)

if θ0∗ (t) < θ(t,−) , then (θ ∗ (t), π ∗ (t)) = (θ(t,−) , πˇ (θ(t,−) )), if θ0∗ (t) > θ(t,+) , then (θ ∗ (t), π ∗ (t)) = (θ(t,+) , πˇ (θ(t,+) )).

Proof Since inf

sup

˜ θ (t)∈ (t) π(t)∈ (t)

= =

inf

˜ θ (t)∈ (t)

i (θ(t), π(t))

i (θ(t), πˇ i (θ(t)))

inf

[ p(t) + πˇ i (θ(t))(μi − σi θ (t))  ∞ 1 + πˇ i2 (θ(t))σi2 + (1 − θ (t))λ(t) (eαz − 1)F(dz, dt)], 2 0

˜ θ (t)∈ (t)

(3.17)

when the constraint is inactive, the result is obvious. If the constraint is active, we just prove 1 and 3, while the proof of 2 is similar to 1. The definition of θˆi (·) shows that θˆi (π) is a decreasing function of π . Thus, if π(t,+) < π0∗ (t), we have (0,∗) (0,∗) (t)) = θi (t) θˆi (π) > θˆi (πi

(3.18)

for any π ∈ [π(t,−) , π(t,+) ]. Denote gi (θ(t)) := i (θ(t), πˇ (θ (t))), then we have gi (θ) = gi0 (θ) in the interval [θˆi (π(t,+) ), θˆi (π(t,−) )] ∩ [θ(t,−) , θ(t,+) ] and it follows from Lemma 3.3, together with (3.18), that gi (θ) increases with θ in this interval. On the other hand, if θ (t) < θˆi (π(t,+) ), then πˇ i (θ(t)) = π(t,+) and if θ (t) > θˆi (π(t,−) ), then πˇ i (θ(t)) = π(t,−) . We clare that in these two cases gi (θ) still increases with θ. In fact, when (θ,π ) π < πi(0,∗) (t), it follows from the definition of i (·, ·) that ∂i∂θ > 0, i,e, i (θ, π) increases (0,∗) with θ . Thus π(t,+) < πi (t) implies that both i (θ, π(t,+) ) and i (θ, π(t,−) ) increase with θ. The arguments above shows that gi (θ) increases with θ in [θ(t,−) , θ(t,+) ]. As a result, gi (θ) attains its minimum at θi∗ (t) = θ(t,−) . (0,∗) For 3, if θi(0,∗) (t) < θ(t,−) , then π(θ ˇ ) < πi (t) for all θ ∈ [θ(t,−) , θ(t,+) ], thus we have (0,∗) ∗ ∗ θ (t) = θ(t,−) , π (t) = π(θ ˇ (t,−) ). The proof of the case θi (t) > θ(t,+) is similar.  As (πi∗ (t), θi∗ (t)) denotes the N ash equilibrium at time t and Z (t) = ei , i = 1, · · · , N , we let (π ∗ (t), θ ∗ (t)) = (πi∗ (t), θi∗ (t)) if Z (t) = ei , i = 1, · · · , N , it is easily seen that ˜ ), ˜ which solves the original Problem 3.1. The results of Theorem 3.1 (π ∗ (·), θ ∗ (·)) ∈ ( , show that (1)

(0,∗)

(0,∗)

(·) is beyond the risk constraint, i.e., πi (t) > π + or when the risk with πi (0,∗) πi (t) < π − , t ∈ T , the statements of 1 and 2 show that risk is stabilized by π(t,−) < πˇ (t) < πt,+ ;

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(2)

the utility is improved if the risk constraint is inactive, but the constraint on θ is active (from (3)).

Remark 3.1 When there is no constraint and p(t), λ(t), μ(t), σ (t) are constants, our result coincides with Zhang & Siu (2009). We transform the stochastic differential game of (2.25) into a deterministic game of Problem (3.6), which is much simpler than the original one. Furthermore, although the regime-switching model in this work is Markovian, there is no need to assume that the parameters p(t), λ(t), μ(t), σ (t) are also Markovian, which can be generalized by Gprogressively measurable process and our approach still works.

4. Conclusion We considered an optimal investment problem of an insurer in the presence of risk constraint and regime-switching. By adopting a robust approach, we incorporated model uncertainty. We imposed a risk constraint on the investment taken by the insurer using the notion of convex risk measures as well as a constraint on the uncertainty aversion of the insurer on the ‘true’investment model. The goal of the insurer was to maximize the expected utility on terminal wealth in the worst-case scenario subject to the risk constraint. The optimal investment problem of the insurer was then formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. Under the assumption of an exponential utility, we transformed the original game problem into one where the only source of uncertainty is due to the modulating Markov chain. The problem was then equivalent to a pathwise minimax problem. By solving the pathwise minimax problem, we obtained a closed-form expression of the Nash equilibrium of the game. Unlike the HJBI dynamic programming approach, our approach does not require smoothness assumptions on the value function. Indeed, in many practical situations, the smoothness assumptions on the value function may hardly be satisfied. In these situations, one may consider our approach to discuss the optimal investment problem. Our approach works well whether the risk constraint is present or not. Moreover, our technique can be applied to a general model when the parameters of the model are not Markov. This is impossible to achieve using the HJBI dynamic programming approach. Acknowledgements The authors would like to thank the referee for helpful comments and suggestions. The first and second authors were supported by the Research Grants Council of HKSAR (PolyU 5001/11P) and the research committee of the Hong Kong Polytechnic University.

References Artzner, P., Delbaen, F., Eber, J. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, 203–228. Barrieu, P. & El-Karoui, N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics 9(2), 269–298.

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Barrieu, P. & El-Karoui, N. (in press). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: R. Carmona (Ed.), Indifference pricing, Princeton University Press. Bingham, R. (2000). Risk and return: underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk. Refereed Paper/Article Proceedings of the Casualty Actuarial Society Casualty Actuarial Society - Arlington, Virginia 2000: LXXXVII, 31–78. Cont, R. (2006). Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance 16(3), 519–547. Derman, E. (1996). Model risk. In: Quantitative strategies research notes. Goldman sachs. New York. Elliott, R. J. & Aggoun, L. (1994). Hidden Markov models: estimation and control. Berlin: Springer. Elliott, R. J. & Siu, T. K. (2010). A stochastic differential game for optimal investment of an insuerer with regime switching. Quantitative finance 11(3), 1–16. Evans, L. C. & Souganidis, P. E. (1984). Differential games and representation formulas for solutions of Hamilton-JacobiIsaacs equations. Indiana University Matimatics Jorunal 33, 773–797. Elliott, R. J. & Siu, T. K. (2010). On risk minimizing portfolios under a markovian regime-switching black-scholes economy. Annals of Operations Research 176(1), 271–291. Elliott, R. J., Siu, T. K. & Badescu, A. (2010). On mean-variance portfolio selection under a hidden Markovian regimeswitching model. Economic Modeling 27(3), 678–686. Follmer, H. & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics 6, 429–447. Frittelli, M. & Gianin, E. R. (2002). Putting order in risk measures. Journal of Banking and Finance 26, 1473–1486. Liu, J. Z., Bai, L. H. & Yiu, K. F. C. (2012). Optimal investment with a value-at-risk constraint. Journal Of Industrial And Management Optimization 8(3), 531–547. Liu, J. Z., Yiu, K. F. C. & Teo, K. L. (2011). Optimal Portfolios with stress analysis and the effect of a CVaR constraint. Pacific Journal of Optimization 7, 83–95. Mataramvura, S. & Oksendal, B. (2008). Risk minimizing portfolios and HJB equations for stochastic differential games. Stochstics and stochstics Reports 80, 317–337. Sass, J. & Haussmann, U. G. (2004). Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Finance and Stochastics 8, 553–577. Yin, G. & Zhou, X. Y. (2004). Markowitz’s mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits. IEEE Transactions on Automatic Control 49, 349–360. Yiu, K. F. C. (2004). Optimal portfolios under a value-at-risk constraint. Journal of Economic Dynamics and Control 28, 1317–1334. Yiu, K. F. C., Liu, J. Z., Siu, T. K. & Ching, W. C. (2010). Optimal Portfolios with Regime-Switching and Value-at-Risk Constraint. Automatica 46, 979–989. Zariphopoulou, T. (1992). Investment-consumption models with transaction fees and Markov-chain parameters. SIAM Journal on Control and Optimization 30(3), 613–636. Zhang, Q. & Yin, G. (2004). Nearly-optimal Asset Allocation in Hybrid Stock Investment Models. Journal of Optimization Theory and Applications 121(2), 197–222. Zhang, X. & T. K., Siu (2009). Optimal investment and reinsurance of an insurer with model uncertainty. Insurance: Mathematics and Economics 45, 81–88. Zhou, X. Y. & Yin, G. (2003). Markowitz’s mean-variance portfolio selection with regime switching: a continuous time model. SIAM Journal on Control and Optimization 42(4), 1466–1482.

Quantitative Finance, Vol. 11, No. 10, October 2011, 1547–1564

Optimal investment under dynamic risk constraints and partial information WOLFGANG PUTSCHO¨GLy and JO¨RN SASS*z yUniCredit Bank Austria AG, Risk Integration, Risk Architecture & Risk Methodologies, Julius Tandler Platz 3, 1090 Vienna, Austria zDepartment of Mathematics, University of Kaiserslautern, PO Box 3049, 67653 Kaiserslautern, Germany (Received 3 April 2008; in final form 6 July 2009) We consider an investor who wants to maximize expected utility of terminal wealth. Stock returns are modelled by a stochastic differential equation with non-constant coefficients. If the drift of the stock returns depends on some process independent of the driving Brownian motion, it may not be adapted to the filtration generated by the stock prices. In such a model with partial information, due to the non-constant drift, the position in the stocks varies between extreme long and short positions making these strategies very risky when trading on a daily basis. To reduce the corresponding shortfall risk, motivated by Cuoco, He and Issaenko [Operations Research, 2008, 56, pp. 358–368.] we impose a class of risk constraints on the strategy, computed on a short horizon, and then find the optimal policy in this class. This leads to much more stable strategies that can be computed for both classical drift models, a mean reverting Ornstein–Uhlenbeck process and a continuous-time Markov chain with finitely many states. The risk constraints also reduce the influence of certain parameters that may be difficult to estimate. We provide a sensitivity analysis for the trading strategy with respect to the model parameters in the constrained and unconstrained case. The results are applied to historical stock prices. Keywords: Portfolio optimization; Utility maximization; Risk constraints; Limited expected shortfall; Hidden Markov model; Partial information

1. Introduction We formulate a financial market model which consists of a bank account with stochastic interest rates and n stocks whose returns satisfy a Stochastic Differential Equation (SDE) with a stochastic drift process. The investor’s objective is to maximize the expected utility of terminal wealth over a finite time horizon. If the drift process is not adapted to the filtration of the driving Brownian motion and the investor can only use the information he gets from observing the stock prices, this leads to a model with partial information. To compute strategies explicitly under this realistic assumption, we have to specify a model for the drift

of the stocks, typically with some linear Gaussian Dynamics (GD) (see for example Lakner 1998, Pham and Quenez 2001, Brendle 2006, Putscho¨gl and Sass 2008) or as a continuous-time Markov chain with finitely many states. The filter for the first model is called a Kalman filter. The latter model was proposed in Elliott and Rishel (1994) and we refer to it as the Hidden Markov Model (HMM); the corresponding filter is called the HMM filter. It satisfies a lot of stylized facts observed in stock markets (cf. Ryde´n et al. 1998 for related regime switching models). Efficient algorithms for estimating the parameters of this model are available (cf. Elliott 1993, James et al. 1996, Hahn et al. 2007a). It has also been used in the context of portfolio optimization (e.g. in Sass and

*Corresponding author. Email: [email protected] Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2011 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/14697680903193413

1548

W. Putscho¨gl and J. Sass

Haussmann 2004, Martinez et al. 2005, Rieder and Ba¨uerle 2005, Putscho¨gl and Sass 2008). The filters can be described as the solution of one Stochastic Differential Equation (SDE) in the HMM case or by one SDE and an ordinary differential equation for the second moment in the Kalman case. For more models which allow for finite-dimensional filters we refer to Sekine (2006). We also allow in our stock models for non-constant volatility (cf. Hobson and Rogers 1998, Haussmann and Sass 2004, Hahn et al. 2007b). Various risk measures have been applied recently to measure and control the risk of portfolios. Usually the risk constraint is static, i.e. a risk measure like Value-at-Risk (VaR) or some tail-expectation-based risk constraint like Limited-Expected-Loss (LEL) or Limited-Expected-Shortfall (LES) has to hold at the terminal trading time (see, for example, Duffie and Pan 1997, Basak and Shapiro 2001, Gundel and Weber 2008, Gabih et al. 2009). In Cuoco et al. (2007), a risk constraint is applied dynamically for a short horizon while the investor strives to solve the optimization problem with a longer time horizon. An optimal consumption and investment problem with a dynamic VaR constraint on the strategy is studied in Yiu (2004). In both papers dynamic programming techniques are applied and then numerical methods are used to solve the resulting Hamilton–Jacobi–Bellman equation. Their findings indicate that dynamic risk constraints are a suitable method for reducing the risk of portfolios. In Pirvu (2007), the model of Yiu (2004) is generalized. For risk management under partial information we also refer to Runggaldier and Zaccaria (2000). Also in practice (motivated by Basel Committee proposals) it is common to reevaluate risk constraints frequently (e.g. on a daily or weekly basis) with a short time horizon (cf. Jorion 2000). Motivated by Cuoco et al. (2007), we impose a slightly different class of risk constraints on the strategy, computed on a short horizon. An additional motivation for using dynamic risk constraints is that they can be specified in such a way that they limit the risk caused by trading at discrete times. Due to the non-constant drift in the models with partial information, portfolio optimizing strategies might have extreme long and short positions. This is no problem when trading continuously, since these strategies adapt to the continuous price movements. But if we trade only, say, daily, this can already lead to severe losses, e.g. having a long position of 400% leads to bankrupty when stock prices fall by more than 25%, as we observe for the market data in section 9.3. Also extreme short positions pose a problem. In section 5 we will see that dynamic risk constraints can limit these positions and thus put a bound on this discretization error. More precisely, these dynamic risk constraints lead to convex constraints on the strategy. These may depend on time, e.g. for non-constant volatility models. A verification result for finding an optimal strategy can then be provided analogously to the classical theory for convex constraints using implicitly the separation principle for filtering which states that we can do filtering first and optimization afterwards (cf. Gennotte 1986). In special

cases also the existence of a solution can be guaranteed. For logarithmic utility an analytic solution can be derived in a fairly general market model under partial information. Further, the risk constraints reduce the influence of certain parameters which may be difficult to estimate. We investigate the impact of inaccuracy that a parameter, e.g. the volatility of the stocks, has on the strategy and compare the results of the constrained case with those of the unconstrained case. Finally, the results are applied to historical stock prices. The results under dynamic risk constraints indicate that they are a suitable remedy for reducing the risk and improving the performance of trading strategies. The paper is organized as follows. In section 2 we introduce the basic model and the risk neutral measure which we need for filtering and optimization. In section 3 we introduce the optimization problem of maximizing expected utility from terminal wealth. We show how to use time-dependent convex constraints in section 4. In section 5 we apply risk constraints dynamically on the strategy and show how to derive optimal strategies under those constraints using the results of section 4. We introduce Gaussian dynamics for the drift in section 6 and a hidden Markov model in section 7. We investigate the impact of inaccuracy that a parameter has on the strategy in section 8, where we also compare the results of the constrained case with those of the unconstrained case. Finally, we provide a numerical example in section 9 where we illustrate the strategies in the constrained case and also apply them to real stock data. Notation. The symbol > will denote transposition. For a vector v, Diag(v) is the diagonal matrix with diagonal v. For a matrix M, diag(M) is the vector consisting of the diagonal of the matrix M. We use the symbol 1n for the n-dimensional vector whose entries all equal 1. The symbol 1nd denotes the n  d-dimensional matrix whose entries all equal 1. The symbol Idn denotes the n-dimensional identity matrix. Moreover, F X ¼ ðF X t Þt2½0, T  stands for the filtration of augmented -algebras generated by the F -adapted process X ¼ (Xt)t2[0,T]. We write x for the negative part of x: x ¼ max {x, 0}, and xþ for the positive part of x: xþ ¼ max {x, 0}. We denote the kth component of a vector a by ak. The kth row and column of a matrix A are denoted by (A)k . and (A). k, respectively.

2. The basic model In this section we outline the basic market model. We have to start with general conditions that allow us to change from the original measure to the risk neutral measure. Filtering and optimization will then be done under the risk neutral measure. Let ( , A, P) be a complete probability space, T40 the fixed finite terminal trading time, and F ¼ (F t)t2[0,T ] a filtration in A satisfying the usual conditions, i.e. F is right-continuous and contains all P-null sets. We can invest in a money market with constant interest rate r

1549

Optimal investment under dynamic risk constraints and n risky securities (stocks). The corresponding discount factors  ¼ (t)t2[0,T] read t ¼ exp (rt). The ðnÞ > price process S ¼ (St)t2[0,T], St ¼ ðSð1Þ t , . . . , St Þ , of the stocks evolves as dSt ¼ DiagðSt Þðt dt þ t dWt Þ,

S0 ¼ s0 ,

where W ¼ (Wt)t2[0,T] is an n-dimensional standard Brownian motion with respect to P. The return process R ¼ (Rt)t2[0,T] associated with the stocks is defined by dRt ¼ Diag(St)1 dSt, i.e. dRt ¼ t dt þ t dWt : We assume that the Rn-valued drift process  ¼ (t)t2[0,T] is progressively measurable with respect to the filtration F and the Rnn-valued volatility matrices ( t)t2[0,T] are progressively measurable w.r.t. F S and  t is non-singular for all t 2 [0, T ]. Definition 2.1: We define the market price of risk  ¼ (t)t2[0,T] by t ¼ t1 ðt  r 1n Þ and the density process Z ¼ (Zt)t2[0,T] by ! Z Zt 1 t 2 > ks k ds , t 2 ½0, T : ð1Þ Zt ¼ exp  s dWs  2 0 0 Assumption 2.2: Suppose Z T  kt k þ k t k2 dt 5 1

and

0

Z

kt k dt 5 1,

Lemma 2.3:

~

~

We have F S ¼ F R ¼ F W ¼ F R .

Proof: Due to the Lipschitz and linear growth condi~ t and of tions, the system consisting of dR~ t ¼ ðt ÞdW ~ Þ. equation (2) has a unique strong solution ðR, ~ R~ W In particular, F t  F t for all t 2 [0, T ]. To show the other inclusion, note that t }  t(!) ¼ (t(!)) is continuous and hence for At ¼  t  > t Z t  Z th 1 Aijt ¼ lim Aijs ds  Aijs ds h&0 h 0 0  1  ~i ~j i j ~ ¼ lim ½R , R t  ½R , R~ th , h&0 h where [X, Y ] denotes the quadratic covariation process ~ of X and Y. In particular, At is F R t -measurable. Choosing a fixed algebraic scheme to compute the root  t of At, we ~ can assume – without l.o.g. – that  t is F R t -measurable. ~ 1 W ~ t ¼ ð t Þ dR~ t shows that F  F R~ for all Thus, dW t t ~ ~ t 2 [0, T ]. Therefore, F W ¼ F R . Using that xi(x) is continuous and
limh&0 1h ½S i , S j t  ½S i , S j th ij At ¼ , Sti Stj ~

similar arguments as above imply F S ¼ F W . Constant r ~ implies F R ¼ F R . œ

T

2

where  and  are Rm-valued. Further, we demand that ,  and  as well as x } Diag(x)(x) satisfy the usual Lipschitz and linear growth conditions. We cite the next lemma from proposition 2.1 of Hahn et al. (2007b) but provide a more detailed proof here.

ðP  a.s.Þ

0

and that Z is a martingale with respect to the filtration F and the probability measure P. Further we demand ðnÞ for t ¼ ðð1Þ t , . . . , t Þ " # n  h i 4 X ðiÞ 5 1, E kt k 5 1, t 2 ½0, T  and E 0 i¼1

i ¼ 1, . . . , n:

Next, we introduce the risk neutral probability measure P~ by dP~ ¼ ZT dP, where Z is defined as in equation (1). ~ We denote by E~ the expectation operator under P. ~ t ¼ dWt þ t dt is Girsanov’s theorem guarantees that dW ~ a P-Brownian motion with respect to the filtration F . Thus, also the excess return process R~ ¼ ðR~ t Þt2½0, T  , ~ t, dR~ t ¼ dRt  r1n dt ¼ ðt  rÞdt þ t dWt ¼ t dW ~ and the price process has under P~ is a martingale under P; dynamics ~ t Þ: dSt ¼ DiagðSt Þðr1n dt þ t dW We model the volatility as t ¼ ðt Þ

Remark 1: The filtration F S is the augmented natural ~ ~ and ðW, ~ F S Þ is a P-Brownian filtration of W motion. 2 ~ So every claim in L ðP, F ST Þ can be hedged by classical martingale representation results. Thus, the model is complete with respect to F S. Example 2.4: For one stock (n ¼ 1) a class of volatility models which satisfy our model assumptions was introduced by Hobson and Rogers (1998). As the factor processes introduced in equation (2) we use the offset functions ( j ) of order j, Z1 tð j Þ :¼  eu ðR~ t  R~ tu Þ j du, t 2 ½0, T , j ¼ 1, . . . , m: 0

Contrary to Hobson and Rogers (1998), we use in the definition of the offset functions the excess return of the stock instead of the discounted log-prices. The offset function of order j can be written recursively as j ð j  1Þ ð j2Þ ~  dtð j Þ ¼ jtð j1Þ dR~ t þ t d ½R t  tð j Þ dt 2 ! j ð j  1Þ ð j2Þ 2 ðjÞ ~t t  t dt þ jtð j1Þ t dW t ¼ 2

in terms of the m-dimensional factor process  ¼ (t)t2[0,T] with dynamics

for j ¼ 1, . . . , m, where (1) :¼ 0 and (0) :¼ 1. Here, [] denotes the quadratic variation. In the special case m ¼ 1 we have

~ t, dt ¼ ðt Þdt þ ðt ÞdW

~ t  tð1Þ dt: dtð1Þ ¼  t dW

ð2Þ

1550

W. Putscho¨gl and J. Sass

In Hobson and Rogers (1998) the model rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   t ¼ ðtð1Þ Þ :¼  1 þ " tð1Þ

2

^

is considered, which we call the HR-model. The minimum and maximum volatility are given by 40 and  4 , respectively, and "  0 scales the influence of the offset function. For more details and further examples we refer to Hahn et al. (2007b). 3. Optimization We consider the case of partial information, i.e. we can only observe the interest rates and stock prices. Therefore, only the events of F S are observable and the portfolio has to be adapted to F S. Definition 3.1: A trading strategy  ¼ (t)t2[0,T] is an n-dimensional F S-adapted, measurable process satisfying Z T  2 > jp> s s j þ k s ps k ds 5 1 a.s. 0

In the definition above, t denotes the wealth invested in the stocks at time t. We denote the corresponding fraction of wealth invested in the stocks at time t by pt ¼ pt =Xpt , t 2 [0, T ]. For initial capital x040 the wealth process Xp ¼ ðXpt Þt2½0,T  corresponding to the self-financing trading strategy  is well defined and satisfies > p dXtp ¼ p> t ðt dt þ t dWt Þ þ ðXt  1n pt Þr dt,

X0p ¼ x0 : ð3Þ

Definition 3.2: A trading strategy  is called admissible for initial capital x040 if Xtp  0 a.s. for all t 2 [0, T ]. We denote the class of admissible trading strategies for initial capital x0 by A(x0). Definition 3.3: A utility function U :¼ [0, 1) ! R [ {1} is strictly increasing, strictly concave, twice continuously differentiable, and satisfies limx!1 U 0 (x) ¼ 0 and limx#0U 0 (x) ¼ 1. Further, I denotes the inverse function of U 0 . Assumption 3.4:

We demand that I satisfies

Ið yÞ  cð1 þ ya Þ

and jI 0 ð yÞj  cð1 þ yb Þ

ð4Þ

for all y 2 (0, 1) and for some positive constants a, b, c. Well known examples for utility functions are the logarithmic utility function U(x) ¼ log(x) and the power utility function U(x) ¼ x/ for 51,  6¼ 0. For a given utility function we can now formulate the following optimization problem. Optimization Problem 3.5: We consider the problem of maximizing the expected utility from terminal wealth, i.e. for given U i h maximize E UðXTp Þ over p 2 Aðx0 Þ under the condition E½U ðXTp Þ 5 1.

Under partial information with a stochastic drift this problem has first been addressed in Lakner (1995, 1998). His results show that the optimal terminal wealth can be expressed in terms of the conditional density

 t ¼ E Zt jF St ,

which is the filter for the martingale density. It will be convenient to denote by ~ ¼ t t , t 2 [0, T ], the discounted conditional density. In the subsequent sections we will see that the computation of the corresponding optimal strategy is based on the filter ^ t ¼ E½t jF St  for the drift t. Definition 3.6: We introduce the function X : (0, 1) } (0, 1] by i h ð5Þ X ð yÞ ¼ E ~T Ið y~T Þ :

Theorem 3.7: Suppose that X ( y)51 for every y 2 (0, 1). Then there exists a unique number y 2 (0, 1) such that X ( y ) ¼ x0. The optimal terminal wealth reads X T ¼ Ið y ~T Þ:

ð6Þ

If Ið y ~T Þ 2 ID1,1 then the unique optimal trading strategy for optimization problem 3.5 is given by  i h  S > 1 ~ X ð Þ E D p t ¼ 1 t T t t T F t  i h > 1 ~ 0 ~ ~  S ¼ 1 t T ðt Þ E I ð y T Þ y Dt T F t :

For a drift process with linear Gaussian dynamics or given as a continuous time Markov chain explicit solutions are provided in Lakner (1998) and Sass and Haussmann (2004), respectively. Here D denotes the Malliavin derivative; for a definition of the space ID1,1 and an introduction to Malliavin calculus we refer to Ocone and Karatzas (1991). Proof: Lemma 6.5 and theorem 6.6 in Lakner (1995) yield the first statement and equation (6), respectively. The only difference is that we have to look at the dis~ since we consider also a non-zero counted density , interest rate r. That E½U ðXpT Þ 5 1 follows as in Karatzas and Shreve (1998, theorem 3.7.6). Using martingale representation arguments, Lakner (1995, theorem 6.6) further shows that the optimal investment strategy  is uniquely given by ZT ~ t: T XT ¼ x0 þ t ð t Þ>  t dW 0

On the other hand, for T X T ¼ x0 þ

Ið y ~T Þ Z

T 0

2 ID1,1 Clark’s formula

~ t X j F S > dW ~t E½D T t

holds, see Ocone and Karatzas (1991, proposition 2.1). œ By comparison we get the representation for  .

4. Time-dependent convex constraints Under full information, convex constraints on the strategy have been examined in detail in Cvitanic´ and

1551

Optimal investment under dynamic risk constraints Karatzas (1992), Cvitanic´ (1997) and under partial information in Martinez et al. (2005) and Sass (2007). The latter only consider time-independent constraints and we cannot use the first results directly, since the model with partial information does not satisfy their assumptions (Brownian filtration). But following Sass (2007) we can apply filtering techniques and transform our market model to a model under full information w.r.t. F S, which is a Brownian filtration – see remark 1. Then we can adapt the theory of time-dependent constraints as outlined in Cvitanic´ and Karatzas (1992, section 16) to our model. Using the filter h i ^ t ¼ E t jF St ,

we can reformulate the model with respect to the innovation process V ¼ (Vt)t2[0, T ] defined by Zt Zt Zt 1 ^ ^ s ds: Vt ¼ Wt þ  1 ð   Þds ¼  dR   1 s s s s s s  0

0

0

ð7Þ

We assume F V ¼ F S. For F S F V we would need instead of the standard martingale representation a representation for F S-martingales with respect to F V (cf. Sass 2007, remark 3.4). We can now write our model under full information with respect to F S as dRt ¼ ^ t dt þ  t dVt ,

t 2 ½0, T :

The whole theory from time-independent constraints carries over to the case of time-dependent constraints with some minor modifications (cf. Cvitanic´ and Karatzas 1992, section 16). Next, we will impose constraints on t, the fraction of wealth invested in the stocks at time t. We define the random set-valued process K0,T ¼ (Kt)t2[0,T], where Kt represents the constraints on portfolio proportions at time t and is given for ! 2 by a non-empty closed convex set Kt(!)  Rn that contains 0. Definition 4.1: A trading strategy  is called K0,Tadmissible for initial capital x040 if Xtp  0 a.s. and pt 2 Kt for all t 2 [0, T ]. We denote the class of K0,Tadmissible trading strategies for initial capital x0 by A0,T(x0). The constrained optimization problem then reads as follows. Optimization Problem 4.2: We consider the problem of maximizing expected utility from terminal wealth, i.e. for given U and x040 h i maximize E UðXpT Þ over p 2 A0,T ðx0 Þ under the condition E½U ðXpT Þ 5 1.

For each t we define the support function t: Rn } R [ {þ1} of Kt by t ð yÞ ¼ t ð yjKt Þ ¼ sup ðx> yÞ, x2Kt

y 2 Rn :

For each (t, !) the mapping y } t(yjKt(!)) is a lower semi-continuous, convex function on its effective domain 6 ;. Then, K~ t is a convex cone, K~ t ¼ fy 2 Rn : t ð yÞ 5 1g ¼ called the barrier cone of Kt. Example 4.3: (i) The unconstrained case corresponds to Kt ¼ Rn; then K~ t ¼ f0g and t  0 on K~ t for all t 2 [0, T ]. (ii) Typical time-dependent constraints might be given by Kt ¼ fx 2 Rn : ltðiÞ  xi  uðiÞ t , i ¼ 1, . . . , ng, ðiÞ where lt , uðiÞ t take values in (1, 0] and [0, 1), and t ð yÞ ¼ respectively. Then K~ t ¼ Rn Pn ðiÞ ðiÞ ðiÞ þ ðiÞ  ðu ð y Þ  l ð y Þ Þ. t t i¼1 (iii) Time-dependent constraints might also be given by closed convex sets Kt B"t(0) for some "t40, where B"t(0) denotes the ball with centre 0 and radius "t. If Kt is also bounded, then K~ t  Rn . We introduce dual processes  : [0, T ]  ! R, t 2 K~ t , which are F St -progressively measurable processes and satisfy E

"Z

0

T



2

#

kt k þ t ðt Þ dt 5 1:

We denote the set of dual processes by H and the subset of uniformly bounded dual processes by Hb. Assumption 4.4: For all t 2 H, (t(t))t2[0,T] is also F St -progressively measurable. For example 4.3 (iii), whether assumption 4.4 holds depends on the boundary of Kt. Example 4.3 (ii) is a special case of example 4.3 (iii) for which assumption 4.4 S is satisfied if ltðiÞ , uðiÞ t are F t -progressively measurable and square-integrable (cf. Cvitanic´ and Karatzas 1992, section 16). For each dual process  2 H we introduce a new interest new rate process rt ¼ r þ t ðt Þ and the corresponding Rt discount factor is given by t ¼ t expð 0 s ðs ÞdsÞ. Further, we consider a new drift process ^ t ¼ ^ t þ t þ t ðt Þ1n . Then the new market price of risk reads ^ t  r 1n þ t Þ and the new ^ t  rt 1n Þ ¼  1 t ¼  1 t ð t ð density process  ¼ ðt Þt2½0, T  is given by ! Zt Zt 1 2 k k ds , t 2 ½0, T : t ¼ exp  ðs Þ> dVs  2 0 s 0 Moreover, we introduce theRnotation ~t ¼ t t . If  is a t martingale under P and 0 ks k2 ds 5 1 a.s. for all  t 2 [0, T ], then P defined by dP ¼ T dP would be a ~  ¼ dVt þ  dt a Brownian probability measure and dW t t motion under P. Note that  would be a martingale, if  2 Hb. Thus, we consider a new market with bond prices S0, and stock prices Si, for i ¼ 1, . . . , n and  ¼ (1, . . . , n) given by  Z t ð0Þ S0,  ð Þds , ¼ S exp s s t t 0 Z t    i ð Þ þ  exp  Sti, ¼ SðiÞ s s t s ds : 0

1552

W. Putscho¨gl and J. Sass

The wealth process X, then satisfies Zt p > ~ t Xp, ¼ x þ s Xp, 0 t s ðs Þ s dWs , 0

5. Dynamic risk constraints Xp, 0 ¼ x0 :

Further, we introduce the functions X  as the analogue of equation (5) where ~ is substituted by ~ . Given a dual process  we strive to solve the optimization problem in the new market under no constraints in the class of admissible trading strategies for initial capital x0 which is denoted by A0,T ðx0 Þ.

~ t( j Kt(!)) Assumption 4.5: Let K0,T be such that K~ t  K, is continuous on K~ for all t 2 [0, T ], ! 2 , and that all ~ constant, K-valued processes  belong to H. Weaker conditions can be formulated for the case where 0 2 = Kt or K~ t is not constant (cf. Cvitanic´ and Karatzas 1992, assumption 16.2). For constraint sets as in example 4.3, assumption 4.5 is satisfied (cf. Cvitanic´ and Karatzas 1992, section 16). We can now formulate a proposition like that of Cvitanic´ (1997, lemma 11.6) that allows us to compute optimal strategies. Proposition 4.6: Suppose x040 and E½U ðXpT Þ 5 1 for all  2 A0,T(x0). A trading strategy  2 A0,T(x0) is optimal for optimization problem 4.2 if, for some y 40,  2 H with X  51 for all y40, XTp ¼ Ið y ~T Þ,



X  ð y Þ ¼ x0 ,



where ~T ¼ ~T . Further,  and  have to satisfy the complementary slackness condition t ð t Þ

þ

ðpt Þ>  t

¼ 0,

t 2 ½0, T :

The proof works analogously to the proofs in Cvitanic´ and Karatzas (1992) (cf. Cvitanic´ 1997) and also the direct proof in Sass (2007, proposition 3.2). In the proof it is shown that y ,  as given in the proposition solve the dual problem h i ~ y ~ Þ , ~ y Þ ¼ inf E Uð Vð T 2H

In this section we want to apply risk constraints on the wealth dynamically. Some motivation for dynamic risk constraints has been given in the introduction. In Cuoco et al. (2007) the risk constraint is applied dynamically under the assumption that the strategy is unchanged for a short time, i.e. the portfolio manager trades continuously to maintain a constant proportion of his wealth invested in the risky assets for this short time horizon. In Yiu (2004) the constraint on the strategy is applied dynamically for a short time horizon assuming that the wealth invested in the risky assets remains constant during this period. Our approach works also for these approximations. However, we make a slightly different approximation which is of special interest in particular when optimizing under partial information. When applying the optimal strategies to market data we can only trade at discrete times, e.g. daily, and we observe that very big short and long positions may occur when using a non-constant drift model. For daily trading and utility functions with low risk aversity this may result in bankruptcies. Therefore, instead of assuming strategy or wealth to be constant, we use the assumption that the number of shares remains constant between the discrete trading times for computing the risk. Then the risk constraints allow us to measure and control the risk caused by discretization. We can write the dynamics of the wealth process as > p p dXpt ¼ N> t DiagðSt Þðt dt þ t dWt Þ þ ð1  1n t ÞXt r dt,

where Nt :¼ Xpt DiagðSt Þ1 pt represents the number of stocks in the portfolio. Limited Expected Loss (LEL) constraint. The wealth process satisfies under risk neutral dynamics ~ t þ Xp r dt: dXpt ¼ ðpt Þ> Xpt t dW t

ð8Þ

where U˜ is the convex dual function of U n o ~ yÞ ¼ sup UðxÞ  xy , y 4 0: Uð x40

These proofs are carried out for time-independent constraints Kt  K; for the differences resulting from time-dependent constraints, we refer to Cvitanic´ and Karatzas (1992, section 16). Remark 2: Cvitanic´ and Karatzas (1992, theorem 13.1) provides conditions that guarantee the existence of an optimal solution under full information. If F V ¼ F S holds then the same result holds under partial information as in the case of full information. For power utility U(x) ¼ x/ with  2 (0, 1) the conditions of Cvitanic´ and Karatzas (1992, theorem 13.1) are satisfied if assumption 4.4 and 4.5 hold. For logarithmic utility, we give in what follows an explicit representation of the optimal solution under time-dependent constraints.

If we cannot trade in (t, t þ Dt), then the difference DXpt ¼ XptþDt  Xpt reads   DXpt ¼ Xpt 1  ðpt Þ> 1n expðr DtÞ þ Xpt ðpt Þ>  exp

"Z

t

tþDt 

þ

 1 > r  diagð s  s Þ ds 2

Z

tþDt t

#

~ s  Xp  s dW t

    ¼ Xpt exp r Dt  Xpt þ exp r Dt ðpt Þ> Xpt 

"



1 exp  2 þ

Z

Z

tþDt

t

tþDt t

diagð s  > s Þds 

#

~ s  1n ,  s dW

1553

Optimal investment under dynamic risk constraints where we write exp(a) ¼ (exp(a1), . . . , exp(an))> for a ¼ (a1, . . . , an)>. Next, we impose the relative LEL constraint i h ð9Þ E~ ðDXpt Þ jF St  "t ,

with "t ¼ LXpt . Then, any loss in the time interval [t, t þ Dt) could be hedged with a fraction L of the portfolio value at time t, when we assume that – even if we restrict ourselves to trading at discrete times – we can still trade at these times with options available in the continuous-time market. Note that for hedging the loss we then only need European call and put options. We define  h o i n  KLEL :¼ pt 2 Rn  E~ ðDXpt Þ jF St  "t : t For n ¼ 1 we obtain KLEL ¼ ½lt , ut , where we can find ut t l and t numerically as the maximum and minimum of  for which inequality (9) is satisfied (cf. section 9.1).

time t. Again we need for hedging only European call and put options. We define  h n o i  :¼ p 2 Rn E~ ðDXpt þ qt Þ jF St  "t : KLES t

Analogously to the LEL constraint we obtain for n ¼ 1 the interval KLES ¼ ½lt , ut . Figure 2 shows the bounds on pt t for various values of L1 and L2 and for  ¼ 0.2 and r ¼ 0.02. Figure 6 shows KLES for n ¼ 2 where we used  L1 ¼ 0.01, 0:3 0:15 L2 ¼ 0.05, r ¼ 0.02 and  ¼ . 0:15 0:3 The next lemma guarantees that also in the multidimensional case we are again in the setting of convex constraints. Lemma 5.1:

Proof: It suffices to prove the lemma for KLES . Suppose t 0 t , 0t 2 KLES and X ¼ X 4 0. We want to show for t t t  2 (0, 1) that  t ¼ t þ ð1  Þ0t 2 KLES , i.e. t h i E~ ðDXt t þ qt Þ jF St  "t ,

Remark 3: For constant volatility and interest rates, ut and lt are time-independent and we can then use the results from Sass (2007) directly to solve optimization problem 4.2. Figure 1 illustrates the bounds on the strategy for various values of L and Dt. Figures 3 and 4 show the bounds for fixed values of L and for  ¼ 0.2 and r ¼ 0.02. Figure 5 shows KLEL for n ¼ 2 where we used  0:3 0:15 . L ¼ 0.01, r ¼ 0.02 and  ¼ 0:15 0:3

L1 Xpt

10 Boundson η π

Limited Expected Shortfall (LES) constraint. Further, we introduce the relative LES constraint as an extension to the LEL constraint h i E~ ðDXpt þ qt Þ jF St  "t , ð10Þ

KLEL are convex. and KLES t t

0

10

−10 2 5

1

L2 Xpt .

and qt ¼ For L2 ¼ 0 we obtain the with "t ¼ LEL constraint. Then, any loss greater than a fraction L2 of the portfolio value in the time interval [t, t þ Dt) could be hedged with a fraction L1 of the portfolio value at

L1

0

L2

0

Figure 2. Bounds on  (LES).

8 L = 2% L =1.2% 6 Upper boundon η π

Bounds on η π

10

0

L = 0.4%

4

2 5

−10 2

L (in

1 % of W ealth)

0

0

Figure 1. Bounds on  (LEL).

d (in ∆t

) ays

0 0

1

2

3

4

∆ t (in days)

Figure 3. Upper bound on  (LEL).

5

1554

W. Putscho¨gl and J. Sass Reduction of risk. In the introduction we motivated the constraints above as a remedy to reduce the discretization error which comes from investing at discrete trading times only while the market evolves continuously. In this section we illustrate how these constraints are related to certain convex risk measures of the terminal wealth. This analysis shows that by imposing the dynamic risk constraints motivated from discrete trading, we indeed control the risk associated with our portfolio. We show this by an example for the LES constraint and interest rate r ¼ 0. To this end, suppose that we have split [0, T ] in N intervals [tk, tkþ1] of length Dt ¼ N1T. By Xk we denote the wealth at tk obtained by trading at discrete times t0, . . . , tk1. So XN is the terminal wealth when we apply the continuous-time optimal strategy at these discrete times only. Then Xkþ1 ¼ Xk þ DXk, where DXk is defined as for DXtp for t ¼ tk (preceding equation 9). Therefore,

0

Lower bound on η π

−2

−4

−6

L = 2% L= 1. 2% L = 0. 4%

−8 0

1

2 3 ∆ t (in days)

4

5

Figure 4. Lower bound on  (LEL).

2

XN ¼ x0 þ

N 1 X

ð11Þ

DXk :

k¼0

We impose the risk constraints (10) dynamically. Since for the choices "t ¼ L1 Xtp , qt ¼ L2 Xtp all terms are proportional to Xtp , (10) does not depend on Xtp . Therefore, DXk satisfies the LES constraint  h i  E~ ðDXk þ L2 Xk Þ  F Stk  L1 Xk , k ¼ 0, . . . , N  1:

η2

1

0

ð12Þ

−1

PN1

By adding L2 k¼0 Xtk  q for q40 on both sides of (11), taking the negative part, using that x } x is subadditive ~ we get from the LES and taking expectation w.r.t. P, constraints " ! # N 1 X E~ XN  q þ L2 Xk

−2 −2

−1

0 η1

1

2

k¼0

Figure 5. KLEL for n ¼ 2.

"

¼ E~

where we use the notation  t DXt ¼ Xt expðrDtÞ  1 þ expðrDtÞð t Þ> :

 ðx0  qÞ þ

 Z 1 tþDt  exp  diagð s  > s Þds 2 t   Z tþDt ~ s  1n  s dW þ

ðDXk þ L2 Xk Þ

k¼0

N 1 X

E~ ½ðDXk þ L2 Xk Þ 

k¼0

 ðx0  qÞ þ

N 1 X

~ k : L1 E½X

k¼0

t

0

¼ DXt t þ ð1  ÞDXt t :

Due to the convexity of x } x and the linearity of E~ we obtain h i E~ ðDXt þ qt Þ jF St i h 0 ¼ E~ ðDXt t þ ð1  ÞDXt t þ qt þ ð1  Þqt Þ jF St i i h h  E~ ðDXt þ qt Þ jF St þ ð1  ÞE~ ðDXt þ qt Þ jF St  "t þ ð1  Þ"t ¼ "t :

x0  q þ

! #

N 1 X

h

~ Then Suppose now that XN is integrable under P. S ~ ~ Xk ¼ E½XN j F k , in particular E½Xk  ¼ x0 . So we have shown " ! # N 1 X S ~ NjF  ðXN Þ :¼ E~ XN  q þ L2 E½X tk

k¼0

~ N   qÞ þ NL1 E½X ~ N :  ðE½X

ð13Þ

We define the acceptance set ~ : ðX Þ  ðE½X ~   qÞ þ NL1 E½X ~ g: R ¼ fX 2 L1 ðPÞ

1555

Optimal investment under dynamic risk constraints Then, since (i) inf{m 2 R : m 2 R} ¼ 051, (ii) Y  X, X 2 R implies Y 2 R and since (iii) R is convex, we can define a convex risk measure by ðX Þ :¼ inffm 2 R : m þ X 2 Rg,

~ X 2 L2 ðPÞ,

see Foellmer and Schied (2004, section 4.1) for corresponding results, definitions and motivation. So, by imposing dynamic risk constraints we also put a bound on a convex risk measure of the terminal wealth. The limiting cases of inequality (13) have a good interpretation. For example, for L1 ¼ N1, 40, and L2 ¼ 0, inequality (13) corresponds to a bound on the expected shortfall

This is a risk constraint used in the literature quite often (cf. the references provided in the introduction). Another extreme choice would be the portfolio insurer problem. Say we want to be sure that we get back at least a fraction 2 (0, 1) of our initial capital, i.e. we require ð14Þ

Then, by defining L2 as (1  L2)N ¼ and choosing L1 ¼ 0 we get from our dynamic LES constraints (12) at each time tk that Xkþ1  ð1  L2 ÞXk ¼ DXk þ L2 Xk  0 P~  a.s., k ¼ 0, . . . , N  1:

Other constraints. To apply a Value-at-Risk constraint we proceed as follows. Under the original measure, DXpt is given by   DXpt ¼ Xpt exp r Dt  Xpt þ ðpt Þ> Xpt   Z tþDt 1 Þ ds  exp s  diagð s  > s 2 t   Z tþDt  s dWs  expðr DtÞ : þ t

~ N  qÞ   ðx0  qÞ þ x0 : ðXN Þ ¼ E½ðX

PðXN  x0 Þ ¼ 1:

positions which cannot be adjusted when trading only in discrete time.

ð15Þ

Since P~ is equivalent to P, this also holds under P and thus we get by iteration XN  ð1  L2 ÞXN1  ð1  L2 Þ2 XN2      ð1  L2 ÞN x0 ¼ x0 : Therefore, the dynamic risk constraints in constraints (12) imply equality (14). Remark 4: Note that the static risk constraint obtained in (13) from the dynamic constraints only illustrates the relation to convex risk measures on the terminal wealth, since the bounds might be very poor. Instead, we get a sharp bound for the following constraint: by comparing (12) and (15) we see that we control the maximum shortfall in one period, n o max E~ ½ðXkþ1  ð1  L2 ÞXk Þ  : k ¼ 0, . . . , N  1  L1 x0 ,

which has a straightforward interpretation. But a definition corresponding to that for  above would not lead to a convex risk measure since  would not be monotone. One can construct examples where a wealth process Y with Yk  Xk (P~  a.s.) might not be preferred to X. The reason is that Y might have a much higher variance ~ k  ð1  L2 ÞYk1 5 Xk  ð1  L2 ÞXk1 Þ 4 0 and thus PðY can occur. Note that the stronger dependency on the variance is good in our case since it yields narrower bounds on the possible long and short positions, and the discretization error is mainly due to extreme

We impose the relative VaR constraint on the loss ðDXpt Þ ,   ð16Þ P ðDXpt Þ  LXpt jF St , t ¼ ^ t  ,

for some prespecified probability . Since the probability is computed under the original measure P we need the (unknown) value of the drift. As we optimize under partial information and cannot observe the drift directly we consider for the drift t the filter for the drift ^ t . At time zero we have no information and we can assume a stationary distribution for the drift. Under full information we would not need to make such an approximation, under partial information this allows us to keep the model tractable. In the one-dimensional case n ¼ 1 the risk constraint equation (16) results in an upper bound ut and lower bound lt on the strategy pt and we can find numerically the maximum and minimum of pt for which equation (16) is satisfied. Note that for n41 the set KVaR may not be convex. For L ¼ 0.1, ¼ 0.05, r ¼ 0.02 and   2 0 0:5 10 B¼ , 2 10 0:5 0 0 1 20:16 0:08 20 0:08 B 20 40:08 20 0:08 C B C Q¼B C, @ 20 0:08 20:16 0:08 A ¼



20 0:08  0:04 0:05 , 0:05 0:05

20

40:08

Figure 7 illustrates the set KVaR at time t ¼ 0 where we consider a stationary distribution for the drift. The construction of the example is based on a Markov chain with extreme states and a correlation structure which – without further noise – does not allow to have very long positions in both stocks simultaneously. Due to the volatility matrix with low volatility level and high correlation these features carry over to the stock model. Therefore, if we are very long in one stock, the risk can only be reduced by choosing a short position for the other stock, which can be seen very well in Figure 7. At time t40 when we also have information about the filter for the drift we can find a modified example for which KVaR is not convex either.

1556

W. Putscho¨gl and J. Sass hold at the terminal time only and we can trade dynamically, the strategies under the LEL and VaR constraint exhibit quite different risk profiles (cf. Basak and Shapiro 2001).

6

4

Optimization. Under dynamic risk constraints we have to solve optimization problem 4.2 for specific constraint sets Kt, e.g. Kt ¼ KLES , t 2 [0, T ]. Using proposition 4.6 t we can show in the one-dimensional case, where Kt ¼ ½lt , ut , lt 5 0 5 ut , the following corollary.

2

η2

0

−2

Corollary 5.2: Suppose n ¼ 1 and that the boundaries l and u are F S-progressively measurable and square integrable. Then for U(x) ¼ log(x) the optimal risky fraction for the constrained problem is given by 8 u  if ot 4 ut , > > h i < t ct :¼ pt ¼ ot if ot 2 lt , ut , > > : l t if ot 5 lt ,

−4

−6 −5

0 η1

5

Figure 6. KLES for n ¼ 2.

where

ot :¼ pt ¼ 20

is the optimal risky fraction without constraints. So we cut off the strategy obtained under no constraints if it exceeds u or falls below l. For completeness, we provide a direct proof based on proposition 4.6.

10

η2

1 ð^  rÞ:  2t t

0

−10

−20 −20

−10

0 η1

10

20

Figure 7. KVaR for n ¼ 2.

Analogously to the previous case it is possible to apply a large class of other risk constraints on the strategy like a Conditional Value-at-Risk (CVaR) constraint. The CVaR at a given confidence level is the expected loss given that the loss is greater than or equal to the VaR (see, for example, Rockafellar and Uryasev 2000, 2002). Note that the (relative) LEL/LES and the (C)VaR constraints do not result in a substantially different risk behaviour. This is due to the assumption that we cannot trade in the time interval for which the constraint is applied. A similar result would hold if the strategy (i.e. the fraction of wealth invested in the risky assets) were assumed to be constant in the interval in which the risk constraint is applied. For a static risk constraint, i.e. if the constraint has to

Proof: For these constraints, assumptions 4.4 and 4.5 are satisfied (cf. the references given after those assumptions). For U ¼ log, we have X ( y) ¼ y1 for all dual processes , hence y ¼ x1 0 . The dual process corresponding to the strategy in the corollary is 8 2 u u o o > <  t ðt  t Þ, if t 4 t ,  t ¼ 0, if 0t 2 ½lt , ut , t 2 ½0, T , > : 2 l  t ðt  ot Þ, if ot 5 lt ,

and one can verify directly that ct is optimal in the auxiliary market given by  , since the market price of ^ t  r   t Þ ¼  t ct . As in the unconrisk is t ¼  1 t ð strained Merton problem the corresponding terminal wealth then has the same form as in proposition 4.6. Finally, we can compute t ð t Þ ¼ sup ð t Þ ¼ ct  t , 2½lt ,ut 

so the complementary slackness condition holds.

h

Also in the multidimensional case, under assumptions 4.4 and 4.5 the problem can be solved up to a pointwise minimization which characterizes the dual process (cf. Cvitanic´ 1997, example 12.1). For example, in cases such as example 4.3 (iii), we have K~ ¼ Rn and thus Rockafellar (1970, theorem 10.1) implies that  is continuous in y, so assumption 4.5 holds. If t is, for example, also continuous in t, this would yield assumption 4.4. To get constraint sets and support functions of such a form is mainly due to conditions on the volatility

1557

Optimal investment under dynamic risk constraints matrices  t, e.g. some boundedness from above and away from 0 would be needed. The strategies corresponding to logarithmic utility under no constraints maximize the average rate of returns resulting in risky strategies. The dynamic risk constraints result in higher risk aversion. In the unconstrained case it is possible to use Malliavin calculus to obtain rather explicit representations of the strategies for general utility. If it is possible under constraints to find the optimal dual process  and to show that T Ið y ~T Þ lies in the domain ID1,1 of the Malliavin differential operator, then we can use Clark’s formula (cf. Ocone 1984) to obtain  i h   1 1    ~  S F pt ¼ t Ið y Þ D ð > Þ E   t T T t , t

where E denotes expectation under P, and y is such that the budget constraint is satisfied. Clearly, this is a difficult task. One approach would be to formulate the dual problem (8) as a Markovian stochastic control problem for controls . This can be done in terms of the innovations process (7), if we can compute the filter by solving a finite number of stochastic differential equations (see Sass 2007, p. 233, for the HMM). If this problem can be solved in a first step, one might be able to derive conditions on  which guarantee T Ið y ~T Þ 2 ID1,1 . This is subject for future research.

In this section we model the drift as in Lakner (1998) as the solution of the SDE t, dt ¼ ð  t Þ dt þ  dW

ð17Þ

 is an n-dimensional Brownian motion with where W respect to (F , P), independent of W under P, and ,  2 Rnn,  2 Rn . We assume that  is non-singular and that 0 follows an n-dimensional normal distribution with known mean vector ^ 0 and covariance matrix 0. We are in the situation of Kalman filtering with signal  and observation R, and ^ t ¼ E½t j F St  ¼ E½t j F R t  and t ¼ E½ðt  ^ t Þðt  ^ t Þ> j F St  are the unique F S-measurable solutions of h i  1 1 d^ t ¼   t ð t  > dRt , ^ t þ  dt þ t ð t  > t Þ t Þ ð18Þ

_ t ¼

 t  t > þ > ,

ð19Þ

with initial condition ð^ 0 , 0 Þ and ^ is conditionally Gaussian (cf. Liptser and Shiryayev 1978, theorem 12.7). Proposition 6.1: ~ F W ¼ F V.

1 þ t ð t  > t Þ  t dVt , dRt ¼ ^ t dt þ  t dVt ,

has a strong solution since it satisfies the usual Lipschitz and linear growth conditions (cf. Liptser and Shiryayev 1978, p. 29, Note 3). œ Hence, we can use the results in section 4 that guarantee the existence of a solution for certain utility functions and constraint sets. The conditional density  can be repre^ The next sented in terms of the conditional mean . theorem corresponds to Lakner (1998, theorem 3.1). Theorem 6.2: the SDE

The process 1 ¼ ðt1 Þt2½0, T  satisfies

1 ~t, dt1 ¼ t1 ð^ t  r1n Þ> ð > dW t Þ

and we have the representation Zt 1 ~s dW t ¼ exp  ð^ s  r1n Þ> ð > s Þ 0

1 þ 2

Z

t 0

^s k 1 s ð

2

!

 r1n Þk ds :

Assumption 6.3: Suppose that k 1 t k, t 2 ½0, T , is uniformly bounded by a constant c140. For the constants a, b of assumption 3.4 let

6. Gaussian dynamics for the drift

1 t ð t  > t Þ t

equation (18) in terms of the innovation process V and the system  h i 1 > 1  ^ Þ þ  ð  Þ þ   dt d^ t ¼   t ð t  >  t t t t t

For the GD we have F S ¼ F R ¼ ~

~

Proof: Lemma 2.3 ensures F S ¼ F W ¼ F R and equation (7) yields F V 7 F S. The Ricatti equation (19) has a continuous unique solution. Further we can write

trð0 Þ þ Tkk2 

1 Tc21 c2 maxt2½0,T ket k2 ,

where c2  max{360, (8a  3)2  1, (16b þ 1)2  1}. Note that if  is a positive symmetric matrix then maxtT ketk2  n (cf. Lakner 1998, p. 84). Depending on the utility function, this assumption ensures that the variance of the drift  is small compared to the variance of the return R. The following lemma states that this is sufficient for the assumptions of sections 2 and 3 to hold. Lemma 6.4: For Z as in equation (1), where the drift  is defined as in equation (17), assumption 2.2 and the conditions of theorem 3.7 are satisfied. Proof: The verification of the first part of assumption 2.2 can be done as in the proof of Lakner (1998, lemma 4.1). The proof of the second part follows from the finiteness of the corresponding moments of the Gaussian distribution. With some modifications based on the nonconstant volatility and the dependence of assumption 6.3 on the utility function via the constants a and b, it can be shown similarly as in Lakner (1998, lemma 4.1) along the lines of lemmas A.1 and A.2 in Lakner (1998), that q Zs 2 Lq(P) for q  5, Z1 s 2 L ðPÞ for q  4, and ~ ~ for p  maxf4, 2a, 4ðb  1Þg, s 2 L5 ðPÞ, s1 2 Lp ðPÞ s 2 ½0, T :

1558

W. Putscho¨gl and J. Sass

Following the arguments in Lakner (1998) or Putscho¨gl and Sass (2008) this implies Ið y ~T Þ 2 ID1,1 for y40. As in Putscho¨gl and Sass (2008) it can also be shown that a ~ E½sup t2½0,T t  5 1, which guarantees X (y)51 for all y40. œ Remark 5: A special case of GD is the Bayesian case, ðnÞ where we assume that the drift t  0 ¼ ðð1Þ 0 , . . . , 0 Þ is an (unobservable) F 0-measurable Gaussian random variable with known mean vector ^ 0 and covariance matrix 0. Then we can even solve the multidimensional filtering equation explicitly (cf. Liptser and Shiryayev 1978, theorem 12.8), and the solution reads  1 Zt > 1 ^ t ¼ 1nn þ 0 ð s  s Þ ds 0   Zt > 1  ^ 0 þ 0 ð s  s Þ dRs ,  t ¼ 1nn þ 0

0 Zt 0

1 ð s  > ds s Þ

1

0 :

7. A hidden Markov model for the drift In this section we model the drift process  of the return as a continuous-time Markov chain given by t ¼ BYt, where B 2 Rnd is the state matrix and Y is a continuoustime Markov chain with state space the standard unit vectors {e1, . . . , ed} in Rd. The state process Y is further characterized by its rate matrix Q 2 Rdd, where Qkl ¼ limt!0 1t PðYt ¼ el jY0 ¼ ek Þ, k 6¼ l, is the jump rate or transitionP rate from ek to el. Moreover, k ¼ Qkk ¼ dl¼1,l6¼k Qkl is the rate of leaving ek. Therefore, the waiting time for the next jump is exponentially distributed with parameter k, and Qkl/k is the probability that the chain jumps to el when leaving ek for l 6¼ k. For filtering we need the risk neutral probability measure P~ introduced in section 2. Let us introduce the martingale density process Z as in equation (1) with t ¼ Yt where t ¼ t1 ðB  r1nd Þ. Hence, the process Z satisfies dZt ¼ Zt ðs Ys Þ> dWs : Then (tYt)t2[0,T] is uniformly bounded, Z ¼ (Zt)t2[0,T] is a martingale under P~ and assumption 2.2 is satisfied. We are in the situation of HMM filtering with signal Y and observation R. We denote the normalized filter Y^ t ¼ E½Yt jF St . Besides the conditional density t ¼ E½Zt jF St  ~ 1 Yt jF S . we need the unnormalized filter E t ¼ E½Z t T Theorem 7.1 : The unnormalized filter satisfies Zt Zt ~ E t ¼ E½Y0  þ Q> E s ds þ DiagðE s Þ> s dWs , t 2 ½0, T : 0

0

The normalized filter is given by Y^ t ¼ t E t , where t1 ¼ 1> d Et.

This is Haussmann and Sass (2004, theorem 4.2), which extends Elliott (1993, theorem 4) to non-trivial volatility.

We cite the next proposition 4.5).

proposition

from

Sass

(2007,

Proposition 7.2: For the HMM we have F S ¼ F R ¼ ~ F W ¼ F V. Hence, we can use the results in section 4 which guarantee existence of a solution for certain utility functions and constraint sets. Next, we cite the following corollary about the conditional density  from Haussmann and Sass (2004, corollary 4.3) and the subsequent lemma from Haussmann and Sass (2004, lemma 2.5, proposition 5.1). Corollary 7.3: The processes  and 1 ¼ ðt1 Þt2½0, T  are ~ continuous F S-martingales with respect to P and P, respectively. Moreover,

 S and t1 ¼ E~ Z1 t jF t Zt ~ s, t1 ¼ 1 þ ðs E s Þ> dW

t 2 ½0, T :

0

Lemma 7.4: For all q  1 and t 2 [0, T ] we have t 2 Lq(P) ~ and t1 2 Lq ðPÞ. Due to the boundedness of , for uniformly bounded t1 a result like lemma 6.4 holds for the HMM without any further assumptions.

8. Sensitivity analysis In this section we investigate the impact of inaccuracy that a parameter , e.g.  ¼ i, has on the strategy  and compare the results for the unconstrained case with several constrained cases. There are several possibilities to compute the sensitivity. The finite difference method or resampling method  consists in computing E ½pt on a fine grid for  and using a forward or central difference method to get an estimator  of the gradient of E ½pt (cf. Glasserman and Yao 1992, L’Ecuyer and Perron 1994). This method yields biased estimators and the computations may be time-consuming. If  is not smooth, Malliavin calculus may be applied (under some technical conditions) and the sensitivity may be written as an expression where the derivative of  does not appear (cf. Gobet and Munos 2005). If the interchange of differentiation and expectation is justified we may put the derivative inside the expectation and thus get an unbiased estimator. This method has been proposed in Kushner and Yang (1992) and is called the pathwise method. We provide a sensitivity analysis for the HMM using the pathwise  method. In particular, we want to compute ðq=q&ÞE ½pt : In the following we compute the sensitivity for the HMM and logarithmic utility w.r.t. various parameters. For the GD an analysis can be performed analogously. Since we want to analyse the HMM we use constant

1559

Optimal investment under dynamic risk constraints volatility; for ease of notation we set the interest rates equal zero. In the unconstrained case we consider i q h oi q h E t ¼ E ð > Þ1 ^ t : q& q& Using the filter for the HMM we obtain i q h > 1 ^ i q h > 1 E ð Þ BYt ¼ E ð Þ Bt E t q& q& i q ~ h > 1 ¼ E ð Þ BE t : q&

The pathwise derivative of E with respect to  under P~ which we denote by E_ reads Zt E_ t ¼ ðQ_ > E s þ Q> E_ s Þds 0

þ

Z t 0

 _ > dW ~ s: DiagðE_s Þ> þ DiagðE s Þ

The SDE for ðE, E_ Þ> satisfies the Lipschitz and linear growth conditions that ensure the existence of a strong ~ > E_ t j2  5 1 for t 2 [0, T ]. Thus, we can solution and E½j1 d further compute (cf. Gobet and Munos 2005)

  q ~ > 1 q  > 1  ð Þ B E t þ ð > Þ1 BE_ t : E ð Þ BE t ¼ E~ q& q& ð20Þ In the one-dimensional constrained case we consider below the derivative fails to exist at ot ¼ ut and ot ¼ lt . In our examples this event has probability 0, hence "  #   q h ci ~ q  2 B E t þ  2 BE_ t  1fct 2ðlt ,ut Þg : E t ¼ E q& q&

the jump rates i can take large values and are difficult to estimate (cf. Hahn et al. 2007a). Hence, the sensitivities with respect to different parameters cannot be compared directly. Analysis for E0. We choose a state of the normalized filter as starting value for the unnormalized filter, i.e. for d ¼ 2 we have E t0 ¼ (x, 1  x)> for x 2 [0, 1]. The sensitivities equation (20) and equation (21) for  ¼ E t0 are illustrated in figure 8. We see that the influence of the start value E t0 is high at the beginning but decreases quickly. Analysis for r. Depending on the filter, the trading strategy may be positive or negative, i.e. we may be long or short in the risky assets. Since the trading strategy is basically inversely proportional to the square of the volatility it is more reasonable to consider the effect of the volatility on the absolute value of the strategy instead of the strategy itself, i.e. we consider the sensitivity of jct j. In general, the derivative fails to exist at jct j ¼ 0, but this event has probability 0. Hence, jct j is almost surely differentiable with respect to . The sensitivities equation (20) and equation (21) for  ¼  are illustrated in figure 9. The volatility affects the strategy considerably. Analysis for Q. For d ¼ 2 the rate matrix has the entries Q11 ¼ Q12 ¼ 1 and Q21 ¼ Q22 ¼ 2. The sensitivities equation (20) and equation (21) for  ¼ 1 and  ¼ 2 are illustrated in figures 10 and 11, respectively. The results indicate that the impact of the rate matrix on the strategy is not very high.

ð21Þ

Table 1. Parameters for the numerical example. Time horizon, T Dt Initial wealth, X0p Parameters for the drift: State vector, B Rate matrix, Q Initial state for the filter E, E 0 Volatility, 

0.25 (three months) 0.004 (daily values) 1 (0.5, 0.4)>   15 15 15 15 >

(0.5, 0.5) (stationary distribution of Y ) 0.2

Analysis for B. For d ¼ 2 we have B ¼ (b1, b2). The sensitivities equation (20) and equation (21) for  ¼ b1 and  ¼ b2 are illustrated in figures 12 and 13, respectively. The strategy is very sensitive with respect to the states

10 8

Unconstrained L =2%

6 Sensitivity

In what follows, we present a sensitivity analysis for the parameters involved in computing the strategy; the parameters used for the calculations are provided in table 1. For the constrained case we consider the LEL risk constraint. The results should always be considered with respect to the absolute value of the parameters, e.g. for  we obtain rather large values – meaning that t would change by these values if  were to increase by one, however  taking typically small values is unlikely to be misspecified by a value close to one. On the other hand,

L =1% L =0. 5%

4 2 0 −2

10

20

30

40

Time(in days)

Figure 8. Sensitivity w.r.t. E t0.

50

60

1560

W. Putscho¨gl and J. Sass 0

15

Unconstrained −10

L = 2% L = 1%

Sensitivity

Sensitivity

10 −20

5

Unconstrained −30

L = 0. 5%

L = 2% L = 1% L = 0. 5%

−40

10

20

30 40 Time (in days)

50

0

60

10

Figure 9. Sensitivity w.r.t. .

30 40 Time (in days)

50

60

50

60

Figure 12. Sensitivity w.r.t. b1. −2

0

−4

Unconstrained −0.1

Unconstrained L = 2%

L = 2% −6 Sensitivity

L = 1% Sensitivity

20

L = 0. 5% −0.2

L = 1% L = 0 . 5%

−8 −10

−0.3 −12 −0.4

10

20

30

40

50

60

−14

10

20

Figure 13. Sensitivity w.r.t. b2.

Figure 10. Sensitivity w.r.t. 1.

0.4

9. Numerical example

0.3

Unconstrained

9.1. Implementation

L = 2%

To find the bounds lt and ut for one of the constraints equations (9), (10) and (16) we can proceed as follows. For non-constant volatility we approximate R tþDt 1 > expf 12 t diagð Þ dsg by expf diagð s  > t  t ÞDtg and s 2 R tþDt ~ ~ ~t ¼ expf t  s dWs g by expf t DWt g where DW o ~ tþDt  W ~ t . If the constraints are satisfied for  then W t the constraints are not binding and ct ¼ ot (cf. corollary 5.2). Let us consider the case in which the constraints are not satisfied. The constraints are certainly satisfied for t ¼ 0. Hence, we can use a bisection algorithm with start-interval ½ot , 0 for ot 5 0 or ½0, ot  for ot 4 0 to obtain lt and ut , respectively. The strategy is then given by ct ¼ lt or ct ¼ ut , respectively. To compute the expected values in equations (9), (10) and (16) we can use Monte Carlo methods. We only need to do the necessary simulations once, then it is possible to compute the expected value for different values of t very easily and the bisection algorithm works quite efficiently.

L = 1% Sensitivity

40

Time (in days)

Time (in days)

L = 0. 5% 0.2

0.1

0

30

10

20

30 40 Time (in days)

50

60

Figure 11. Sensitivity w.r.t. 2.

of the drift. The constraints help to reduce the influence of the state vector. For a short discussion on the relevance for portfolio optimization, please refer to section 9.2.

1561

Optimal investment under dynamic risk constraints 120

1.6

110

1.4

100

1.2

90

1 X

π

0.8

80 S 70

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

Time

0.8

1

Figure 15. Wealth X.

Figure 14. Stock S.

For the unnormalized filter E we compute robust versions which we denote by E D (cf. Sass and Haussmann 2004). By robust we mean that the differential equations define versions that depend continuously on the observation path (cf. Clark 1978, James et al. 1996). Subsequently we consider an equidistant discretization (ti)i ¼ 0, . . . , N of the interval [0, T ], where ti ¼ iDt and Dt ¼ T/N. Then E D is given by  E Dti ¼ ti ½Idd þ DtQ> E Dti1 :

0.6 Time

σ 0.35

0.3

0.25

Here, we use the notation

ðd Þ ti ¼ Diagð ð1Þ t , . . . , ti Þ,  i   > 1 ðkÞ 2 ðkÞ ðkÞ ~ bti DWti  kbti k Dt , ti ¼ exp 2

0.2 0

0.2

0.4

0.6

0.8

1

0.8

1

Time

Figure 16. Volatility .

1 Ek ~ ti ¼ W ~ ti  W ~ ti1 and bðkÞ where DW ti ¼ ti B .

9.2. Simulated data We illustrate the strategy using simulated data for logarithmic utility. We consider the HMM for the drift and the LEL risk with L ¼ 0.01. The setup of this example is given in table 1 except that we use the HR-model with parameters  ¼ 7,  ¼ 0.2, " ¼ 70, 0 ¼ 0. Figures 14–19 illustrate the strategy. For heavy market movements, i.e. for sharp increases or pronounced drops in stock prices, also the offset function increases or decreases (figure 17). Big absolute values of the offset function result in high volatility (figure 16). Since in volatile markets investing in a risky asset becomes riskier, the bounds on the strategy, which represents the fraction of wealth invested in the risky asset, become narrower as can be seen from comparing figure 16 with the bounds l and u in figure 19. For logarithmic utility the optimal strategy without constraints is essentially proportional to the filter ^ ¼ BY^ plotted in figure 18. Under dynamic risk constraints the optimal strategy for logarithmic utility then corresponds to the strategy under no constraints capped or floored

0.1

0

−0.1

ξ −0.2 0

0.2

(1)

0.4

0.6 Time

Figure 17. Offset (1).

at the bounds resulting from the dynamic risk constraints as can be seen in figure 19. Summarizing, in times of heavy market movements the volatility increases and the bounds on the strategy become

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W. Putscho¨gl and J. Sass Table 2. Numerical results (logarithmic utility). UðX^ T Þ

0.4

0 Drift

−0.4

Filter 0

0.5 Time

1

^ Figure 18. Drift  and Filter BY.

2 1 0 −1 −2

η η η

Median

St.dev.

Bankrupt

0.1195 0.0826 1.5175 0.0824 0.0277

0.2297 0.4745 1.0849 0.5024 0.9228

0 2 79 2 13

0.0988 0.0294 0.0988 0.1242

0.1591 0.1767 0.1595 0.2004

0 0 0 0

LEL risk constraint (L¼1.0%): Merton 0.0927 0.0967 GD 0.0531 0.0474 Bayes 0.0921 0.0962 HMM 0.1513 0.1387

0.2871 0.3274 0.2894 0.3626

0 0 0 0

LES risk constraint (L1¼1.0%, L2¼5%): Merton 0.0497 0.0885 GD 0.2389 0.2431 Bayes 0.0410 0.0870 HMM 0.1352 0.1283

0.4264 0.5643 0.4555 0.5884

0 0 0 0

LES risk constraint (L1¼0.1%, L2¼5%): Merton 0.0956 0.0975 GD 0.0395 0.0350 Bayes 0.0950 0.0968 HMM 0.1505 0.1402

0.2735 0.3086 0.2752 0.3434

0 0 0 0

LES risk constraint (L1¼0.1%, L2¼10%): Merton 0.0546 0.0900 GD 0.1996 0.2079 Bayes 0.0494 0.0884 HMM 0.1419 0.1343

0.4155 0.5222 0.4317 0.5519

0 0 0 0

Unconstrained: b&h 0.1188 Merton 1 (0.0331) GD 1 (1.2600) Bayes 1 (0.0204) HMM 1 (0.0026)

0.2

−0.2

Mean

u l

0

0.5 Time

1

Figure 19. Strategy .

more narrow, thus avoiding extreme positions which might even lead to bankruptcy if we can only trade on a discrete time scale. Therefore the strategies become more robust. During big changes in the markets also the parameters describing the markets might change. And for constant coefficients it is known that the impact of parameter misspecification can be much worse than infrequent trading (cf. Rogers 2001). The effect on the strategy through parameter misspecification is reduced significantly by dynamic risk constraints as is indicated by the sensitivity analysis in the previous section. Overall this makes the strategy more robust or ‘less risky’. The relation to rigorous risk measures was discussed in section 5. 9.3. Historical data For the numerical example we consider logarithmic utility in the one-dimensional case and the LEL risk constraint. We assume constant volatility and constant interest rates. We consider now 20 stocks of the Dow Jones Industrial Index and use daily prices (adjusted for dividends and splits) for 30 years, 1972–2001, each year consisting of approximately 252 trading days where a stock was chosen only if it had such a long history. For the interest rates we consider the corresponding historic fed rates. For each stock we use parameter estimates for the HMM based on a Markov chain Monte Carlo (MCMC) method

LEL risk constraint (L¼0.5%): Merton 0.1003 GD 0.0252 Bayes 0.1002 HMM 0.1285

(cf. Hahn et al. 2007a). Also the parameters for the GD are obtained from a multiple-block MCMC sampler based on time discretization similar to the sampler described in Hahn et al. (2007a). The parameter estimates are based on five years with starting years 1972, 1973, . . . , 1996. For the interest rate we use the average fed rate of the fifth year. Based on these estimates we compute in the subsequent year the optimal strategy. Hence, we perform 500 experiments whose outcomes we average. As a starting value for the filter when computing the strategy we use the stationary distribution in the first year and then the last value of the filter that was obtained by the optimization in the preceding year. We start with initial capital X0 ¼ 1. We compare the strategy based on HMM, GD and Bayes with the Merton strategy, i.e. the strategy resulting from the assumption of a constant drift, and with the buy and hold (b&h) strategy. We compute the average of the utility which is 1 if we go bankrupt at least once. In this case we also list the average utility where we only average over those cases in which we don’t go bankrupt. The results are presented in table 2, where we used the LEL risk constraint and computed the mean, median, and the standard deviation. We face two problems when applying results of continuous-time optimization to market data: model and discretization errors. The 13 bankruptcies for the HMM mainly fall on

Optimal investment under dynamic risk constraints Black Monday 1987, where single stocks had losses up to 30%. Owing to the non-constant drift, very big short and long positions may occur for the HMM (cf. Sass and Haussmann 2004). For GD the positions are even more extreme, since the drift process is unbounded. For daily trading without risk constraints and risky utility functions this may result in bankruptcies; and also the Merton and Bayes strategies go bankrupt twice. The latter is due to the fact that the parameters for the Merton strategy are estimated over a relatively short period and can result in some cases also in quite long positions which led to bankruptcy in 1987. The HMM is less prone to parameter misspecification, since changes in the drift parameter are included in the model, and with more states it could also explain big price movements. But it is still a model with continuous asset prices and exhibits extreme positions which make the strategy less robust when trading at discrete times only. The results under dynamic risk constraints suggest that they are a suitable remedy for the discretization error. We can improve the performance for all models compared to the case of no constraints and the strategies don’t go bankrupt anymore. In particular the HMM strategy with risk constraints clearly outperforms all other strategies.

10. Conclusion In this paper we show how dynamic risk constraints on the strategy can be applied to make portfolio optimization more robust. We derive explicit trading strategies with risk constraints under partial information. Further, we analyse the dependence of the resulting strategies on the parameters involved in the HMM. The results under dynamic risk constraints indicate that they are a suitable remedy for reducing the risk and improving the performance of trading strategies.

Acknowledgements The authors thank the Austrian Science Fund FWF, Project P17947-N12, the German Research Foundation DFG, Heisenberg Programme, for financial support, and two anonymous referees for their stimulating comments and suggestions. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of their employers.

References Basak, S. and Shapiro, A., Value-at-risk-based risk management: optimal policies and asset prices. Rev. Finan. Stud., 2001, 14, 371–405. Brendle, S., Portfolio selection under incomplete information. Stochast. Process. Appls., 2006, 116, 701–723. Clark, J., The design of robust approximations to the stochastic differential equations of nonlinear filtering. In Proceedings of the 2nd NATO Advanced Study Institute Conference on

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Ocone, D., Malliavin’s calculus and stochastic integral representations of functionals of diffusion processes. Stochastics, 1984, 12, 161–185. Ocone, D. and Karatzas, I., A generalized Clark representation formula, with application to optimal portfolios. Stochast. Stochast. Rep., 1991, 34, 187–220. Pham, H. and Quenez, M.-C., Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab., 2001, 11, 210–238. Pirvu, T., Portfolio optimization under the value-at-risk constraint. Quant. Finan., 2007, 7, 125–136. Putscho¨gl, W. and Sass, J., Optimal consumption and investment under partial information. Decis. Econ. Finan., 2008, 31, 137–170. Rieder, U. and Ba¨uerle, N., Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab., 2005, 42, 362–378. Rockafellar, R.T., Convex Analysis, 1970 (Princeton University Press: Princeton, NJ). Rockafellar, R. and Uryasev, S., Optimization of conditional value-at-risk. J. Risk, 2000, 2, 21–41.

Rockafellar, R. and Uryasev, S., Conditional value-at-risk for general loss distributions. J. Banking Finan., 2002, 26, 1443–1471. Rogers, L.C.G., The relaxed investor and parameter uncertainty. Finan. Stochast., 2001, 5, 131–154. Runggaldier, W. and Zaccaria, A., A stochastic control approach to risk management under restricted information. Math. Finan., 2000, 10, 277–288. Ryde´n, T., Teras´ virta, T. and Asbrink, S., Stylized facts of daily return series and the hidden Markov Model. J. Appl. Econom., 1998, 13, 217–244. Sass, J., Utility maximization with convex constraints and partial information. Acta Applic. Math., 2007, 97, 221–238. Sass, J. and Haussmann, U., Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Finan. Stoch., 2004, 8, 553–577. Sekine, J., Risk-sensitive portfolio optimization under partial information with non-Gaussian initial prior, 2006 (preprint). Yiu, K., Optimal portfolios under a value-at-risk constraint. J. Econ. Dynam. Contr., 2004, 28, 1317–1334.

OPTIMAL RISK CLASSIFICATION WITH AN APPLICATION TO SUBSTANDARD ANNUITIES Nadine Gatzert,* Gudrun Schmitt-Hoermann,† and Hato Schmeiser‡

ABSTRACT Substandard annuities pay higher pensions to individuals with impaired health and thus require special underwriting of applicants. Although such risk classification can substantially increase a company’s profitability, these products are uncommon except for the well-established U.K. market. In this paper we comprehensively analyze this issue and make several contributions to the literature. First, we describe enhanced, impaired life, and care annuities, and then we discuss the underwriting process and underwriting risk related thereto. Second, we propose a theoretical model to determine the optimal profit-maximizing risk classification system for substandard annuities. Based on the model framework and for given price-demand dependencies, we formally show the effect of classification costs and costs of underwriting risk on profitability for insurers. Risk classes are distinguished by the average mortality of contained insureds, whereby mortality heterogeneity is included by means of a frailty model. Third, we discuss key aspects regarding a practical implementation of our model as well as possible market entry barriers for substandard annuity providers.

1. INTRODUCTION Substandard annuities pay higher pensions to individuals with impaired health.1 These contracts are increasingly prominent in the U.K. insurance market, where, according to Towers Watson (2012), about 15% of annuities sold are based on enhanced rates. Since its development in the 1990s, the market for substandard annuities in the United Kingdom has experienced impressive growth.2 Starting from £420 million in 2001, sales of substandard annuities in the U.K. market had exceeded £3,019 million in 2011.3 Enormous growth potential remains, with up to 40% of annuitants estimated to be eligible for increased pension payments. Outside the United Kingdom, however, substandard annuities are surprisingly rare. In the United States, for instance, only 11 providers out of 100 insurers issuing singlepremium immediate annuities offer substandard annuity products.4 Only about 4% of annuities sold in the U.S. market are based on enhanced rates.5 It is not obvious why the substandard annuity market is so small, especially given that such a risk classification generally increases a company’s profitability.6 Furthermore, substandard annuities may make private pensions available for a broader range of the

* Nadine Gatzert is the Chair for Insurance Economics at the Friedrich-Alexander-University of Erlangen-Nu¨rnberg, Lange Gasse 20, 90403 Nu¨rnberg, Germany, [email protected]. † Gudrun Schmitt-Hoermann works at a reinsurance company in Munich. ‡ Hato Schmeiser is Chair for Risk Management and Insurance, at the University of St. Gallen, St. Gallen, Switzerland, [email protected]. 1 See LIMRA and Ernst & Young (2006, p. 10). The term ‘‘substandard annuity’’ includes enhanced, impaired life, and care annuities (for a detailed description, see Section 2). In the U.K. market, all types of substandard annuities are sometimes referred to as ‘‘enhanced annuities.’’ 2 This may to some extent be due to mandatory partial annuitization of retirement income in the United Kingdom. See LIMRA and Ernst & Young (2006, p. 6). 3 See Towers Watson (2012). 4 See LIMRA and Ernst & Young (2006, p. 7). 5 See LIMRA and Ernst & Young (2006, p. 18). 6 See Doherty (1981).

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population and could thus improve retirement incomes for insureds with a reduced life expectancy.7 Thus, there must be important reasons behind the reluctance of many insurers to enter the substandard annuity market. The aim of this paper is to develop a model to determine the optimal risk classification system8 for substandard annuities that will maximize an insurance company’s profits. We further include the costs of insufficient risk assessment (underwriting risk) that occurs when insureds are assigned to inappropriate risk classes. This extension is crucial, because underwriting risk is considered to be the most significant risk factor in the issuance of substandard annuities and thus should be taken into account when making informed decisions. In addition, we provide qualitative background information about underwriting and classification methods and describe underwriting risks for different types of substandard annuity products. We also discuss key aspects regarding a practical implementation of our model as well as possible market entry barriers. Because the risk classification model is formulated in a rather general way, it can as well be applied to other classification problems. Selling substandard annuities is a challenging task, and several factors in the process will influence a provider’s profitability. First, a reasonable classification system must be established based on insureds’ life expectancy. Second, adequate underwriting guidelines are necessary to ensure that each applicant is assigned to the proper risk class. Distinctive features of risk classes include medical conditions or lifestyle factors, such as smoking, weight, geographical location, education, or occupation. Resulting classification costs need to be taken into consideration when pricing the contract. Finally, demand for the product is determined by the annuity amount paid to insureds in each risk class. The literature on substandard annuities primarily deals with practical issues of substandard annuity markets. Ainslie (2000), Weinert (2006), and LIMRA and Ernst & Young (2006) provide detailed studies of substandard annuities in the United Kingdom and the United States. Information on the development, size, and potential of substandard annuity markets, different product types, underwriting methods and challenges, mortality and risk classification issues, the impact on the standard annuity market, tax considerations, distribution channels, and reinsurance can be found in Ainslie (2001), Brown and Scahill (2007), Cooperstein et al. (2004), Froehling (2007), Hamdan and Rinke (1998), Richards and Jones (2004), and Rinke (2002). Junus et al. (2004), Nicholas and Cox (2003), and Turner (2001) focus on the underwriting of substandard annuities. The impact of individual underwriting on an insurance company’s profit is examined in Hoermann and Russ (2008), based on actuarial pricing. In Ranasinghe (2007) underwriting and longevity risk for impaired lives are assessed by means of a provision for adverse deviation. Regarding risk classification within the insurance sector, Williams (1957) provides an overview of insurance rate discrimination, including its definition, various forms, economic effects, and government regulation. Doherty (1981) examines the profitability of rate classification for an innovating insurer and the associated market dynamics. This paper is an extension of previous work (Doherty 1980), in which the author investigates rate discrimination in the fire insurance market. Christiansen (1983) draws a parallel to substandard annuities when analyzing the ‘‘fairness’’ of rate discrimination. Zaks et al. (2008) show the existence of an equilibrium point, when, in a portfolio consisting of several risk classes with respective price-demand functions, the premium amount and the number of policyholders in each risk class are iteratively updated. A great deal of the literature is dedicated to risk classification controversies concerning social issues. Some authors argue that competition by risk classification is inefficient, particularly if it becomes purely selective, that is, if it makes insurance expensive or unaffordable for persons representing high risks for insurers. In contrast, others regard risk classification as essential in avoiding adverse selection. De Jong and Ferris (2006) provide the background for this

7 The introduction of substandard annuities may also have an impact on the costs for standard annuities because of selection effects, and the overall benefit may therefore also depend on relative price elasticity of demand across different groups of annuitants; see, e.g., Thomas (2008). 8 According to Actuarial Standard of Practice No. 12, a risk classification system is a ‘‘system used to assign risks to groups based upon the expected cost or benefit of the coverage or services provided’’ (Actuarial Standards Board 2005).

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discussion. Other authors addressing this topic include Abraham (1985), De Wit (1986), Feldman and Dowd (2000), Rothschild and Stiglitz (1997), Thiery and Van Schoubroeck (2006), Thomas (2007), and Van de Ven et al. (2000). In addition, De Jong and Ferris (2006) provide a demand model to investigate the effects of changes in risk classification systems. The authors determine the impact of unisex pricing in the U.K. annuity market on the expected purchase of annuity amounts depending on a person’s individual mortality level, which is described by a frailty factor. Various authors assess risk classification in insurance from a social utility point of view (see, e.g., Bond and Crocker 1991; Crocker and Snow 1986, 2000; Hoy 1989, 2006; Sheshinski 2007; Strohmenger and Wambach 2000; Van der Noll 2006; Villeneuve 2000). Promislow (1987) measures the inequity that arises from considering only certain factors—and ignoring others—when setting insurance rates. There are several papers that deal with practical issues of risk classification. Kwon and Jones (2006) develop a mortality model that reflects the impact of various risk factors. Derrig and Ostazewski (1995), Horgby (1998), and Lohse (2004) focus on risk classification based on fuzzy techniques. The work of Werth (1995) provides a broad overview of preferred lives products. Leigh (1990) reviews the underwriting of life and sickness benefits, and Walters (1981) develops standards for risk classification. Today the impact of genetics on risk classification has taken on added importance (see, e.g., Brockett et al. 1999; Brockett and Tankersley 1997; Hoy and Lambert 2000; Hoy and Ruse 2005; Macdonald 1997, 1999; O’Neill 1997). A substantial body of literature is also found on rate classification in non-life insurance, especially in the automobile sector. For instance, Schwarze and Wein (2005) consider the third-party motor insurance industry and empirically test whether risk classification creates information rents for innovative insurers. Cummins et al. (1983, pp. 27–62) and Driver et al. (2008) focus on the economic benefits of risk classification. The authors argue that in many cases risk classification contributes to economic efficiency, limits adverse selection, reduces moral hazard, and encourages innovation and competition within the insurance market. In this paper we contribute to the literature by providing a comprehensive analysis of challenges and chances for life insurers offering substandard annuity products. Toward this end, we combine the two strands of literature, that on substandard annuities and that on risk classification. From an insurer’s viewpoint, we solve the problem of optimal risk classification for substandard annuities taking into consideration classification costs and underwriting risk, which has not been done to date. In addition, we provide a detailed discussion on the background of substandard annuities and limitations regarding risk classification within annuity products. In this paper Section 2 provides practical background information about different types of substandard annuities and also describes underwriting and classification issues. In Section 3 we develop a model to determine the optimal number and size of risk classes, as well as the optimal price-demand combination for each risk class that will maximize an insurer’s profit. Risk classes are distinguished by the average mortality of individuals in a certain class relative to the average population mortality. We account for mortality heterogeneity and use a frailty model to derive individual probabilities of death. The profit is maximized based on given price-demand dependencies in population subgroups and classification costs. When taking into account costs of underwriting risk, a modified risk classification system might be optimal, depending on the underwriting quality. In Section 4 we provide a detailed description of market entry barriers and risks for substandard annuity providers and aspects concerning the practical application of our model. The paper concludes with a summary in Section 5.

2. SUBSTANDARD ANNUITIES

AND

UNDERWRITING

In general, there are three types of substandard annuities, as opposed to the standard annuity: enhanced annuities, impaired (life) annuities, and care annuities.9 Usually all three are immediate annuities for a single lump-sum payment, where the annual annuity amount depends on the insured’s

9

See, e.g., Ainslie (2001, p. 16), Brown and Scahill (2007, pp. 5–6), and Cooperstein et al. (2004, pp. 14–15).

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health status. In the following, we start with a definition of each annuity type and then go on to provide information about key aspects of annuity underwriting. Market size and underwriting risks are discussed in the last part of the section. Enhanced annuities pay increased pensions to persons with a slightly reduced life expectancy.10 Most applicants are between 60 and 70 years of age.11 Calculation of enhanced annuities is based on environmental factors, such as postal code or geographic location, and lifestyle factors, such as smoking habits, marital status, or occupation, as well as disease factors, including diabetics, high blood pressure, high cholesterol, or being overweight.12 When impairments are considered, this type of annuity is sometimes referred to as an impaired (life) annuity.13 Impaired life annuities are typically related to health impairments such as heart attack, cancer, stroke, multiple sclerosis, lung disease, or kidney failure for annuitants in an age range of 60 to 85.14 Care annuities are aimed at seriously impaired individuals between age 75 and 90 or persons who already have started to incur long-term-care costs.15 Risk assessment for care annuities is based on geriatric symptoms such as frailty or restricted mobility, which are measured in terms of activities of daily living (ADL) and instrumental activities of daily living (IADL); cognitive skills may also be taken into account.16 For standard annuities, the annuity amount is calculated based on the average mortality of one class comprising all insureds. Payments depend on an annuitant’s age and gender. Based on so-called singleclass underwriting, the insurer decides whether to accept or reject an applicant.17 Substandard annuities require adjustment of the underlying pricing assumptions based on an individual’s impairment level, necessitating the provision of medical information. It is the applicant’s responsibility to provide sufficient evidence that he or she is eligible for increased annuity payments. Upon receipt of this evidence, the reduction in life expectancy is quantified by the insurer’s underwriting.18 Life expectancy can be measured either in terms of average life expectancy (ALE) or in terms of the maximum realistic life expectancy (MRLE), corresponding to the 50% or 90% quantile of the remaining lifetime, respectively.19 The modified annuity amount is determined either by an age rate-up or by a rating factor.20 The former involves an adjustment of the insured’s actual age for calculation purposes. For instance, a 60-year-old impaired male may be rated to have the life expectancy of a 65-year-old and would thus receive the annuity amount based on being age 65. The rating factor is applied to the standard mortality table.21 For example, an extra mortality of 100% would mean multiplying average mortality probabilities by a factor of ‘‘2.’’ Different types of underwriting techniques are employed depending on the applicant’s health status and the type of annuity requested. In the underwriting process for enhanced annuities, applicants are assigned to different risk classes depending on their health status or individual mortality. This multiclass underwriting22 is the most common method used in pricing substandard annuity products. Impairments or lifestyle factors are assessed by a health questionnaire.23 Certain rules are applied (the rules-based approach) to determine the rating factor, the reduction in ALE, or the age rate-up.24 This 10

See Richards and Jones (2004, p. 20) and Weinert (2006, p. 6). See Ainslie (2001, p. 16) and Cooperstein et al. (2004, p. 14). 12 See, e.g., Ainslie (2001, p. 17) and Brown and Scahill (2007, p. 5). 13 See Brown and Scahill (2007, pp. 5–6). Definitions of enhanced and impaired life annuities sometimes overlap. See, e.g., Nicholas and Cox (2003, p. 5). 14 See, e.g., Ainslie (2001, pp. 16–17). 15 See Ainslie (2001, pp. 15–16) and Cooperstein et al. (2004, p. 15). 16 See Brown and Scahill (2007, p. 6) and Junus et al. (2004, p. 7). 17 See Rinke (2002, p. 5). 18 See LIMRA and Ernst & Young (2006, p. 32). 19 See LIMRA and Ernst & Young (2006, pp. 32–33) and Nicholas and Cox (2003, pp. 5–6). 20 See Junus et al. (2004, p. 4) and LIMRA and Ernst & Young (2006, pp. 32–34). 21 See Cooperstein et al. (2004, p. 16), LIMRA and Ernst & Young (2006, pp. 33–34), and Richards and Jones (2004, p. 22). 22 See Rinke (2002, pp. 5–6). 23 See Brown and Scahill (2007, p. 5). 24 See Ainslie (2001, pp. 16–17), Cooperstein et al. (2004, p. 14), and Nicholas and Cox (2003, p. 5). 11

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underwriting approach—in contrast to full individual underwriting25 —is appropriate only for mild impairments and lifestyle factors that correspond to extra mortalities between 25% and 50% and thus result in only slight annuity enhancements of about 10% to 15%.26 Impaired life annuities require a more extensive assessment of an applicant’s health status because of the larger potential increase in the annuity amount, e.g., up to 50% for extra mortalities up to 150% depending on the issue age.27 In addition to the health questionnaire, a doctor’s report may be considered, implying a mixture between rules-based and individual underwriting.28 Based on this information, the applicant is assigned a risk class (multiclass underwriting). Sometimes, full individual underwriting is required for impaired life annuities.29 However, there is a tradeoff between the additional costs of such and the increased accuracy thus derived.30 According to Nicholas and Cox (2003), the impaired life expectancy is measured in terms of both ALE and MRLE.31 Care annuities are individually underwritten based on a doctor’s report,32 meaning that individual life expectancy is calculated for each applicant and no risk classes are established.33 To obtain a more precise specification, the MRLE is used.34 Extra mortalities between 250% and 300% yield annuity enhancements of up to 125%.35 The market for enhanced annuity products is large, whereas the market for impaired life annuities is of moderate size. Care annuities have a small niche market.36 Some providers focus solely on restricted market segments, whereas others cover the full range of standard and substandard annuity products.37 Sometimes companies merely add one substandard (enhanced or impaired life) product to their standard annuity portfolio.38 It is often claimed that accurate underwriting is the crucial factor in offering substandard annuities. In particular, there is substantial risk that the underwriting will not correctly assess an applicant’s mortality level.39 LIMRA and Ernst & Young (2006) list several causes of underwriting risk, such as the pressure of competition, the lack of adequate underwriting procedures and experience, and insufficient mortality data. The last factor is also discussed in Lu et al. (2008), who emphasize the risk of making ratings based on small-sample medical studies. The lack of mortality data—especially for higher age groups—may be partly responsible for the slow development of the substandard annuity market. This problem could be solved by outsourcing underwriting to reinsurers, who have more data.40 There is also the danger for the point of view of an annuity provider that the life expectancies of impaired persons can improve dramatically because of developments in the medical field. Therefore, it is vital that underwriters carefully monitor the mortality experience in their book of business as well as developments in medical research.41 The former is also important with respect to adverse selection, especially when insurers offer both standard and substandard annuities.42 Another risk factor has to do 25

See Cooperstein et al. (2004, p. 14), Richards and Jones (2004, p. 20), and Rinke (2002, p. 6). See Ainslie (2001, p. 16), Cooperstein et al. (2004, p. 14), and Weinert (2006, p. 8). 27 See Ainslie (2001, p. 16) and Weinert (2006, p. 8). 28 See Ainslie (2001, pp. 16–17) and Brown and Scahill (2007, pp. 5–6). 29 See Richards and Jones (2004, p. 20). 30 See Rinke (2002, pp. 5–6). 31 See Nicholas and Cox (2003, p. 6). 32 See Brown and Scahill (2007, p. 6) and Richards and Jones (2004, p. 20). 33 See Rinke (2002, pp. 5–6). 34 See Nicholas and Cox (2003, p. 6). 35 See Ainslie (2001, p. 16). 36 See Ainslie (2001, p. 16) and Cooperstein et al. (2004, p. 15). 37 See Froehling (2007, p. 5). 38 See Weinert (2006, p. 12). 39 See, e.g., LIMRA and Ernst & Young (2006, p. 31) and Richards and Jones (2004, p. 20). 40 See Cooperstein et al. (2004, p. 13). 41 See Cooperstein et al. (2004, p. 16), Junus et al. (2004, p. 5), Nicholas and Cox (2003, p. 8), Richards and Jones (2003, p. 20), Sittaro (2003, p. 9), and Weinert (2006, p. 17). 42 See Cooperstein et al. (2004, p. 16). 26

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with using lifestyle characteristics as a basis for underwriting; this practice can increase the risk of adverse selection if an insured improves his or her life expectancy by changing behavior, for example, by quitting smoking or losing weight.43 In this paper we focus on the large enhanced and impaired life annuity market, where insureds are categorized in risk classes with differing mortality by means of rating factors. We determine the optimal risk structure for an insurer offering substandard annuities and explicitly model and integrate costs related to underwriting risk, which is of great concern to insurers.

3. THE MODEL FRAMEWORK 3.1 Basic Model We consider a general population consisting of N 僆 ⺞ potential risks, that is, potential policyholders of a given gender and at a specific age x 僆 {0, . . . , ␻}, where ␻ is the limiting age of a population mortality table describing the average mortality in the general population.44 The entry q⬘x thus specifies the average one-year probability of death for a person age x out of the general population, where the prime indicates population mortality.45 Mortality heterogeneity in the general population is considered by means of a frailty model.46 To obtain individual probabilities of death, we apply a stochastic frailty factor to the average mortality probability. The one-year individual probability of death qx(d) for an x-year-old is thus given as the product of the individual frailty factor d 僆 ⺢⫹ 0 and the probability of death q⬘ x from the population mortality table: qx(d) ⫽

冦

d  q⬘, d  q⬘x ⬍ 1 x 1, x ⫽ min[x ˜ 僆 {0, . . . , ␻} ⬊ d  q⬘˜x ⱖ 1] for x 僆 {0, . . . , ␻}. 0, otherwise

(1)

If the resulting product is greater than or equal to 1 for any ages ˜ x, the individual probability of death is set equal to 1 for the youngest of those ages; for all other ages ˜ x, it is set to 0. For d ⬍ 1, we let q␻(d) ⬊⫽ 1. The frailty factor specifies an individual’s state of health. A person with a frailty factor less than 1 has an above-average life expectancy, a frailty factor greater than 1 indicates that the individual is impaired with a reduced life expectancy, and a frailty factor equal to 1 means the person has average mortality. The individual frailty factor d is a realization of a random variable D.47 The distribution FD of D represents the distribution of different states of health and thus of different life expectancies in the general population. For its characteristics, we follow the assumptions in Hoermann and Russ (2008): We let FD be a continuous, right-skewed distribution on ⺢⫹ 0 with an expected value of 1, such that the mortality table describes an individual with average health. As probabilities of death approaching zero are not realistic, the probability density function fD is flat at zero with fD(0) ⫽ 0. We assume that the insurer is able to distinguish a maximum of H different subpopulations that aggregate to the total general population (see Fig. 1). Subpopulations differ by health status of contained risks and are ordered by their mortality, where h ⫽ 1 is the subpopulation with the lowest mortality and h ⫽ H is the subpopulation with the highest mortality. Risks belong to a given subpopulation if their individual mortality lies in a corresponding frailty factor range. Subpopulation h com-

43

See LIMRA and Ernst & Young (2004, pp. 42–43). A detailed overview of different mortality regimes and its impact on the pricing of annuities can be found in Milidonis et al. (2011). 45 To be directly applicable to insurance data, we use a discrete model. By substituting annual mortality probabilities by the continuous force of mortality, a continuous model could also be employed. General results remain unchanged, however. 46 See Hoermann and Russ (2008). 47 See, e.g., Jones (1998, pp. 80–83), Pitacco (2004, p. 15), and Vaupel et al. (1979, p. 440). 44

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Figure 1 Segmentation of the General Population into H Subpopulations Depending on Mortality Level (Described by the Frailty Distribution) Frailty Distribution

Percentage of General Population

Subpopulation

1

H

Subpopulation h

➝

Low Mortality (Frailty Factor)

➝

High Mortality (Frailty Factor)

prises all persons with a frailty factor lying in the interval [dLh , dhU ), where dhL defines its lower and dhU defines its upper limit for h ⫽ 1, . . . , H and dUh ⫽ dLh⫹1, h ⫽ 1, . . . , H ⫺ 1. All intervals combined— corresponding to the H subpopulations—aggregate to the positive real axis, that is, 傼Hh⫽1[dLh , dUh ) ⫽ [0, ⬁). Thus, the whole range of positive real-valued frailty factors is covered. The number of risks in subpopulation h is denoted by Nh. Risks in all subpopulations sum up to the total number of risks N of the general population; that is, 兺Hh⫽1 Nh ⫽ N. Nh depends on the frailty distribution. It can be derived as the percentage of risks out of the general population with a frailty factor between dLh and dUh . Thus, it is calculated as the product of the total number of risks N and the probability of the frailty factor lying in the interval [dLh , dhU ). The latter can be expressed in terms of the frailty distribution FD, leading to Nh ⫽ N  P(dLh ⱕ D ⬍ dhU ) ⫽ N  (FD(dUh ) ⫺ FD(dLh )). Each subpopulation h is further characterized by two functions. First, its cost function gh(n) describes costs of the insurance of survival risk for individuals in subpopulation h. Second, its price-demand function fh(n) specifies how many risks n would acquire one unit of annuity insurance for a given price Ph ⫽ fh(n). Both functions are defined for the number of insureds n ⫽ 1, . . . , Nh in each subpopulation h ⫽ 1, . . . , H. Because the insurer cannot distinguish beyond given subpopulations, they are treated as homogeneous with respect to the mortality. The cost function is independent of the number of sales, that is, gh(n) ⫽ P Ah , n ⫽ 1, . . . , Nh, where P hA describes the actuarial premium for covering the cost of one unit of annuity insurance for the average potential insured in subpopulation h. The actuarial premium is based on the average mortality in a subpopulation, which can be derived from the frailty distribution. The average frailty factor d¯ h for subpopulation h is given as the truncated expected value of frailty factors in the corresponding interval [dLh , dhU ): d¯ h ⫽ E(D  1{D 僆 [dLh , dUh )}) ⫽

冕

dU h

dLh

z  fD(z)dz,

where 1{.} represents the indicator function. Therefore, the average k-year survival probability for a person age x in subpopulation h is given by

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Figure 2 Constant Cost Function gh and Linear Price-Demand Function fh (in Terms of the Price P and as a Function of the Demand n) in Subpopulation h

( )

Cost Function gh n Price-Demand Function

P

PhR

fh (n )

PhA

0

Nh

0

n

Notes: P Rh ⫽ reservation price, P hA ⫽ actuarial premium for (one unit of) annuity insurance, Nh ⫽ number of risks in subpopulation h.

写 (1 ⫺ q

t⫺1 h t px ⫽

(d¯ h ))

x⫹l

l⫽0

with qx(d¯ h ) as in equation (1). The actuarial premium P Ah for one unit of annuity insurance equals the present value of future annuity payments and thus results in

冘

␻⫺x

P Ah ⫽

t

p xh vt,

t⫽0

where v denotes the discount factor. Because the average frailty factor is increasing for ascending risk classes, that is, d¯ 1 ⬍ d¯ 2 ⬍    ⬍ d¯ H, the actuarial premium is decreasing, that is, P A1 ⬎ P A2 ⬎    ⬎ P AH. The price-demand function fh(n) is monotonously decreasing in the number of risks n; that is, the lower the price, the more people there are willing to buy insurance. Its first derivative f⬘(n) with respect h to the demand n is hence negative for all n. In addition, we assume that the reservation price (the unique choke price) PRh , that is, the price for which the demand is zero (PRh ⫽ fh(0)), increases with decreasing mortality probabilities in a subpopulation. Consequently, a subpopulation with high life expectancy contains individuals who would be expected to pay more for one unit annuity insurance compared to individuals in a subpopulation with low life expectancy. As the actuarial fair value of one unit annuity insurance is higher for healthy persons, it makes sense that their willingness to pay will also be higher. In addition, Turra and Mitchell (2004) found that annuities are less attractive to poorer risks with uncertain out-of-pocket medical expenses. In terms of reservation prices, this means that PR1 ⬎ P2R ⬎    ⬎ PHR . At a price of zero, in contrast, everyone in the general population would purchase insurance, that is, fh(Nh ) ⫽ 0, h ⫽ 1, . . . , H. An illustration of the determinants for one subpopulation h is provided in Figure 2; for illustration purposes, a linear price-demand function is displayed.48

48

See, e.g., Baumol (1977, pp. 401–402).

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3.2 Optimal Risk Classification We now consider profit-maximizing insurers who intend to introduce rate-discriminating annuity products. A risk class is comprised of insureds subject to a specific range of risks in regard to their remaining life expectancy. Thus, annuity prices will vary by risk class. For instance, an insured with reduced life expectancy will obtain a higher annuity for a given price or, vice versa, pay a lower price for one unit of annuity insurance, as described in the previous section. A combination of population subgroups to risk classes is called a classification system. Offering a standard annuity product corresponds to addressing the total general population. This is consistent with a classification system that aggregates all existing population subgroups to one single risk class. In our setting, subgroups are sorted by decreasing average life expectancies, and, hence, only adjacent subgroups can be merged with each other into a risk class. Let M be the set of all possible classification systems. A classification system m 僆 M consists of Im risk classes. However, conducting risk classification is associated with classification costs. This includes the costs for distinguishing between risk classes as described in the previous section, for example, costs for establishing underwriting guidelines for each additional risk class (beyond the total general population). We assume these costs to be proportional to the number of distinctions and thus set them to 49 When offering only one standard class, that is, Im ⫽ 1, no classification k(Im ⫺ 1), where k 僆 ⺢⫹ 0. costs are incurred. In each classification system m with Im risk classes, a risk class i, i ⫽ 1, . . . , Im is composed of Si m subpopulations, where 兺Ii⫽1 Si ⫽ H. To simplify notation, in the following, we omit the index m when focusing on a specific risk class i (within a classification system m). Hence, all H subpopulations of the general population are assigned to Im risk classes. Subpopulations contained in one risk class are ordered by increasing average mortality; this is indicated by the index s. The total number of individuals in risk class i is given as the sum of the number of persons Ns in each contained subpopulation s:

冘 N. Si

Ni ⫽

s

s⫽1

As for each subpopulation, a risk class i is characterized by its price-demand function fi(n) and its cost function gi(n) for n ⫽ 1, . . . , Ni. If risk class i contains exactly one subpopulation h (and thus Si ⫽ 1), then fi ⫽ fh and gi ⫽ gh. Otherwise, fi and gi are aggregated functions of the price-demand and cost functions of the Si subpopulations. The aggregation process is complex and must be conducted stepwise by means of inverse functions. f ⫺1 denotes the inverse function of fh, which is defined on the h interval [0, PRh ]. The price-demand function fi of risk class i is aggregated based on the price-demand functions fs (s ⫽ 1, . . . , Si ) of the Si underlying subpopulations. The aggregated function fi will exhibit breaks when it becomes equal to the reservation price PRs of one of the contained subpopulations s, because each reservation price represents the point at which the next subpopulation will start buying the policy. In each risk class i, we have Si reservation prices. Hence, there are Si intervals on the x axis, on each of which the aggregate price-demand function is defined differently. The intervals Iv are given by

Iv ⫽

49

冋冘 冋冘

冦

v

s⫽1

s⫽1

v

s⫽1

冘f

v⫹1

f s⫺1(PvR ), R v

册

f (P ), Ni ⫺1 s

⫺1 s

冊

R (Pv⫹1 )

for v ⫽ 1, . . . , Si ⫺ 1 (2) for v ⫽ Si.

For a more general specification, one might define classification costs by any monotonously increasing function in the number of risk classes Im. Thus, disproportionately high classification costs can be represented, for instance.

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Figure 3 Aggregate Cost and Price-Demand Function (in Terms of the Price P and as a Function of the Demand n) in Risk Class i Consisting of Two Subpopulations s ⴝ 1, 2 Cost and price-demand function in subpopulation s = 1 P R 1

P

Aggregate functions fi and gi in risk class i Cost Function gi ( n ) P

Cost and price-demand function in subpopulation s = 2

Cost Function g1 ( n ) Price-Demand Function f1 ( n )

Cost Function g 2 (n )

P

Price-Demand Function

f2 (n)

Price-Demand Function

P1R

fi ( n )

P2R

P2R

P1A

P1 A P2A

0 0

n

N1

0 0

n

N2

0

N1 + N2

0

n

Notes: P Rs , s ⫽ 1, 2 ⫽ reservation price, P As , s ⫽ 1, 2 ⫽ actuarial premium for (one unit of) annuity insurance, Ns , s ⫽ 1, 2 ⫽ number of risks in subpopulation s.

On each interval Iv, the aggregated price-demand function is defined by

冉冘 v

fi(n) ⫽ fi( f ⫺1 i (Pi )) ⫽ fi

冊

f s⫺1(Pi ) , if n 僆 Iv, v ⫽ 1, . . . , Si.

s⫽1

Thus, for any given price Pi ⬊⫽ fi(n), the number of insured risks n in a risk class is the sum of the number of insured risks in each contained subpopulation s for which f ⫺1 s (Pi ) is defined. Graphically, the aggregate price-demand function of a risk class is received by horizontal addition of the pricedemand functions of belonging subpopulations. It starts at the highest reservation price PRs⫽1, and whenever the function passes another reservation price, there is a bend as the demand of another subpopulation is added. Figure 3 sketches the aggregation process for risk class i consisting of two subpopulations, Si ⫽ 2. In this setting, the cost function in a subpopulation is constant in the number of insureds. This means that no inverse function of the cost function exists. Therefore, the aggregate cost function gi of risk class i must be derived based on the associated price-demand function fi. It is defined piecewise in sections analogously to fi (see eq. (2)). For a given demand n in risk class i, the price is given by fi(n) ⫽⬊ Pi. For this price, the corresponding number of persons in each contained subpopulation s is ⫺1 determined by the inverse function f ⫺1 s ( fi(n)) ⫽ f s (Pi ) ⫽⬊ ns, if existent. The number of persons in each subpopulation is then weighted with the corresponding costs gs(Pi ) ⫽⬊ P As in that subpopulation. v ⫺1 v Finally, average costs are determined for the total of n ⫽ 兺vs⫽1 f ⫺1 s ( fi(n)) ⫽ 兺s⫽1 f s (Pi ) ⫽ 兺s⫽1 ns insureds. The resulting formula for the aggregate cost function on the interval Iv is thus descriptive and given by 1 gi(n) ⫽ n

冘 v

s⫽1

1 f ( fi(n))  gs( fi(n)) ⫽ n ⫺1 s

冘 v

s⫽1

1 f (Pi )  gs(Pi ) ⫽ n ⫺1 s

冘n P v

s

A s

s⫽1

if n 僆 Iv, v ⫽ 1, . . . , Si. Because the cost function is lower for higher subpopulation indices s, that A ⫽ gs⫹1(n), and because only adjacent subpopulations may be merged into risk is, gs(n) ⫽ P As ⬎ P s⫹1 classes, the aggregate cost function is generally decreasing (as illustrated in Fig. 3). The profit ⌸ i in risk class i is calculated as the difference of earnings and costs, that is,

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⌸ i(n) ⫽ Ei(n) ⫺ Ci(n) ⫽ n  fi(n) ⫺ n  gi(n), n ⫽ 1, . . . , Ni, where Ei(n) ⫽ n  fi(n) describes the earnings and Ci(n) ⫽ n  gi(n) describes the costs for n insured risks. Hence, a risk class is profitable as long as the market price Pi ⫽ fi(n) is higher than the actuarial premium P Ai ⫽ gi(n), setting aside any additional (classification) costs. The total profit from classification system m is given as the sum of the profit in each risk class i, i ⫽ 1, . . . , Im less classification costs k(Im ⫺ 1). It is denoted by

冘 ⌸ (n ) ⫺ k(I Im

⌸(n1, . . . , nIm) ⫽

i

i

m

⫺ 1)

i⫽1

with ni being the number of insured risks in risk class i. The insurer aims to determine the optimal classification system and optimal price-demand combinations in the corresponding risk classes such that the profit is maximized: max

max

m僆M {(n1,...,nIm)}

⌸(n1, . . . ,nIm ).

(3)

The optimal number and composition of risk classes depend on classification costs. If classification costs are zero, profit is maximized for the maximum number of distinguishable risk classes, that is, if each risk class i corresponds to a subpopulation h for all h ⫽ 1, . . . , H.50 In the presence of classification costs, this pattern changes, depending on the costs. The maximization process must be undertaken as follows. For each classification system m 僆 M, optimal price-demand combinations must be derived for each risk class i ⫽ 1, . . . , Im, which is done by setting the first derivative of the profit ⌸(n1, . . . , nIm ) with respect to the number of risks ni equal to zero:51 ⭸⌸(n1, . . . , nIm) ⭸⌸ i(ni ) ⫽ ⫽ fi(ni ) ⫺ gi(ni ) ⫹ ni( f⬘(n i i ) ⫺ g⬘(n i i )) ⫽ 0. ⭸ni ⭸ni Because fi and gi are defined in Si sections, for each interval Iv, v ⫽ 1, . . . , Si as specified in equation (2), the number of insureds in risk class i is implicitly given by f (nv) ⫺ gi(nvi ) nvi ⫽ ⫺ i iv , for nvi 僆 Iv, v ⫽ 1, . . . , Si.52 v f⬘(n i i ) ⫺ g⬘(n i i) To determine the optimal price-demand combination for risk class i, we need to compare the profit based on the number of risks niv for each section v ⫽ 1, . . . , Si. It is maximized for53 niv* ⫽ arg max ⌸ i(nvi ). nvi 僆Iv,v⫽1,...,Si

Figure 4 is an illustration of the optimal price-demand combination in the case of a risk class i that equals a subpopulation h. Since the number of underlying subpopulations is Si ⫽ 1 in this case, the functions exhibit no breaks, which means that no distinctions need be made between different intervals. The total profit from each classification system m 僆 M is then determined by taking into account classification costs k(Im ⫺ 1). The classification system that yields the highest profits for the insurer is optimal, that is, 50

See, e.g., Doherty (1980, 1981). See, e.g., Baumol (1977, pp. 416–417), assuming that the functions are differentiable. 52 A solution nvi is valid only if it lies in the corresponding interval Iv . 53 In this context, we assume that price discrimination is not possible within a subpopulation. Hence, for policyholders in one subpopulation— i.e., policyholders who face an identical risk situation–the same price P*i will be asked for (for this point cf. the chapter ‘‘Fairness in Risk Classification’’ in Cummins et al. 1983, pp. 83–92). In an extreme case, where first-degree price discrimination is possible, an insurer would like to charge a different price (customer’s reservation price) to each policyholder in one subpopulation (see, e.g., Pindyck and Rubinfeld 2008, pp. 393–403, and Baumol 1977, pp. 405–406). 51

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Figure 4 Optimal Profit-Maximizing Price-Demand Combination in Risk Class i Cost Function gi ( n ) Price-Demand Function f i ( n )

P

Pi R

Pi * Π i ( ni* )

Pi A

0 0

ni*

n

Ni

Notes: P Ri ⫽ reservation price, P Ai ⫽ actuarial premium for one unit of annuity insurance, n*i ⫽ optimal number of insureds, P*i ⫽ optimal market price, ⌸i (n*) ⫽ maximum profit, Ni ⫽ number of risks in risk class i. i

m* ⫽ arg max ⌸(n1, . . . , nIm), m僆M

implying a maximum total profit of

冘 ⌸ (n ) ⫺ k(I Im*

v* ⌸(nv* 1 , . . . , nIm*) ⫽

i

v* i

m*

⫺ 1).

i⫽1

From this formula, in some cases, one can derive the maximum classification costs that can be incurred such that the optimal risk classification system will still be profitable: ⌸ (n ) 冘 )⫽0⇔k⫽ . Im*

i

⌸(n1v*, . . . , nIv*m*

v* i

i⫽1

Im* ⫺ 1

However, it needs to be verified that the classification m* is still optimal under the modified classification costs k, because a different classification with less or more risk classes may be optimal then. If the optimization problem yields the same result, an insurer may compare this amount with the estimated costs of increased underwriting effort.

3.3 Optimal Risk Classification and Costs of Underwriting Risk One of the main reasons why insurers are reluctant to engage in risk classification has to do with the costs of underwriting risk as laid out in Section 2. The effect of costs of underwriting risk in connection with risk classification on an insurer’s profit situation has never been modeled. In the following, we propose a model that allows a general assessment of costs of underwriting risk. We start from the insurer’s optimal risk classification system m* as presented in Section 3.2, including the optimal number of risk classes Im* as well as optimal price-demand combinations (n*, P*) i i for i ⫽ 1, . . . , Im* that satisfy equation (3). Here, and in the following, we omit the superscript v to facilitate notation. We model underwriting risk by assuming that a policyholder who actually belongs to risk class i is wrongly assigned to a higher risk class j ⬎ i with error probability pi j ⱖ 0(兺jⱖi pi j ⫽ 1). For j ⫽ i, the underwriting classification is correct. We define costs of underwriting risk in terms of underwriting errors, which lead to a reduction of profit. In practice, of course, mistakes in classifi-

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cation can be either to the advantage or disadvantage of the insurer. Our approach can be interpreted as the excess negative effect. Furthermore, one can generally assume that the error probability decreases with increasing distance between i and j as it becomes more likely that individuals will be wrongly classified in adjacent risk classes. Because the policyholder is actually in risk class i with the associated price-demand function fi(n), the optimal price-demand combination targeted by the insurer to maximize profit is given by n*i and P*, representing the optimal number of policyholders n*i buying insurance for the price P*. However, i i errors in underwriting imply that wrongly classified policyholders out of risk class i are charged the lower price P*, which is optimal only for insureds in risk class j. The latter are representatives of a j higher risk class with higher average mortality probability and thus lower costs of insurance. As a first effect, this error leads to a reduction of profits ⌸ i in risk class i, since for the percentage pi j, j ⫽ i of individuals out of risk class i, the lower price P*j ⬍ P*i is charged: A ⌸ i ⬊⫽ ⌸ i(n*) ⫽ n*f ⫺ n*g ⫽ n*(P* i i i(n*) i i i(n*) i i i ⫺ Pi ) ⱖ

冘 p n*(P* ⫺ P ). ij i

j

A i

jⱖi

As a second effect, the requested premium P*j creates a new demand ni j ⫽ n*i ⫹ ⌬ni j with probability pi j, which in turn leads to additional changes in the insurer’s profit. Here ni j denotes the number of policies sold in risk class i for the price P*, which is optimal in risk class j: j ni j ⫽ f i⫺1(P*). j ⌬ni j thus describes the difference between the actual number ni j and the optimal number n*i of insureds in risk class i. The above notation is simplified in that it does not explicitly consider that fi and its inverse function may be defined in sections. The profit in risk class i with insureds wrongly classified to risk class j changes to ⌸ i(ni j) ⫽ ni j fi(ni j) ⫺ ni j gi(ni j) with probability pi j. The expected profit in risk class i is then given as the average profit when wrongly classifying insureds to risk classes j ⫽ i ⫹ 1, . . . , Im, weighted with the respective underwriting error probabilities for each risk class:

˜i ⫽ ⌸

冘 p ⌸ (n ). ij

i

ij

(4)

jⱖi

Figure 5 illustrates the effect of underwriting errors by means of two risk classes. Because original price-demand combinations in risk class i are optimal for a profit-maximizing in˜ i after accounting for underwriting errors will be lower than the original surer, the modified profit ⌸ ˜ i ⬍ ⌸ i. The difference between the two amounts is the cost of underwriting risk in risk profit, that is, ⌸ class i:

˜ i. εi ⫽ ⌸ i ⫺ ⌸ The cost will be influenced by various factors, such as the extent of underwriting error probabilities, the distance between two risk classes in terms of the difference between optimal risk class prices, and the cost function in each risk class. We illustrate the effect of these different factors on the cost of underwriting risk for a case including only one erroneous classification from i to j. With ni j ⫽ n*i ⫹ ⌬ni j, one can reformulate the expression for costs of underwriting risk as follows:

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Figure 5 Change of Profit in Risk Class i When Insureds Are Wrongly Classified to Risk Class j ⬎ i with Probability pij ⴝ 1 Cost Function gi ( n ) Price-Demand Function f i ( n )

P

Pi

R

PjR

Pi * Π i ( ni* ) Pj*

Pi A

0 0

Cost Function g j ( n ) Price-Demand Function f j ( n )

P

Pj*

Π i ( nij )

PjA

ni*

nij

Ni

n

0

n*j

0

n

Nj

Notes: P Ri , P jR ⫽ reservation price, P iA, P jA ⫽ actuarial premium for (one unit of ) annuity insurance, n*, n*j ⫽ optimal number of insureds, P *, i i P j* ⫽ optimal market price, Ni , Nj ⫽ number of risks in risk class i or j, respectively; ⌸i(n*) ⫽ maximum profit in risk class i, ⌸(ni j ) ⫽ profit in i risk class i for demand ni j .

˜ i ⫽ ⌸ i(n*) εi ⫽ ⌸i ⫺ ⌸ ⫺ (pi j⌸ i(ni j) ⫹ (1 ⫺ pi j)⌸ i(n*)) ⫽ pi j(⌸ i(n*) ⫺ ⌸ i(ni j)) i i i ⫽ pi j(n*f ⫺ n*g ⫺ ni j fi(ni j) ⫹ ni j gi(ni j)) i i(n*) i i i(n*) i ⫽ pi j(n*f ⫺ n*g ⫺ (n* i i(n*) i i i(n*) i i ⫹ ⌬ni j)fi(ni j) ⫹ (n* i ⫹ ⌬ni j)gi(ni j)) ⫽ pi j(n*([ fi(n*) ⫺ fi(ni j)] ⫺ [gi(n*) ⫺ gi(ni j)]) ⫹ ⌬ni j[gi(ni j) ⫺ fi(ni j)]) i i i ⫽ pi j(n*([P* ⫺ [gi(n*) ⫺ gi(ni j)]) ⫹ ⌬ni j[gi(ni j) ⫺ P*]). i i ⫺ P*] j i j As discussed above, the first term is positive, [P*i ⫺ P*] ⱖ 0, because the price in a higher risk class j j is lower compared to the price in risk class i. Therefore, the greater the distance between two risk classes, that is, the bigger the difference between P*i and P*, the higher the costs of underwriting risk. j At the same time, however, the error probability pi j will decrease with increasing distance between j and i, which dampens the effect of the difference between P*i and P*j on the overall cost εi. The second term represents the difference between the costs for the optimal number of policyholders n*i and the actual number ni j. Because the cost function in risk class i is aggregated, it may be decreasing. In general, the cost for ni j ⬎ n*i individuals may thus be lower than the cost for n*i individuals, that is, [gi(n*) ⫺ gi(ni j)] ⱖ 0. Hence, the difference will reduce costs of underwriting risk. The last i term represents the difference between the actual costs for the insurer and the actual price paid, If the price paid does not cover actual costs in risk class i, costs of underwriting risk [gi(ni j) ⫺ P*]. j will be even larger. If the price paid does exceeds the costs of covering the insurer’s expenses, the term will be negative, thus implying a reduction in costs of underwriting risk. ˜ i can even become negative if costs in risk class i are higher than the price P*j paid by The profit ⌸ the insured (P*j ⬍ gi(ni j)) and if, for example, the underwriter classifies insureds in risk class i to risk class j with probability pi j ⫽ 1. In this special case, the insurer will suffer a loss from erroneous underwriting, which can be seen as follows:

˜ i ⫽ ⌸ i(ni j) ⫽ ni j(P*j ⫺ gi(ni j)) ⬍ 0. ⌸ ⬍0

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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 16, NUMBER 4

In general, costs of underwriting risk should be taken into consideration when making risk classifications. In particular, error probabilities will differ depending on the classification system. For instance, the more risk classes that are established, the smaller will be the differences between them, and the greater the probability of wrongly classifying insurance applicants. Furthermore, it can be assumed that the probability of a wrong classification diminishes with decreasing number of risk classes Im contained in a classification system m. In the presence of underwriting risk, the insurer again faces the problem of finding the optimal classification system. An optimal classification system m** that takes underwriting risk under consideration may differ from the optimal classification system m* that does not. The extent of the difference will depend on the error probability distribution and on classification costs. Calculations can be conducted based on empirically observed underwriting error probabilities or by making reasonable assumptions. The optimal classification system m** solves the following equation based on the formula for modified expected profits in equation (4):

˜ i(n1, . . . , nIm), m** ⫽ arg max ⌸ m僆M

implying an optimal total profit of

˜ 1, . . . , nI ⌸(n

冘 ⌸˜ (n ) ⫺ k(I

Im**

)⫽

m**

i

i

m**

⫺ 1).

i⫽1

Depending on the error probability distribution, the optimal classification system will comprise more or fewer risk classes, which will, of course, have an impact on classification costs. It is vital for an insurer to take all these factors into consideration so as to avoid losses from underwriting risk. By using our proposed approach and given estimation of error probabilities or sound assumptions on empirical erroneous underwriting, these risks can be quantified and appropriately taken into account.

3.4 Numerical Examples: The Interaction between Price and Demand, Optimal Risk Classification, and Costs of Underwriting Risk To illustrate the theoretical model and the interactions between price and demand, optimal risk classification, and costs of underwriting risk, a numerical example is presented in this part. We hereby assume a maximum number of h ⫽ 2 subpopulations with N1 ⫽ 100 and N2 ⫽ 100, implying a total population of N ⫽ 200. The reservation prices are given by PR1 ⫽ 400 and P2R ⫽ 250, that is, PR1 ⬎ P2R, as subpopulations are ordered by increasing average mortality. In addition, the actuarial costs are assumed to be P A1 ⫽ 100, P 2A ⫽ 50, and the classification costs are first set to k ⫽ 0 and then increased. For k ⫽ 0, maximum risk classification is optimal. In this setting, the set M comprises two possible classification systems: m ⫽ 1 denotes the case where each subpopulation represents one risk class, hence I1 ⫽ 2 (case with risk classification), and m ⫽ 2, where the two subpopulations are merged to one single risk class, that is, I2 ⫽ 1 (case without risk classification). In the latter case, the two price-demand functions and the cost functions need to be aggregated. For ease of illustration, we assume linear price-demand functions for both subpopulations, which allow the derivation of closed-form solutions. The slope and intercept of the price-demand function are determined by the number of risks Nh and the reservation price PRh in subpopulations h ⫽ 1, 2: PR Ph ⫽ fh(n) ⫽ ah  n ⫹ bh ⫽ ⫺ h  n ⫹ PRh . Nh For the input parameters assumed above, we thus obtain P1 ⫽ f1(n) ⫽ ⫺4  n ⫹ 400 and P2 ⫽ f2(n) ⫽ ⫺2.5  n ⫹ 250.

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Hence, for any change in the reservation price or the number of risks in each subpopulation, the price-demand functions change. The inverse of the price-demand function is defined by f ⫺1 h (Ph ) ⫽

1 b N  Ph ⫺ h ⫽ ⫺ Rh  Ph ⫹ Nh, h ⫽ 1, 2, ah ah Ph

⫺1 that is, f ⫺1 1 (P1) ⫽ ⫺0.25  P1 ⫹ 100 and f 2 (P2) ⫽ ⫺0.4  P2 ⫹ 100, and the first derivative is given by

PR f⬘(n) ⫽ ah ⫽ ⫺ h , h ⫽ 1, 2. h Nh To establish the classification system m ⫽ 2, the aggregated price-demand function fi(i ⫽ 1 ⫹ 2) of the two subpopulations must be derived. On each interval Iv, v ⫽ 1, 2, R ⫺1 R ⫺1 R ⫺1 R I1 ⫽ [0, f ⫺1 1 (P2 ) ⫹ f 2 (P2 )] ⫽ [0, 37.5], I2 ⫽ [ f 1 (P2 ) ⫹ f 2 (P2 ), N1 ⫹ N2] ⫽ [37.5, 200],

the aggregated price-demand function is defined by

冉冘 冊 冘 冘冉 冊 冘 冉 冒冘 冊 冘 冒冘 v

fi(n) ⫽ fi( f (Pi )) ⫽ fi

f ⫺1 s (Pi ) ,

⫺1 i

s⫽1

v

⇒n⫽

v

f s⫺1(Pi ) ⫽

s⫽1

s⫽1

v

⇔ fi(n) ⫽ Pi ⫽

1

s⫽1

1 b  Pi ⫺ s as as v

1 as

n⫹

s⫽1

s⫽1

v

bs as

ai

v

⫽ Pi

s⫽1

1 ⫺ as

冘 ab , v

s

s⫽1

s

1 , if n 僆 Iv, v ⫽ 1, 2. as

bi

In the present setting, depending on the interval, we thus obtain fi(n) ⫽ ai  n ⫹ bi ⫽ ⫺4  n ⫹ 400, n 僆 I1, fi(n) ⫽ ai  n ⫹ bi ⫽ ⫺1.54  n ⫹ 307.69, n 僆 I2, and the derivative is given by f⬘(n) ⫽ ai, n 僆 Iv, v ⫽ 1, 2. The aggregate cost function can be shown i to have the following form:

冘 a P ⫽ 冘 a

gi(n) ⫽

1 n

v

f ⫺1 s (Pi )  gs(Pi ) ⫽

s⫽1

v

i

s⫽1

s

A s

⫹

1  n

1 n

冘冉 v

s⫽1

冊

1 b  Pi ⫺ s as as

 P As ⫽

1 n

冘 冉a1  (a  n ⫹ b ) ⫺ ab 冊  P v

s

i

s⫽1

s

i

A s

s

冘 (b ⫺ ab )  P , n 僆 I , v ⫽ 1, 2, v

i

s

A s

v

s⫽1

⫽g1

s

⫽g2(n)

and its derivative is thus given by 1 g⬘(n) ⫽⫺ 2 i n

冘 (b ⫺ ab )  P v

i

s⫽1

s

s

A s

1 ⫽ ⫺  g2(n), n 僆 Iv, v ⫽ 1, 2, n

implying that the derivative is zero in case of individual risk classes and in the first interval of the aggregated function (because bi ⫽ bs, s ⫽ 1). As described in the previous subsection, the optimal number of risks in each risk class for classification systems m ⫽ 1 and 2, respectively, can be derived by a closed-form solution:

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Figure 6 The Interaction of Classification Costs and Profit with (m ⴝ 1) and without (m ⴝ 2) Risk Classification for a Constant Total Population of N ⴝ 200, N2 ⴝ N ⴚ N1 a) Profit in classification systems m = 1, 2

b) Maximum classification costs

12,000

Maximum classification costs k 4,500

10,000

4,000 3,500

8,000

3,000

6,000

Profit m = 1 Profit m = 2

4,000

2,500 2,000 1,500 1,000

2,000

500 -

10

25

50

75

100 125 150 175 190 N1

10

25

50

75

100 N1

125

150

175

190

Notes: Difference between profit of m ⫽ 1 (risk classification, k ⫽ 0) and m ⫽ 2 (no risk classification) represents the maximum amount of classification costs k such that risk classification is still profitable.

nvi ⫽

fi(nvi ) ⫺ gi(nvi ) ⫺ai  nvi ⫺ bi ⫹ g1 ⫹ g2(nvi ) ⫽ , v v f⬘(n 1 2 v i i ) ⫺ g⬘(n i i) ai ⫺ ⫺ v g (ni ) ni

冉

⇔ ai  nvi ⫹

冊

1 2 v 1 g (ni )  niv ⫽ ⫺ai  niv ⫺ bi ⫹ g1 ⫹ g2(niv) ⇔ 2  ai  niv ⫽ ⫺bi ⫹ g1 ⇔ niv ⫽ v ni 2

b 1 P 冘 冘 冒 a  冘 a a 1 a 1 b P ⇔n ⫽ ⫹ ⇔ n ⫽ 冉冘 ⫹ 冘 冊, 2 a 2 a a 1 1冒 冘 a 1 P ⇔ n ⫽ 冉 冘 N ⫹ 冘 冊 , for n 僆 I , v ⫽ 1, . . . , S . 2 a v i

冢

v

v

s

s⫽1

s⫽1

s

s

A s

s⫽1

s

v

i

s⫽1

v

v i

s

v

A s

s

s⫽1

s⫽1

冣

v

i

v i

v

s⫽1

v

v

A s

s⫽1

s

s

v i

s

冉

冊

⫺bi g1 ⫹ , ai ai

i

s

Using the closed-form solution above for the optimal number of risks in each risk class for each interval and ensuring integer values by rounding down,54 Figure 6a displays the profits for the classification system m ⫽ 1 (full risk classification) in case classification costs are zero (k ⫽ 0) and for the classification system m ⫽ 2, where no risk classification is conducted and only one risk class is offered by merging the two subpopulations into one single risk class. We thereby assume that the size of the total population remains constant with N ⫽ 200, and the number of risks in subpopulation h ⫽ 1, N1, is varied, that is, N2 ⫽ N ⫺ N1. As can be seen in Figure 6a, risk classification is always preferable if classification costs are zero (k ⫽ 0). However, conducting risk classification is typically associated with classification costs. Thus, Figure 6b displays the corresponding maximum amount of classification costs k(Im ⫺ 1) ⫽ k(2 ⫺ 1) ⫽ k such that risk classification is still profitable. These costs are obtained by calculating the difference between the profit if risk classification is conducted (m ⫽ 1) and the profit if no risk classification is conducted (m ⫽ 2). If classification costs are higher than the values displayed in Figure 6b, risk 54

Note that in general, closed form solutions to the optimization problem with constraints (e.g., solution must be an integer value) are not available, such that numerical optimization procedures must be applied.

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Figure 7 The Interaction between Optimal Demand, Prices, and Resulting Profits from Risk Classification (m ⴝ 1, k ⴝ 0) for Different Reservation Prices P R1 of Subpopulation 1, Given a Constant Total Population (N ⴝ 200, N2 ⴝ N ⴚ N1) a) Optimal demand in risk classes 1 and 2

b ) Optimal prices in risk classes 1 and 2

100

400

80

300

60 200

40 100

20 -

0

50

100 N1

150

0

200

50

100 N1

150

200

Optimal price h = 2

Optimal demand h = 2

Optimal price h = 1, reservation price = 400

Optimal demand h = 1, reservation price = 400

Optimal price h = 1, reservation price = 300

Optimal demand h = 1, reservation price = 300

c) Corresponding total profit of risk classification 12,000 10,000 8,000 6,000

Total profit, reservation price = 400

4,000

Total profit, reservation price = 300

2,000

0

50

100 N1

150

0200

classification is no longer profitable, and insurers would abstain from classifying risks and instead only offer one product for all policyholders. One can also observe in Figure 6 that the interaction and relation between optimal risk classification and classification costs are nonlinear. In particular, classification costs can almost be up to k ⫽ 4,000 for N2 ⫽ N1 ⫽ 100 and still ensure that risk classification is preferable to offering one single standard tariff. However, the classification costs k need to be considerably lower elsewhere in the considered cases, that is, for N1 lower or higher than 100. Figure 6 further shows an increase in profits if the portion of risks in subpopulation h ⫽ 1 is higher as compared to subpopulation h ⫽ 2. This is due to the specific price-demand functions, particularly the reservation price in subpopulation 1, which, with PR1 ⫽ 400, is very high compared to the actuarial costs of 100 (subpopulation 2 has a reservation price of only 250 and actuarial costs of 50). To illustrate the interaction of the reservation price and the optimal demand from risk classification, Figure 7 shows the optimal demand (number of contracts) and prices in the case of risk classification as well as the corresponding profit of the insurer if the reservation price of subpopulation 1 is lowered from PR1 ⫽ 400 to PR1 ⫽ 300.

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Figure 8 Optimal Number of Risks in Each Risk Class, Costs of Underwriting Risk and Profits for p12 ⴝ 1, k ⴝ 0, m ⴝ 1 for Different Reservation Prices a) Optimal number of risks with and without underwriting risks for P1R = 400

b) Corresponding profits in case with and without underwriting risks for P1R = 400

100

12,000

4,000 3,500

80 60

3,000

Costs of underwriting risk (right axis)

2,500

Optimal demand h = 2

2,000 40

1,500 1,000

20

500 0

25

50

75

100

125

150

10,000 8,000

Optimal demand h = 1, reservation price = 400 New demand h = 1 given underwriting risk, reservation price = 400

2,000 25

3,500 80 60 40

100

125

150

9,000

3,000

Costs of underwriting risk (right axis)

2,500

Optimal demand h = 2

20

125

150

6,000

1,500

Optimal demand h = 1, reservation price = 300

4,000

New demand h = 1 given underwriting risk, reservation price = 300

2,000

100

7,000

5,000

500 0

8,000

2,000

1,000

N1

75

d) Corresponding profits in case with and without underwriting risks for P1R = 300

4,000

100

75

50

N1

c) Optimal number of risks with and without underwriting risks for P1R = 300

50

Total profit with underwriting risk, reserv. price = 400

4,000

N1

25

Total profit without underwriting risk, reserv. price = 400

6,000

Total profit without underwriting risk, reserv. price = 300 Total profit with underwriting risk, reserv. price = 300

3,000

1,000 25

50

75

100

125

150

N1

Notes: p12 ⫽ 1 implies that all risks in h ⫽ 1 are wrongly classified to h ⫽ 2; the difference between the profits with and without underwriting risk represents the cost of underwriting risk.

Although the optimal demand in subpopulation 2 remains unchanged if the reservation price of subpopulation 1 is lowered from 400 to 300, the optimal demand in subpopulation 1 is reduced along with a lower optimal price (see Figs. 7b and 7c). This is due to the fact that a reduction in the reservation price of subpopulation 1 reduces the attractiveness of this risk class and at the same time increases the attractiveness of subpopulation 2 from the insurer’s point of view. Thus, the profit decreases in case of PR1 ⫽ 300 if the portion of subpopulation 1 out of the total number of risks N increases. We next consider the impact of underwriting risk in the case of risk classification; that is, we focus only on classification system m ⫽ 1, where each subpopulation represents one risk class. Underwriting risk arises if risks from subpopulation 1 are wrongly assigned to risk class 2 and the costs of underwriting risk are given by the difference between the profit with and without underwriting risk. Figure 8a shows the optimal number of risks in risk classes 1 and 2 for different numbers of risks N1 for an error probability of 100%; that is, all risks in risk class 1 are wrongly classified to risk class 2. It can be seen that in the presence of underwriting risk, the optimal demand for risk class 1 increases considerably because of the lower price (wrongly) required by the insurer, while the optimal demand in case of risk class 2 remains unchanged. In addition, higher portions of risks in risk class 1 (increasing N1) imply an increasing risk exposure for the insurer in terms of underwriting risk, which leads to

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Figure 9 The Interaction between Optimal Risk Classification, Classification Costs, and Underwriting Risk for Different Error Probabilities p12, k ⴝ 0, N1 ⴝ N1 ⴝ 100, m ⴝ 1, P R1 ⴝ 400 12,000

3,000

10,000

2,500

8,000

2,000

6,000

1,500

4,000

1,000

2,000

500

Total profit with underwriting risk, k = 0 Total profit without underwriting risk, k = 0 Maximum amount of classification costs k to reduce underwriting risk (right axis)

-

0

0.25 0.5 0.75 Error probability

1

˜ 1 ⫽ 兺i⬎1 p1j ⌸1(n1j ) ⫽ (1 ⫺ p12)⌸1(n11) ⫹ p12⌸1(n12) (see eq. (4)); maximum classification costs Notes: The total expected profit is given by ⌸ for a given error probability p12 ⬍ 1 are derived by the difference of total profits with underwriting risk for the given error probability p12 ⬍ 1 (k ⫽ 0) and total profits with underwriting risk for an error probability of p12 ⫽ 1. increasing costs of underwriting risk and thus decreasing profits from risk classification for higher N1 (Fig. 8b) even in the case of PR1 ⫽ 400. If the reservation price of subpopulation 1 is lowered to PR1 ⫽ 300 (Figs. 8c and 8d), the costs for underwriting risk decrease because the price-demand function becomes less steep and, thus, less risks are being insured, which in turn reduces the potential for underwriting risk. Hence, although higher reservation prices ceteris paribus considerably improve the profit situation for the insurer for a given number of risks N1 in subpopulation h ⫽ 1, the presence of underwriting risk can lead to a situation where profits become similar to the case where reservation prices are lower, and thus the exposure to underwriting risk is reduced. However, the negative effects of underwriting risk presented Figure 8 can be considerably dampened if error probabilities are less than one and only a fraction p12 ⬍ 1 is wrongly classified. Figure 9 thus displays the expected profits from risk classification (i.e., m ⫽ 1) for different error probabilities p12 and k ⫽ 0 and shows that the expected profit is decreasing for an increasing underwriting risk. However, increasing risk classification costs can lead to lower costs of underwriting risk, which poses a tradeoff for the insurer. Thus, Figure 9 additionally exhibits the maximum amount of classification costs k for p12 ⬍ 1 that would imply the same expected profit as in the case of p12 ⫽ 1 (and k ⫽ 0). Hence, the difference between the total expected profits with underwriting risk for the given error probability p12 ⬍ 1 for k ⫽ 0 and total expected profits with underwriting risk for an error probability of p12 ⫽ 1 for k ⫽ 0 represents the maximum amount of classification costs that can be invested by the insurer to reduce the error probability from one to less than one and thus lower underwriting risk. Note that the classification costs are lower than 4,000, which in the present setting represents the maximum amount of classification costs that still ensure that risk classification is profitable in the first place (see Fig. 6) as compared to offering only one standard product for all risks. If k ⬎ 0 in case p12 ⫽ 1, then the maximum classification costs are reduced accordingly. In case the costs for underwriting risk are transferred to the insureds by increasing the premiums, the demand would decrease.

4. MODEL APPLICATION

AND

MARKET ENTRY

This section discusses additional key issues regarding substandard annuities. For insurers trying to decide whether engaging in risk classification would be a practical and profitable pursuit, or offering

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only a standard tariff, our model can be useful. For practical implementation, estimating price-demand and cost functions for each risk segment is vital. Other important aspects to be considered in making such a decision include market entry barriers and the general risks and advantages of providing substandard annuities. To ensure adequate model application and sound results, the following issues need to be addressed. First, innovating insurers need knowledge about the structure, size, and potential of their target market,55 which includes information about the number and mortality profiles of potential annuitants. Second, estimates must be made of how product price will affect demand in each market segment. Therefore, the maximum number of risks, their reservation price, and price elasticity of demand need to be derived empirically by means of, for example, surveys leading to the respective price-demand function. Third, insurers need to choose the (likely) most profitable market segment in which to conduct business, a choice made easier by employing our model with calibration as specified above. In addition, estimation of classification costs and an assessment of the underwriting quality are crucial, because risk classification and the associated underwriting are considered the most hazardous risk for insurers offering substandard annuity products, as discussed in Section 2. Proper underwriting demands sufficient expertise and, preferably, a sound IT-backed underwriting and classification system,56 which, in turn, has to be accounted for in terms of classification costs. When it comes to underwriting quality, one difficulty is in estimating current mortality probabilities because of a lack of credible mortality data. In addition, the risk of future mortality improvements made possible by developments in the medical field cannot be ignored.57 Moreover, insurers need to make sure that risk factors are not controllable by annuitants, and they must try to prevent insurance fraud, which, at least compared to life insurance, may be fairly challenging. For example, in the life insurance sector, fraud will be detected, once and for all, when the payment becomes due—either the insured is dead or not. Enhanced annuity payments, in contrast, must be made as long as the insured lives, giving much more opportunity for fraud.58 Privacy and regulatory issues with respect to information about risk classification characteristics also need to be kept in mind.59 The above issues need to be taken into account by insurers considering the introduction of risk classification. However, as outlined above, calibration of the model is complex and prone to a relatively high degree of uncertainty such that implementation may present an obstacle to innovation. Moreover, there are other barriers and risks having to do with the target market and product design, as described below. Regarding the question of adverse selection, markets without risk classification may cause individuals with low life expectancies (e.g., because of a former illness) not to purchase an annuity contract. Hence, risk classification in this sector could lead to an increase in the total insured population. If this assumption could be verified empirically, this aspect can be introduced in our model framework as laid out in Section 3 by reformulating the total insured population as a function of the number of risk classes. In addition, adverse selection is, at least to some extent, reflected in our model via the costs of underwriting errors (denoted with ε in the model setup), because ε is caused by information asymmetries between policyholder and insurer. Once established, the substandard annuity market is said to be very competitive.60 Applicants ‘‘shop around’’ for the best rates by submitting underwriting requests to several insurance companies simultaneously.61 Insurers face a tradeoff between staying competitive and maintaining actuarially sound

55

See Ainslie (2001, p. 17). See Weinert (2006, p. 15). 57 See, e.g., Cardinale et al. (2002, p. 16), Cooperstein et al. (2004, p. 13), LIMRA and Ernst & Young (2006, p. 28), and Richards and Jones (2004, p. 20). 58 See Junus et al. (2004, p. 20). 59 See Brockett et al. (1999, p. 11). 60 See LIMRA and Ernst & Young (2004, p. 7). 61 This practice is mainly observed in the United Kingdom. See, e.g., Cooperstein et al. (2004, p. 13) and Ainslie (2001, p. 19). 56

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criteria for qualifying applicants as substandard risks. A provider’s profitability can be negatively affected if the placement ratio, that is, the ratio of sales to underwriting requests, becomes too low.62 In addition, there is competition in the form of other financial products,63 and in some markets there may be not much awareness of substandard annuities. Therefore, if an insurer decides to start with substandard annuities in markets where only standard annuities had been sold so far, it will be in need of a distribution system strong enough to generate sufficient market awareness.64 Product design will need to be attractive to sales force and clients, efficient, and innovative.65 Except for the application itself, which requires the provision of additional health information, sales processes are similar to those of standard products.66 In line with standard annuities, a substandard annuity provider is required to maintain minimum capital requirements and account for longevity and interest rate risk.67 In addition, the impact on existing portfolios needs to be investigated.68 Daunting as these barriers and risks sound, there are also substantial advantages to selling substandard annuities. First, according to Cooperstein et al. (2004) and Towers Watson (2008), the market potential is huge. Turra and Mitchell (2004) also find support for considerable demand for these products. Thus, entering the substandard annuity market is likely to be an attractive alternative for new market players with a solid business plan, giving them the opportunity to reaching a broader population and/or meeting a niche market need.69 For an established market player with an existing standard annuity portfolio, the situation is not as clear-cut,70 although—except for the underwriting—offering substandard annuities may require only modest modifications of organization, product design, and distribution system.71 Yet there is the danger of destabilizing one’s market position by becoming more competitive in the substandard market but, at the same time, less profitable in the standard annuity business, which could result in some reputational damage, too. However, if a standard insurer expects the substandard annuity market to grow, it is advisable to become active in it early on and thus avoid being forced, for defensive reasons, into quickly developing a substandard product later. Early market engagement will allow an insurer to enjoy the benefits of competitive advantage and avoid problems of adverse selection.72 In this context, adverse selection means that standard annuity providers will be left with a greater proportion of healthier lives in their portfolios if those with a reduced life expectancy tend to buy substandard annuities.73 This situation leads to a reduction in profit for standard annuity portfolios of firms that are not classifying risks, which has been quantified by Ainslie (2000) and Hoermann and Russ (2008). Alternatively, standard annuity providers may have to react by increasing their single product price, which in turn will repel future business.

5. SUMMARY In this paper we comprehensively examined key aspects of substandard annuities and developed a model for an optimal risk classification system that includes consideration of underwriting risk. We began with a description of different types of substandard annuity products, their respective underwriting,

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See LIMRA and Ernst & Young (2006, p. 20). See LIMRA and Ernst & Young (2006, p. 22). 64 See, e.g., LIMRA and Ernst & Young (2006, p. 22) and Weinert (2006, p. 15). 65 See Froehling (2007, p. 5), Werth (1995, p. 6), and Weinert (2006, p. 15). 66 See Cooperstein et al. (2004, p. 13) and LIMRA and Ernst & Young (2006, pp. 8, 37–38). 67 See, e.g., Froehling (2007, p. 5), LIMRA and Ernst & Young (2006, p. 7), and Weinert (2006, p. 15). 68 See Werth (1995, p. 6). 69 See LIMRA and Ernst & Young (2006, p. 6). 70 The following points are analogously discussed in Werth (1995) for the case of introducing preferred life products in the life insurance market. 71 See LIMRA and Ernst & Young (2006, p. 21). 72 See, e.g., O’Neill (1997, p. 1088) and Swiss Re (2007, p. 13). 73 See, e.g., Towers Watson (2008). 63

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potential market size, and associated underwriting risk, the last of which is considered crucial for success in the substandard annuity sector. Supported by extant research, we focused on multiclass underwriting implemented by risk classification via rating factors. We proposed a model for a risk classification system in a mortality heterogeneous general population, which is described by a frailty distribution. The optimal number and size of risk classes as well as the profit-maximizing price-demand combination in each risk class were then derived as the solution of an optimization problem. As an extension, we solved for the optimal risk classification system when taking into account costs of underwriting risk. We modeled these costs by assuming error probabilities for wrongly classifying insureds into a higher risk class, thus underestimating the true costs of insurance. We then discussed the practical application of our model, along with market entry barriers and risks and advantages inherent in being a substandard annuity provider. Because of the generality of the model, applications to classification problems other than substandard annuities are possible as well. In conclusion, extended risk classification in annuity markets may not only increase the profitability of insurance companies, but it may contribute to benefit society at large because the introduction of substandard annuities should make it possible for many formerly uninsurable persons to secure for themselves a private pension. In future research, focus should be laid on the impact of firm interaction effects as well as the relevance of risk classification in light of longevity risk and its impact on a firm’s risk situation depending on whether a firm classifies risks or not.

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European Accounting Review, 2014 http://dx.doi.org/10.1080/09638180.2014.906316

Political Connections and Accounting Quality under High Expropriation Risk GEORGE BATTA∗ , RICARDO SUCRE HEREDIA∗∗ and MARC WEIDENMIER∗ ∗ Robert Day School of Economics & Finance, Claremont McKenna College, 500 E. 9th St., Claremont, CA 91711, USA and ∗ ∗ Escuela de Estudios Politicos y Administativos, Universidad Central de Venezuela, Caracas 1053, Venezuela

(Received: June 2012; accepted: January 2014)

A BSTRACT We examine the impact of political connections and accounting quality among Venezuelan industrial firms, which face one of the highest levels of expropriation risk worldwide. Based on prior literature, we expect a negative relationship between expropriation risk and accounting quality as firms manage earnings to avoid ‘benign’ state intervention. We find that politically connected firms have higher accounting quality than non-connected firms, which is consistent with connected firms’ lower risk of expropriation due to connections with high-level government officials or ruling party members. The relationship between accounting quality and political connections appears to be strongly moderated by institutional features like expropriation risk.

1.

Introduction

Recent research in accounting has examined the link between political connections and accounting quality.1 Researchers in this area have posited that political connections may increase accounting quality because connected firms are subject to greater media scrutiny, which could provide for stronger monitoring of earnings manipulation. Connected firms may also have readier access to subsidised financing or government contracts, which may blunt incentives to manage earnings for capital market and contracting purposes. On the other hand, politically connected firms may be shielded from the consequences of poor accounting quality or the revelation of earnings management. Moreover, connected firms may manage their earnings to avoid detection of payments to political insiders to maintain their connected status. Preliminary evidence from this body of research suggests that politically connected firms tend to have lower financial reporting quality. However, there is reason to expect that a country’s political, legal, and media institutions – which affect firms’ financial reporting environment more generally (Leuz, Dhananjay, & Wysocki, 2003) – may moderate the relationship between political connections and accounting quality. For example, lack of transparency may limit media outlets’ role in scrutinising political cronyism. Strong investor-protection laws, accompanied

Correspondence Address: George Batta, Robert Day School of Economics & Finance, Claremont McKenna College, 500 E. 9th St., Claremont, CA 91711, USA. Email: [email protected] Paper accepted by Laurence van Lent 1 Research in this area includes Chaney, Faccio, and Parsley (2011), Guedhami, Pittman, and Saffar (2014), Chen, Ding, and Kim (2010), Correia (2011), Fan, Li, and Yang (2010), Leuz and Oberholzer-Gee (2006), and Ramanna and Roychowdhury (2010). # 2014 European Accounting Association

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by prosecutorial and judicial independence, may impact a connected firm’s ability to escape the consequences of accounting manipulation. In this paper, we examine the moderating role of another important institutional feature, that of expropriation risk. Expropriation risk refers to the risk of governments confiscating value from investors through nationalisation, asset seizure, confiscatory taxation, or exorbitant fines. Based on rankings by the International Country Risk Group (ICRG), a leading political risk consultancy, around 30% of countries possessed a moderate to very high degree of expropriation risk in 2009.2 These countries represented 19% of 2009 world GDP and 28% of capital investment. In countries with high expropriation risk, fear of government expropriation may affect reporting quality. Prior research suggests that firms in high expropriation risk economies manage earnings to avoid offering predatory governments a pretext for expropriating assets. For example, Bushman and Piotroski (2006) find that, especially among so-called ‘code law’ countries, earnings’ sensitivity to bad news is lower in where there is greater risk of asset expropriation. This may result from firms attempting to avoid ‘benign’ state intervention. Bushman, Piotroski, and Smith (2004) also find evidence of that economies characterised by high levels of state intervention in the economy and high levels of expropriation risk tend to score poorly on measures of corporate transparency, as firms attempt to shield themselves from excessive government scrutiny. Durnev and Guriev (2011) find that countries with poor levels of investor legal protection typically feature particularly low corporate transparency for oil and gas dependent companies, which are most vulnerable to government expropriation. There is significant risk of expropriation for all firms operating in economies with high expropriation levels, however, and we expect this to affect average accounting quality. However, politically connected firms in these countries may be less subject to expropriation threats due to managers’, directors’, or owners’ relationships with high-level government officials and ruling party members. Politically unconnected firm’s financial reports may as a result be of relatively poorer quality, and the previously documented negative relationship between political connections and accounting quality may be mitigated or even reversed in economies with high expropriation risk. To explore this question, we examine the relationship between political connections and accounting quality among a panel of listed Venezuelan firms. Venezuela, which had been under the leadership of President Hugo Chavez from 1998 until his death in 2013, adopted a programme of increased state encroachment in the economy, with state expropriation of key private firm assets and restrictions on private enterprise. The state has nationalised or taken large stakes in the transportation, energy, agriculture, media, manufacturing, retail, and oil sectors, and many others besides. Over 400 companies were nationalised between 1999 and 2010, as well as 7.4 million acres of land. This encroachment was accompanied by media clampdowns and political interference in the judicial system, preventing firms from seeking redress, either informally through media campaigns or formally via domestic legal channels. As a result, the ICRG deemed Venezuela as having one of the highest levels of expropriation risk in the world: From 2000 to 2004, Venezuela ranked among the highest decile of countries ranked on expropriation risk. From 2005 to 2013, it ranked among the highest 5%, with only Bolivia, Zimbabwe, and Ecuador ranking equal or lower.3 However, among these four countries, 2

We use ICRG’s contract viability (CV) score as a measure of expropriation risk. ICRG defines CV as ‘the risk of unilateral contract modification or cancellation and, at worst, outright expropriation of foreign owned assets’. 3 AON Corporation’s Political Risk Insurance services in 2012 ranked Venezuela as having Very High political risk, along with Haiti, Belarus, Iran, Iraq, Syria, Yemen, Afghanistan, Pakistan, Sudan, South Sudan, the Democratic Republic of Congo, Somalia, Zimbabwe, and North Korea. The Belgian Export Credit Rating Agency (ONDD) also maintains a list of countries ranked by the risk of expropriation and government action. As of April 2013, Venezuela had the highest

Political Connections and Accounting Quality under High Expropriation Risk 3 Venezuela is unique in two ways: first, in its extensive targeting of both domestic and international firms for expropriation; and second, in its large number of publicly listed industrial firms (33 as of April 2013). These factors provide a sufficiently large data sample for empirical testing of the effects of expropriation risk on accounting quality.4 Moreover, we confirm empirically that official Venezuelan government justifications for expropriation during our sample period frequently related to perceived firm financial performance. Meanwhile, political connections have played an important role in Venezuela’s economy. After a 2002 coup attempt, largely supported by existing business elites, Chavez began using state oil revenue, through the provision of grants, loans, and state purchasing contracts to firms, to elevate a managerial class loyal to his regime. Dubbed the boli-bourgeois (in reference to the businessmen linked to the government, which professes to represent the ideas of Latin American revolutionary Simo´n Bolı´var), this class of connected businessmen have obtained an influential role in the Venezuelan economy. Venezuela thus represents an ideal case study for examining the impact of high expropriation risk on accounting quality and political connections. We measure political connection strength based on assessments by Venezuelan businessmen. These assessments are based on the existence of government equity stakes in firms; business or friendship relationships of owners, managers, or directors with the government or Chavista (i.e. Chavez-supporting) businessmen; or favourable mentions of firms by Chavez-controlled media. Empirical results suggest that politically connected firms in Venezuela do indeed have greater accounting quality than unconnected firms, based on differences in earnings smoothing behaviour, on the incidence of loss and earnings decrease avoidance, and on absolute accruals measures. Importantly, we also find that the effect of political connections on accounting quality is greatest among firms at greater risk of expropriation. These results, robust to a battery of sensitivity tests, confirm our hypothesis on the impact of political connections on accounting quality under high expropriation risk. Our research contributes to the literature in two ways. First, we show how the relationship between political connections and accounting quality can vary when economic activity is vulnerable to expropriation. Whereas prior literature on political connections and accounting quality has tended to avoid examining countries with high expropriation risk, our paper examines this underexplored set of countries and provides nuance to the growing literature on the impact of political connections on accounting quality. Second, we contribute to the literature on the effect of political and legal institutions on accounting quality and corporate governance. Prior literature has examined how the strength of investor-protection laws (Ball, Robin, & Wu, 2001, 2003; Defond, Hung, & Trezevant, 2007; Leuz et al., 2003), as well as legal origin, the extent of state involvement in the economy, and state expropriation risk (Bushman & Piotroski 2006; Bushman et al., 2004), affects the incidence of earnings management, earnings informativeness, timely loss reporting, and financial transparency. In this paper, we show how one key aspect of the institutional environment – the importance of political connections – interacts with expropriation risk to affect accounting quality. We acknowledge that political connections may result from choices made by firms’ executives, owners, or directors, and these choices may be correlated with firm characteristics

risk rating with the ONDD, along with Afghanistan, Argentina, Bolivia, Ecuador, Guinea, Iran, Iraq, Libya, Syria, Yemen, Palestine, Sudan, Somalia, and Zimbabwe. 4 Zimbabwe had 60 listed, domestic non-financial firms as of April 2013, relative to Venezuela’s 33, Ecuador’s 26, and Bolivia’s 18, according to CreditRisk Monitor’s Directory of Public Companies (http://www.crmz.com/Directory/). However, Zimbabwe’s nationalisation threats have primarily targeted subsidiaries of foreign companies and small landowners rather than domestic companies.

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associated with accounting quality for which we have not adequately controlled. However, we emphasise that most of the political connections in Venezuela are based on long-standing school, professional, friendship, or familial ties with prominent government officials or ruling party members.5 More generally, Chaney et al. (2011) find that firms with poorer accruals quality are not more likely to establish political connections, mitigating concerns over the endogeneity of political connection choice.

2.

Literature Review

A number of recent papers in accounting have examined the impact of political connections on accounting quality and transparency. A key feature of most of the literature on political connections’ impact on accounting quality, however, is that it focuses on countries with at best moderate levels of expropriation risk. For example, Correia (2011) and Ramanna and Roychowdhury (2010) both focus on US firms. Of the firms in Chaney et al.’s (2011) sample, less than 5% can be characterised as operating in an environment of moderate-to-high expropriation risk, while only 12% of Guedhami et al. (2014) sample can be similarly characterised. An innovation of our study is that we examine the impact of political connections on accounting in a country that can be deemed to possess one of the highest levels of expropriation risk worldwide, which should serve as a robust test of the effect of expropriation risk on the relationship between political connections and accounting quality. Particularly relevant to our study is Bushman and Piotroski (2006), who find that companies in countries with strong investor-protection laws and independent judiciaries report losses on a more timely basis. However, they find that this conservatism in earnings is less prevalent in code-law countries characterised by a high level of state involvement in the economy, which is consistent with a scenario in which a ‘benevolent’ government seeks to intervene in failing firms. Using cross-country data, Bushman et al. (2004) show the presence of strong state ownership in an economy and high levels of state expropriation risk creates incentives for opaque reporting practices among non-state-owned publicly traded firms. This opacity arises as firms attempt to limit information about the composition, value, and profitability of their productive assets, through reduced disclosure and less timely reports. Results in Bushman and Piotroski (2006) and Bushman et al. (2004) suggest that accounting quality might be strongly affected by the threat of state intervention, with non-stateowned firms offering less conservative and transparent financial statements. However, the incentive for less conservative or transparent financial statements may be less intense for politically connected firms. In addition to receiving preferential financing and contracts from the state, politically connected firms may also benefit from lower expropriation risk, owing to their connections to key players in legislative bodies, government ministries, and the executive branch. In the next section, we outline the recent political history and relations between business and government in Venezuela, and we explore the increasing threat of government expropriation. We also outline the importance of political connections in the Venezuelan political economy and outline a model for understanding the Chavez government’s expropriation motivations. 5

In a cross-country study, Boubraki, Guedhami, Mishra, and Saffar (2012) use whether a firm is headquartered in the capital city as an instrument for the presence of political connections. While presence in, or proximity to, a capital city may generally be associated with political connection strength, we do not believe it serves as an appropriate instrument in our context, as most of the connections we find are based on long-standing school, professional, friendship, or familial ties. Additionally, Chaney et al. (2011) argue that a capital city presence instrument is unlikely to satisfy the exclusion restriction.

Political Connections and Accounting Quality under High Expropriation Risk 5 3.

Background on Venezuela

3.1. Hugo Chavez’ Rise to Power President Hugo Chavez’ rule in Venezuela emerged from the collapse in 1998 of a political agreement called the Pacto de Punto Fijo, which governed domestic political relations for 40 years. Backed by the three major political parties at the time – as well as the military, the Catholic Church, business elites, and trade unions – the programme was structured around the distribution of oil revenues to finance the construction of basic infrastructure and to pay for primary education, health care, and industrial policy. It also allowed for political patronage to be channelled through labour unions, professional associations, and peasant leagues attached to the two main political parties (AD and Copei). Buoyed by high oil prices, the policy was successful through the late 1970s, with Venezuela producing an annual real GDP growth of 6%.6 The situation collapsed in 1983, when the government devalued the bolivar following a sharp drop in oil prices. Double-digit inflation and diminished economic growth soon followed,7 culminating in protests and riots in Caracas and other major cities in 1989. Widespread dissatisfaction among the electorate and plummeting electoral participation compelled many Venezuelans to back Hugo Chavez for president. Chavez was a military officer jailed for two years for helping organise a failed coup against the government in 1992. In 1998, Chavez won the presidency, earning 56% of the vote. His candidacy was initially backed by business elites, who sought direct access to state power, unmediated by the existing party structures of AD and Copei. However, the honeymoon between business elites and Chavez was short-lived. Chavez soon attempted to consolidate his power, following the election of his party members to large majorities in the Constituent Assembly in 1999. This included rewriting the constitution in favour of greater presidential powers, and which also discussed the expropriation of land deemed ‘unproductive’; obtaining from the legislature in 2000 ‘enabling powers’ to rule by decree on a series of policy areas (mostly property rights in the hydrocarbon and agricultural sectors); and gaining control of Venezuela’s election monitoring body. In 2001, Chavez issued a presidential decree that established a legal vehicle for land expropriations, along with 48 new laws designed to tightly regulate Venezuela’s economic environment. As a result, in 2002, many business leaders supported both a general strike and an unsuccessful coup against Chavez’ rule. Since that event, the government has attempted to weaken and control the business sector through two primary means. First, it attempted to elevate a new class of capitalists by placing Chavistas in positions of power in state-owned enterprises and by offering preferential financing and contracts to favoured businessmen, a policy that was supported by high oil prices from 2002 to 2008. The new class of capitalists, dubbed the boli-bourgeois, was meant to provide a countervailing force to the generally anti-Chavez traditional business sector. Many non-Chavista businessmen were forced to leave the country or were accused of corruption by the government.8 Others embraced long-standing family, friendship, and professional relationships with prominent government officials or ruling party members, in order to curry favour with the regime. Second, the government embarked on a programme of state encroachment in the economy through company nationalisations and asset seizures. In 2007, the government nationalised projects in the Orinoco oil field, in which several multinational petroleum companies had majority 6

All Venezuelan economic statistics were obtained from Global Financial Data. Venezuelan average yearly inflation was 32% between 1984 and 1989, while annual real GDP growth averaged only 1.5%. 8 Examples include Guillermo Zuloaga, former owner of the TV station Globovisio´n, and the financier Nelson Mezerhame, who are both residing in the USA as of 2013. 7

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stakes. The government directed additional nationalisation efforts toward utilities, including the nationalisation of Electricidad de Caracas in 2007 (the country’s largest private electricity provider) and CANTV, the country’s major provider of telephone services. The government has also seized some three million hectares of land, and now controls a quarter of the banking sector. From 1999 to 2010, over 400 businesses have been nationalised, with the government now controlling 30% of GDP (Economist, 2010). Chavez’ hand-picked successor and former foreign minister, Nicolas Maduro, has vowed to continue the recently deceased leader’s legacy. 3.2. Motives and Justifications for Expropriation To better understand the link between accounting quality and the expropriation risk, we outline in this section the historical context for the Chavez government’s nationalisation campaign, which often saw firms targeted due to financial performance. As noted above, Bushman and Piotroski (2006) find a lower incidence of loss reporting among firms in countries with high expropriation risk and share of country-level output supplied by state-owned enterprises. They argue that this results from a desire to avoid ‘benign’ state intervention to correct problems at the firm, and is especially strong among Latin American countries like Venezuela that typically follow Napoleanic code law, versus Anglo-Saxon common law, with the former embodying ideologies more comfortable with centralised and activist government (Mahoney, 2001). This tradition of activist government has manifested itself among Latin American countries through the import substitution industrialisation model (ISI) that prevailed throughout the region from the 1930s to the 1980s. ISI was a development programme aimed at helping native industries thrive in a protected environment. The goal was to create industries capable of producing substitutes for expensive imports while simultaneously promoting industrial growth and the expansion of internal economies (Franko, 2007). It was characterised by high tariffs on imported goods, targeted lending to selective industries, and loose monetary policy. Often, it involved an expansion of the state into many areas of the economy, through nationalisations, firm buyouts, expropriations, direct subsidies to certain corporate groups, special credits, and heavy spending. This interventionist philosophy was shared by members of the Venezuelan military, many of whom advocated for a mixed economy model, in which key ‘strategic’ areas of the economy – such as utilities, energy, and materials – remained under state control. Venezuela prior to Chavez thus had a long-standing history of activist government, and Chavez upon his ascendance to the presidency espoused a mixed economy model. Corrales and Penfold (2011), however, suggest that state expansion under Chavez was deployed less so to target strategic sectors than to generate political gains, a motivation amplified after the failed 2002 coup attempt. Expropriations occurred mostly to court certain labour groups, who would be incentivised by promises of enhanced employment and reduced productivity demands post-nationalisation. As a tactic, the government would often encourage Chavistaconnected labour organisations to disrupt work and thus impair a firm’s financial performance, which the regime then used as justification for taking it over (Corrales & Penfold, 2011, ch. 3). More generally, the regime would bring on adverse business conditions for many sectors – through price controls in the context of inflation, exchange rate restrictions, labour unrest, onerous taxation, and inconsistent rules – which led to a rise in the cost of doing business. Companies responded by laying off workers or underutilising capacity, or both. The government would then use this outcome as an excuse for taking over the company. Ultimately, the state became the principal economic agent in a particular region or among a group of workers, which increased the regime’s co-optation capacity. And because of high oil prices (especially between 2002 and 2008), the state could afford to corner the private sector into underperformance, and thus generate demand for more state intervention.

Political Connections and Accounting Quality under High Expropriation Risk 7

Figure 1. Mentions in El Nacional of expropriation instances and threats (by justification). Notes: Figure 1 shows results, by year for 2003–2010, of articles in El Nacional mentioning either actual nationalisations or threats to nationalise, by the justification given for the expropriation. Search terms we used to screen for expropriation-related articles were the following: soberanı´a nacional, sectores estrate´gicos, soberanı´a alimentaria, especuladores, acaparamiento, remarcaje de precios, propiedad social,empresas del estado,empresas de produccio´n social, ganancias excesivas,expropiacio´n, expropiar, nacionalizacio´n, and, nacionalizar.

Confirming this, Figure 1 shows patterns in justifications over time for both actual and threatened nationalisations, based on news articles from El Nacional, one of Venezuela’s main newspapers. We identified nationalisation-related articles based on a set of search terms9 through Lexis-Nexis and recorded the justifications for expropriations that were noted in the article. We find that, indeed, the majority of expropriation justifications in Venezuela over our sample period related to concerns over performance and productivity, with a very large spike in news articles in 2009. However, as evidenced in Figure 1, public justifications for expropriation have also reflected concerns about firms’ excessive profitability, or relatedly, concerns over violations of price controls. Overall, justifications were most frequently related to firms’ performance levels, either criticising them for being too high or too low. After our sample period, the notion of targeting ‘strategic sectors’ became more prominent, though concerns over excessive profitability and underperformance continued. In Figure 2, we plot nationalisations by industry to identify whether there were trends in industries targeted for nationalisation. As expected, nationalisations have increased gradually over time, and there appears to be a spike in 2008 for the consumer staples, energy, and materials industries. The latter two industries represent sectors traditionally considered 9

Search terms were the following: soberanı´a nacional, sectores estrate´gicos, soberanı´a alimentaria, especuladores, acaparamiento, remarcaje de precios, propiedad social,empresas del estado,empresas de produccio´n social, ganancias excesivas,expropiacio´n, expropiar, nacionalizacio´n, and, nacionalizar. These translate as: national sovereignty, strategic sectors, food sovereignty, speculators, hoarding, price gouging, social property, state enterprises, social production companies, profiteering, expropriate, expropriation, nationalisation, and nationalise. We identified articles that described both actual and threatened nationalisations towards particular companies, sectors, or the economy as a whole.

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Figure 2. Mentions in El Nacional of expropriation instances and threats (by industry). Notes: Figure 2 shows results, by year for 2003–2010, of articles in El Nacional mentioning either actual nationalisations or threats to nationalise, by the Global Industrial Classification System (GICS) sector mentioned. Search terms we used to screen for expropriation-related articles were the following: soberanı´a nacional, sectores estrate´gicos, soberanı´a alimentaria, especuladores, acaparamiento, remarcaje de precios, propiedad social,empresas del estado,empresas de produccio´n social, ganancias excesivas,expropiacio´n, expropiar, nacionalizacio´n, and, nacionalizar .

‘strategic’ while the government targeted consumer staples firms later in our sample period as a means of addressing food shortages. The targeting of consumer staples firms and underperformance are related, as the government often assailed these firms for being ‘unproductive’. Thus, as in other economies characterised by heavy state involvement in the economy, Venezuelan expropriation was typically justified by reference to firm performance. The most frequent justification for expropriation was related to firm underperformance, consistent with the experience of code-law countries characterised by heavy state involvement. However, the government in many cases justified expropriation by reference to excessive profits, or relatedly, violation of price controls. This suggests that firms’ incentives in Venezuela were to avoid the appearance of low productivity and to mitigate overall volatility in earnings, so as not to appear either underperforming or excessively profitable. Since accounting reports are prominent indicators of firm performance, accounting information should be an important part of the information used by government actors in making their state intervention decisions.10 And while it is possible the Chavez government simply created pretexts for expropriations unrelated to firm performance, the extraordinary lengths it has undertaken to induce firm underperformance suggests that public justifications required some basis in actual performance indicators, possibly in order to avoid an electoral backlash.

10

As an example, the Venezuelan legislature in 2009 issued a law that required compensation to shareholders of nationalised energy firms equal to the accounting book value of equity.

Political Connections and Accounting Quality under High Expropriation Risk 9 3.3. Protecting Assets from Expropriation Under this strong threat of expropriation, we expect firms to employ various costly means of shielding assets from expropriation, which in Venezuela chiefly comes in the form of nationalisation, rather than through excessive taxation or fines. Our expectation is that politically connected firms Venezuelan firms will be marginally less concerned with expropriation, as their connections with high-level officials and ruling party members should lower expropriation risk, controlling for other characteristics of connected firms that might be associated with accounting quality.11 As a result, the intensity with which they pursue these costly means should be lower.

3.3.1. Earnings management and political connections One potential means of avoiding government expropriation is through earnings management. Overall, then, we predict that Venezuelan firms may have had strong incentives to engage in earnings management, in order to mitigate swings in profitability; optimally, firms would attempt to appear neither overly profitable nor poorly productive. However, we anticipate greater benefits to unconnected firms arising from this earnings management motivation, given their enhanced expropriation risk and the severe consequences of expropriation. Firms’ reporting relationships to governments are thus central to this part of the earnings management calculus. Firms can employ several specific forms of earnings management to avoid the appearance of low productivity and to mitigate earnings volatility. These include untimely loss recognition, managing towards positive earnings and year-over-year earnings increase targets, smoothing earnings, and increased use of accounting accruals. All are similar to earnings management techniques designed to affect capital providers’ perceptions of value and risk. Consistent with Bushman and Piotroski (2006), politically unconnected Venezuelan firms may be less willing to recognise losses in a timely manner, in order to avoid giving the government a pretext for ‘benign’ intervention. Timely loss recognition may thus be higher among politically connected Venezuelan firms. In order to avoid perceptions of low productivity, we also anticipate lower propensity for both small positive earnings and small earnings increases among connected firms. The importance of loss/no loss and earnings decrease/no decrease thresholds has been documented in studies examining firms’ reporting to investors (Burgstahler & Dichev, 1997; DeGeorge, Patel, & Zeckhauser, 1999). Thresholds may result from heuristic cutoffs investors rely on due to high information processing costs. Government actors should be similarly prone to heuristic cutoffs at zero changes in earnings or zero earnings in evaluating a firm’s productivity, given both high information processing costs and the absence of swift market discipline in correcting misperceptions among government actors. 11

The link between political connections and mitigated expropriation risk has been documented in prior literature. Li, Meng, Wang, and Zhou (2008) find that political connections allow Chinese firms to avoid paying ‘extralegal’ fees to government entities. Bodnurak and Massa (2011) find that family firms’ ability to provide political connections positively associated with institutional investor ownership, which they attribute in part to these firms’ lower government expropriation risk. We anticipate a similar relationship among Venezuelan firms. In a recent example, Guillermo Zuloaga, owner of Globovisio´n, a television station, recently agreed to sell Globovisio´n to a financial group known to be close to the Chavez regime. The sale was seen as being forced by the Chavez government, as Globovisio´n had been highly critical of the government, which had vowed to not renew the station’s broadcasting license. The new owner, Juan Domingo Cordero, is seen as a frontman in what would amount to a government takeover of the politically unconnected Globovisio´n (Rueda, 2013). In an additional example, the government’s takeover of several buildings in downtown Caracas was denounced by an opposition party member, who claimed ‘government intends to turn the center of the capital in a space only for its members, for members of the ruling party, into a space where only Chavistas can roam’ (El Mundo, 2010).

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Politically unconnected firms may also attempt to conceal from governments the impact of shocks to firms’ operating cash flows by accelerating the recognition of revenue, deferring expense recognition, or underreporting strong current performance to create reserves for future losses (Leuz et al., 2003). Overall, this should produce a lower incidence of earnings smoothing among connected Venezuelan firms. Finally, in mitigating earnings volatility, we should expect to see unconnected Venezuelan firms increasing their use of accounting accruals. Countering these government reporting benefits for unconnected firms are, potentially, connected firms’ greater capital market and contracting benefits, and their lower relative costs, of earnings management. As Chaney et al. (2011) argue, politically connected firm insiders may manage earnings to obscure or delay reporting any private gains they enjoy from their relationships with government officials and legislators; this may manifest itself in the form of higher accruals for connected firms. Connected firms may also be shielded from legal or regulatory costs of earnings management (though not capital market penalties), whereas unconnected firms may bear the full brunt of government scrutiny upon revelation of accounting improprieties. All accounting quality measures may be then worse among connected firms. Connected firms may also have access to lower-cost financing (which we find evidence of in Section 4.1), which may blunt incentives to engage in earnings management for capital market or contracting purposes. They may also enjoy greater and more stable growth opportunities, due to access to lucrative government contracts or to the Chavez government’s active intervention to enhance their competitive positioning within their industries. Given the importance placed on meeting earnings-related thresholds for capital market or contracting purposes, we may see lower propensities for small positive earnings and small earnings differences from connected firms as a result. We may also see lower evidence of earnings smoothing, given investors’ preferences for smooth earnings paths, and consequently, lower need among connected firms to cater to investors’ preferences. The effect of access to cheaper financing may have ambiguous consequences for timely loss recognition, however: Though connected firms may experience fewer expropriation-related consequences as a result of recognising losses, unconnected firms may be compelled to adopt more conservative accounting policies by the more demanding private, unsubsidised credit market. Unconnected firms may additionally have greater demands placed upon them by private debt capital providers, owing to their potentially lower level and heightened volatility of cash flows. To control for these competing mechanisms for mitigated earnings management behaviour among politically connected firms, we include controls for capital market and debt contracting earnings management incentives. Finally, we examine whether the effect of political connections on accounting quality is greater among firms that are possibly at greater risk of expropriation. Though all firms in Venezuela faced greater risk of expropriation relative to firms operating in other countries, and a very wide range of industries were targeted, we should see an enhanced effect of political connections on accounting quality among higher expropriation risk firms, if expropriation avoidance is chiefly driving our results. 3.3.2. Other means of shielding assets While the focus of this study is on earnings management as a means of mitigating expropriation risk, we also describe (and in our econometric models, control for) other means by which firms can shield assets. First, Venezuelan firms may respond to the threat of nationalisation by increasing leverage ratios, as debt contracts generally have been honoured by the government, even as equity stakes have been nationalised (Businesswire, 2010). This should make

Political Connections and Accounting Quality under High Expropriation Risk 11 debt capital relatively less expensive than external equity financing in Venezuela. Capital structure may therefore also serve as a tool for increasing firm value under the threat of expropriation. While capital structure choices are not the focus of our research, we include leverage ratios in all subsequent econometric tests to account for this possibility.12 To preserve value for investors, firms may attempt to divest corporate assets and return capital to shareholders. Doing so shifts assets from corporation accounts to more dispersed shareholders, which may make assets more difficult to seize. However, it may be difficult to do rapidly and furtively enough to thwart government expropriation. Nonetheless, we include measures of sales growth and the net change in shareholders’ equity in all empirical tests to control for this shareholder value protection strategy. Finally, firms may attempt to ship more liquid assets abroad to keep them out of the purview of government security forces. However, strictly enforced exchange controls have been in place in Venezuela since 2003, making it difficult for firms to readily send assets abroad in an attempt to shield them from government takeover. In the next section we detail how we construct our measure of political connections. We then detail our methodology for measuring earnings management, accounting quality, and expropriation risk.

4.

Data and Methodology

We gathered financial data for Venezuelan public industrial companies from several sources. Our primary source was Thomson One Banker, which provided summarised financial information. When data were not available, we utilised the Mint Global database from Bureau van Dijk. In many cases, we referred to copies of the firms’ originally-issued financial reports, which we obtained from Thomson One Banker or purchased from the Caracas Stock Exchange. In all instances, we obtained financial numbers that were expressed in constant Venezuelan bolivars as of each fiscal year-end date.13 In computations where prior years’ numbers were also required, we adjusted income statement and balance-sheet variables by the appropriate inflation factor based on changes in the Venezuelan Retail Consumer Price Index (CPI), which is available from Global Financial Data.14 We also obtained information on external auditors through inspection of these financial reports. We excluded a small number of firm-years in which financials were reported under International Financial Reporting Standards. The dataset contains information on 29 companies that had been publicly traded at some point between 2000 and 2008, though tests with more restrictive data requirements involve fewer firms. We choose to begin our sample in 2000 because that is when the ‘enabling laws’ were passed that allowed for expropriations by decree. Results are unchanged if we restrict our sample to firm-years beginning in 2002, when Chavez’ campaign against the business sector intensified after the unsuccessful coup attempt. 12

Debt capital may be still remain expensive, as connected firms may enjoy preferential access to debt financing (Khwaja & Mian, 2005), which may raise debt financing costs for connected firms if the pool of loanable funds is scarce. However, our expectation is that external equity capital will be relatively much more expensive, and this may prompt firms to shift leverage ratios upward to increase firm value. 13 Venezuelan accounting rules require firms to revalue non-monetary assets and liabilities, as well as associated expenses like cost of sales and depreciation, based on movements in local price indices. 14 Specifically, all t21 numbers are multiplied by the ratio of the Venezuelan CPI as of the year t fiscal month end to the Venezuelan CPI as of the t21 fiscal month. For stand-alone size measures such as total assets, the number was deflated using the Venezuelan CPI as of December 1997 as a base.

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4.1. Political Connections To construct our measure of political connections, we relied primarily on interviews with 13 highly-placed Venezuelan businesspeople, one of whom served as the point of contact and helped gather information from the other interviewees. The interviews were conducted in mid-2009.15 The respondents represented a range of industrial fields in Venezuela, and their minimum level of professional experience was 25 years. Venezuelan business elites operate in a dense and small network of school, familial, social club, and professional associations ties. Knowledge of different firms’ political connectedness disseminates readily within this dense network, though it often remains non-public. The executives were generally sympathetic to the anti-Chavez opposition, though this is not an uncommon characteristic of non-bolibourgeoisie Venezuelan businessmen. Interviewees made judgments as to whether three sets of parties – top-level executives, board members, or large (i.e. 5% or greater shareownership) blockholders, or any of these parties’ relatives16 – were deemed to have no known connections or be strongly opposed to the Chavez government; or instead to have a weak, mild, or strong relationships with high-level government officials, ruling party members, or bolibourgeoisie businessmen over the 2000 – 2008 period. Because most Venezuelan firms in our sample are family-controlled, executives, board members, and large blockholders are typically either the same parties, or are related through family ties. Interviewees were asked to offer narrative justifications for why they deemed firms to be strongly opposed to the government or to have weak, mild, or strong connections. Executives generally only were able to rate firms in industries they were knowledgeable of; as a result, it was not feasible to calculate measures of inter-rater reliability.17 As noted above, information on Venezuelan political connections are typically non-public. We nonetheless supplemented these executive interviews by searching Venezuelan news media sources, both private and government-sponsored,18 for stories on sample firms, their high-level executives, board members, and large blockholders. We used evidence from these stories to, when possible, corroborate the executive determinations, as well as to ascertain the existence and strength of connections when executives had no knowledge of political connections.19 The political connection rating of each company, along with the years in which data are available for some of our empirical tests, is listed in Appendix 1. Although the executives rated some firm-years as weakly connected, none of those firm-years met our data requirements. Because of the potential subjectivity involved in assessing the strength of the political connection, and because of the relatively small number of firm-years with strong connections, we include mildly and strongly connected firm-years into a single category. The dummy variable CONN 15

Because of the highly sensitive nature of the data, we were asked to withhold the names and identifying details of the businessmen we interviewed. Interviewees feared government harassment or, in the extreme, exile from Venezuela if they were associated with political connection assessments. 16 Relatives included spouses, children, siblings, or cousins. 17 The measures of political connection used in Fisman’s (2001) seminal study also relied on subjective assessments made by consultants from the Castle Group, a political consultancy operating in Indonesia. We were not able to identify the number of consultants who contributed to the assessments used by Fisman, but at present, CastleAsia, the successor firm to the Castle Group, lists six advisory board members. 18 Venezuelan news media sources we utilised include Globovision, Union Radio, El Universal, Agencia Bolivariana de Noticias, Radio Nacional de Venezuela, Venezolana de Televisio´n, El Nacional, El Mundo, Noticiero Digital, Noticias 24, and Aporrea. In addition to news stories, we also searched for interviews with government officials among these sources. 19 To evaluate the existence of connections based on these news sources, we relied on the narrative-defined categories listed in Table 1, and we also searched for stories on Venezuelan government officials either (a) criticising the company, its executives, its board members, or its large blockholders or (b) threatening the company with nationalisation.

Political Connections and Accounting Quality under High Expropriation Risk 13 used in all tests is equal to one if a firm is either strongly or mildly connected, and equal to zero if a firm has no connections or is strongly opposed to the government. Analysis of the executives’ political connection narratives revealed that interviewees’ evidence for political connections, or lack of political connections, fell into a small number of categories, whose frequencies are listed in Table 1. Most commonly, friendship relationships with prominent government and party officials lead to political connections. Respondents also cited as evidence whether individuals associated with the firm ‘did business with’ or ‘had financial dealings with’ the government, government officials, or those closely connected with the government. In all cases, evidence came from only one of the categories listed. Our measure does rely on potentially idiosyncratic assessments of political connections and overall connection strength. However, we believe that in the Venezuelan context, these assessments should produce a more accurate gauge of connection strength than measures relying on publicly available sources, which are severely limited in Venezuela. For example, Bushman et al. (2004) find that Venezuela ranks in the bottom 15% of countries ranked on measures of corporate financial and governance transparency. Moreover, Faccio (2006), in her landmark study of political connections, finds no connections (based on publicly available sources) among the 18 Venezuelan firms in her sample, a finding somewhat at odds with our prior reckoning of political connections’ prevalence in Venezuela.20 However, given potential concerns about the measure’s reliability, in Section 5.1, we assess whether CONN is significantly associated with potential benefits of political connectedness, as a partial validation of our connection measure. Khwaja and Mian (2005) find that connected firms in Pakistan enjoy preferential access to debt financing relative to unconnected firms, so we first assess whether connected firms enjoy lower debt financing costs, controlling for other determinants of these costs. Faccio (2010), additionally, finds that certain connected firms enjoy lower effective tax rates. We therefore assess whether, all else equal, connected firms enjoy lower tax rates than unconnected firms. Finally, although most of the information used to measure political connections is not publicly available, an example of the information used to measure connection strength can be found in International Briquettes Holding (IBH) and Siderurgica Venezolana Sivensa (SVS), both sample firms involved in the metals and mining sector. IBH’s and SVS’ long-standing chairman of the board is Oscar Augusto Machado Koeneke, and Henrique Machado Zuloaga is a long-standing board member of both firms. Oscar Augusto Machado Koeneke and Henrique Machado Zuloaga are the cousin and father, respectively, of Maria Corina Machado, former president of Su´mate, a non-government organisation that has monitored Venezuelan elections since 2004. Su´mate has at time accused the government of election fraud, and in 2005 Venezuela’s interior minister publicly accused Ms. Machado of being an agent of the US Central Intelligence Agency. In 2009 the government ultimately announced a partial nationalisation of both IBH and SVS. For these reasons, we rated IBH and SVS as being strongly opposed to the government for sample years after 2003. 4.2. Earnings Management Measures As described in Section 3.2, we anticipate several forms of earnings management among Venezuelan firms. Given the paucity of usable price data for firms in our sample, we focus on earnings 20

Faccio (2010) suggests that political connection measures derived from public sources may represent more durable ties, as opposed to those related to more ephemeral campaign contributions. While the greater durability of connection measures derived from publicly available sources may hold true in general, we believe it is less likely so in the Venezuelan context, where these connections we identify through non-public sources represent long-standing friendship, familial, and professional ties.

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Political connection categories

Frequency among connected

Interviewees’ evidence of a lack of political connection in firm-year

Frequency among unconnected

Company is nationalised or is partially owned by Venezuelan government in that firm-year

0.02

0.06

Company, director, officer, or blockholder (or relatives thereof) was awarded government contracts, has ‘financial dealings’ with the government, or ‘does business’ with the government Directors, officers, or blockholders (or relatives thereof) have friendship relationships with high-level government officials or members of the (Chavez-led) United Socialist Party of Venezuela Chavez-controlled media favourably showcase the firm or executives

0.18

Directors, officers, or blockholders (or relatives thereof) are part of antigovernment civil society organisations Directors or officers (or a close relation) have criticised the government in media

No evidence for political connection. Treated as unconnected

0.93

Interviewees’ evidence of a political connection in firm-year

0.49

0.01

0.14

Note: Table 1 lists frequencies of categories of evidence for judging the presence and strength of political connections among the 13 Venezuelan executive interviewees.

management measures that can be derived without reference to stock price data. Nearly all tests below include measures of equity issuance, debt issuance, and leverage to control for capital market and debt contracting incentives to manage earnings; the impact of CONN will then better isolate government reporting incentives for earnings management. In regression tests below, we cluster standard errors at the firm level using the Rogers (1993) cluster-robust standard error estimator. 4.2.1. Earnings smoothing We assess whether connected firms are more or less likely to engage in earnings smoothing behaviour. As Lang, Raedy, and Yetman (2003) propose, all else equal, the more variable net income is, the less likely is it that firms are smoothing earnings. We therefore assess whether the standard deviation of changes in net income is greater or lower for connected firms relative to unconnected firms. Since changes in net income may vary for reasons other than income smoothing, we take the standard deviation of residuals from a model explaining net income changes. We use the following regression specification to estimate residuals. These utilise the same controls as those in Lang et al. (2003): DNIit = a + b1 SIZEit + b2 GROWTHit + b3 LEVit + b4 TURNOVERit + b5 CFOit

+ b6 LOCALAUDit + b7 EQUITY ISSit + b8 DEBT ISSit + b9 XLISTit + 1it .

(1)

Political Connections and Accounting Quality under High Expropriation Risk 15 DNI is the change in net income divided by prior-period total assets. GROWTH is salest minus salest21 divided by salest21. SIZE is the logarithm of total assetst. LEV is long-term debtt divided by total assetst. TURNOVER is salest divided by total assetst21. CFO is net operating cash flows from the statement of cash flows in year t divided by total assetst21. LOCALAUD is equal to one if the firm’s auditors in year t were not affiliated with international accounting firms, and zero otherwise. EQUITY_ISS is equal to the per cent change in total shareholders’ equity. DEBT_ISS is equal to the per cent change in total liabilities. XLIST is a dummy variable for cross-listing status within a firm-year. All financial variables are inflation-adjusted, as described in the introduction to Section 4, and winsorised at the 1% level.21 Consistent with Lang et al. (2003) and Barth, Landsman, and Lang (2008), we include industry dummies in the regression, basing ours on more aggregated GICS economic sector codes, given our relatively smaller sample size. We then compare the standard deviation of these residuals for connected and unconnected firms. While standard tests do exist for comparing differences in standard deviations across samples, they do not control for potential error dependencies among observations; in Appendix 2, we describe a bootstrapping procedure for testing for the significance of these differences. As Lang et al. (2003) also note, the variability of net income changes can also be driven by cash flow changes. To control for cash flow variability, we also compare the ratios of the standard deviations of changes in net income and the standard deviation of changes in cash flows. To generate residuals for changes in cash flows we use the same specification as above. We assess differences in the ratios of standard deviations of net income and cash flow changes between connected and unconnected firms, using the bootstrap procedure described in Appendix 2. Because tests of differences in ratios may be skewed by low denominators, we also compare compute the difference between the standard deviation of changes in (residual) net income and the standard deviation of changes in (residual) cash flows between connected and unconnected firms. Finally, we assess whether there are differences in the correlations between accounting accruals (the difference between accounting net income and cash flows) and cash flows. While we expect some degree of negative correlation between accruals and cash flows, as firms employ the accounting revenue recognition and matching principles to smooth periodto-period fluctuations in working capital, large differences in this correlation may suggest an excessive degree of smoothing. We compute accruals as equal to the difference between net income and cash flow from operations for each year. To control for other determinants of accruals and cash flow levels, we regress each variable on the same set of controls described above for changes in net income (though we exclude CFO in the cash flow regression). We then compute the correlation between the accrual and cash flow residuals for both connected and unconnected firms and compare this correlation between the two groups using the bootstrap procedure described in Appendix 2.

4.2.2. Managing towards targets Next, we test whether connected firms manage towards earnings targets. We measure small positive net income (SMALLPOS) as equal to 1 for observations where net income divided 21

The sales growth variable was subject to severe outliers, so we winsorised instances of over 200% sales growth to the 95th percentile of sales growth. Results were unchanged when we only winsorised at the 1% level. Some firm-years had zero revenue, owing to income primarily derived from equity method investments. When prior year revenue was zero, we recorded sales growth as equalling zero.

16

G. Batta et al.

by prior-period assets is between 0 and 0.005, and zero otherwise.22 DeGeorge et al. (1999) also find evidence that firms attempt to avoid year-over-year earnings decreases. We test for the importance of this threshold by creating a variable called SMALLDIFF, which is equal to 1 for observations where the difference between t and t21 net income, scaled by total assets as of the end of year t21, is between 0 and 0.005, and zero otherwise.23 To control for other determinants of small positive return on assets and small return on asset differences, we use the following ordinary least squares (OLS) regression model, consistent with Barth et al. (2008):24 INDit = a + b1 CONNit + b2 SIZEit + b3 GROWTHit + b4 LEVit +5 TURNOVERit +b6 CFOit + b7 LOCALAUDit + b8 EQUITY ISSit + b9 DEBT ISSit

(2)

+b10 XLISTit + 1it , where INDit ¼ SMALLPOSit or SMALLDIFFit, when appropriate. All financial variables are inflation-adjusted, as described in the introduction to Section 4. 4.2.3. Timely loss recognition We also assess whether there are differences in timely loss recognition between politically connected and unconnected firms. Ball and Shivakumar (2005) find that public UK firms have a higher positive correlation between contemporaneous negative cash flows and accruals relative to private UK firms, which Ball and Shivakumar claim face lower demands for financial reporting quality. We first use the Ball and Shivakumar (2005) model of timely loss recognition to assess differences in timely loss recognition. Specifically, we adopt the following regression framework: ACC it = a + b1 CFOit + b2 NEGCFOit + b3 CFOit × NEGCFOit + b4 CONNit + b5 CONNit × CFOit + b6 CONNit × NEGCFOit + b7 CONNit × CFOit

(3)

× NEGCFOit + 1it . NEGCFOit is an indicator variable equal to one when CFOit , 0, and equal to zero otherwise. To assess whether connected firms have asymmetrically stronger timely loss recognition, we assess whether b7 is greater than zero. All financial variables are inflation-adjusted, as described in the introduction to Section 4. Following Lang et al. (2003), we also measure timely loss recognition by assessing whether there are differences in large negative earnings, which we define as having a net income divided 22

We adopt a 0.005 threshold for small positive earnings, versus 0.01 as in Lang et al. (2003). Overall sample mean return on assets (net income divided by beginning-of-period assets, as per Section 4) is quite low, at 0.007, with a median of 0.011. Around 45% of the sample has return on assets plus or minus 0.03 of the mean of 0.01. Given that the distribution is fairly tight and has a mean at exactly at the 0.01 used by Lang et al. (2003), adopting the 0.01 threshold may be a poor indicator of actual earnings management behaviour. When we use this less stringent 0.01 threshold, our results on small positive earnings are similar. 23 For consistency with our small positive earnings measure, we adopt a 0.005 threshold for earnings increases. Results are similar using the less stringent 0.01 threshold, with 14% of firm-years having small earnings increases, versus 9% using the 0.005 threshold. 24 Barth et al. (2008), following Lang, Raedy, and Wilson (2006), use an OLS framework because logit models are prone to severe heteroskedasticity problems in small samples. Lang et al. (2006) use SMALLPOS as a right-hand side variable, as their controls are meant to control for determinants of the cross-listing decision.

Political Connections and Accounting Quality under High Expropriation Risk 17 by prior-period assets less than 20.20. We also include the same set of control variables as in our small positive earnings and small earnings increase regressions. 4.2.4. Abnormal accruals Finally, one chief means by which firms can achieve less timely loss recognition, smooth earnings, avoid losses, and avoid decreases is through the use of accounting accruals. We therefore use the absolute value of abnormal accruals as a general earnings management proxy, consistent with Chaney et al. (2011) and Hribar and Nichols (2007). We run the following basic regression for each firm-year i, using all other Venezuelan firms with available data to estimate the regression, based on a cross-sectional version of the Jones (1991) model of Dechow, Sloan, and Sweeney (1995): ACCnt = a + b1



 1 + b2 (DSalesnt ) + b3 PPEnt + 1nt . ASSETSnt

(4)

ACCit is the difference between net income and operating cash flows from the cash flow statement in year t for firm i, scaled by prior-period total assets; ASSETSit is total assets; DSalesit is the yearly sales difference, scaled by prior-period total assets; and PPEit is net property, plant, and equipment, scaled by prior-period total assets. All financial variables are inflation-adjusted, as described in the introduction to Section 4. To generate abnormal accruals, we then subtract the change in accounts receivable from the statement of cash flows (scaled by prior-period assets) from the change in sales; the difference between actual accruals and those predicted by the model are abnormal accruals. Consistent with the performance-matching procedure of Kothari, Leone, and Wasley (2005), we then subtract the abnormal accruals of the firm with the closest-matched lagged return on assets, which is defined as net income over prior-period assets within each fiscal year, to generate performance-matched discretionary accruals. Hribar and Nichols note that the absolute value of accruals, as estimated from discretionary accruals models, can be high merely due to high operating volatility. To control for operating volatility, we estimate the following model, as in Chaney et al. (2011) and as advocated by Hribar and Nichols (2007): ABS(ACCit ) =a + b1 CONNit + b2 SIZEit + b3 LEVit + b4 (CFO SDit ) + b5 (GROWTH SDit ) + b6 (TURNOVER SDit ) + b7 GROWTHit

(5)

+ b8 LOCALAUDit + b9 EQUITY ISS + b10 DEBT ISS + b11 XLIST + 1it . ABS(ACC) is the absolute value of abnormal accruals. CFO SD is the trailing 5-year standard deviation of CFO, GROWTH SD is the trailing 5-year standard deviation of GROWTH, and TURNOVER SD is the trailing 5-year standard deviation of TURNOVER. In our study, we also include EQUITY_ISS and DEBT_ISS to control for capital market incentives to manage earnings. To assess whether there are differences in earnings quality between politically connected and unconnected firms, we test for whether b1 is different from zero.

4.3. Expropriation Risk As noted in Section 3.2, the Chavez government’s underlying motivation for expropriation was typically to consolidate its power, often by coopting workers at expropriated firms. We posit then that the government would prefer to maximise the political advantage it gains from each

18

G. Batta et al.

nationalisation. This suggests non-managerial employee headcount as a possible measure of expropriation risk, as a greater number of non-managerial employees represents a larger number of ‘votes’ that could be garnered through expropriation, or workers to be coopted into the regime. Unlike US publicly traded firms, Venezuelan firms are not required to disclose employee headcount, managerial or otherwise. However, controlling for firm size in our regressions, proxies for firms’ labour intensity may then be associated with variation in headcount. We proxy for labour intensity by taking the ratio of the reported book value of total assets to each firm’s net PPE. Reasonably, where net PPE composes a larger percentage of total assets, reliance on fixed capital, rather than labour, to produce value should be greater.25 Confirming this, in US industrial firm data within Compustat over our sample period, the correlation between the log of total employees to net PPE and log of total assets to net PPE is quite high, at 0.69.26 In Table 2, we present the top and bottom five GICS industries in Compustat North America, ranked on the median of the log of total assets to net PPE over 2000 – 2008, along with their ranks for median employees to net PPE. The bottom five correspond to industries – such as utilities, energy, and marine transport – that intuitively represent more capital-intensive firms, and also have very low ranks of employee to net PPE. The top five correspond to industries – such as software and IT services – that intuitively represent more labour intensive firms, and also generally have high ranks of employee to net PPE.27 As noted in Section 3.2, firms in the consumer staples, energy, and materials industries experienced a spike in expropriations and threats in 2008, which may be related to the government’s shift to targeting ‘strategic’ industries. We find that all firms deemed high expropriation risk using our proxy for labour intensity were in these sectors in 2008. 5.

Results

5.1. Summary Statistics and Political Connection Measure Validation Table 3 provides summary statistics on characteristics of connected and unconnected firms, as well as univariate correlations among variables. Panel A shows that connected firms have lower absolute abnormal accruals, lower incidence of small positive earnings, a lower incidence of earnings decreases, and a higher incidence of extreme losses. Panel B shows that connected and unconnected firms are about the same size and have about the same degree of financial leverage. They also have generally lower sales growth but higher equity issuance, which provide ambiguous signals on divestment as a channel for expropriation avoidance. Connected firms are slightly less likely to rely on a local versus an international auditor. They also have a 25

A higher ratio of total assets to net PPE may also simply represent cash-rich firms (where most of assets are composed of cash and short-term investments, rather than PPE) that are targeted by the government. This does not appear to be among the government’s espoused expropriation motivations; however, results are similar if we subtract cash and short-term investments from total assets in calculating the ratio. 26 McKay and Phillips (2005) use employees to net PPE as a measure of labour- versus capital-intensity. 27 In untabulated regressions, we evaluate, as a benchmark, whether the assets to net PPE ratio for industrial firms in Compustat North America from 2000 to 2008 is associated with the likelihood of small positive earnings; the likelihood of small earnings increases; and the absolute value of abnormal accruals. We adopt the same empirical specification and controls for our tests above, though we exclude XLIST and LOCALAUD, and adopt a logit specification for binary dependent variables. Among US firms, we find that the logarithm of the ratio of assets to net PPE is actually negatively and significantly associated with both small positive earnings and small earnings increases. Estimating abnormal accruals at the GICS group and fiscal year level with at least 10 observations, we find no significant association with the absolute value of performance-matched abnormal accruals. Results are identical if we restrict tests to only those industries present among Venezuelan sample firms, or substitute in the logarithm of the ratio of employee headcount to net PPE.

Political Connections and Accounting Quality under High Expropriation Risk 19 Table 2.

Ranking of log of total assets to net property, plant, and equipment

Compustat North American firms Median industry Ln (assets/net PPE)

Rank of employees/PPE (out of 61 industries)

Top 5 Software Data processing services Biotechnology Internet software and services IT services

2.820 2.598 2.576 2.572 2.570

4 2 34 9 1

Bottom 5 Water utilities Oil, gas and consumable fuels Marine Electric utilities Gas utilities

0.204 0.313 0.376 0.407 0.422

58 61 55 59 56

Industry

Notes: Table 2 shows rankings of GICS industries in Compustat North America by the median log of total assets to net PPE over 2000–2008. The industry’s associated ranking of median employees to net PPE over the sample period is also shown.

somewhat higher ratio of the logarithm of total assets to net PPE, our proxy for the labour-tocapital ratio (and, ultimately, expropriation risk). Panel C provides univariate correlations. The absolute value of abnormal accruals appears to be highly positively correlated with asset turnover, and highly negatively correlated with profitability, equity issuance, and leverage. The assets to net PPE measure is highly correlated with nearly all variables, though only slightly negatively correlated with the absolute value of abnormal accruals Table 4 shows results of our validation tests for our political connection measure. To see if political connections are tied to measureable political connection benefits, we first assess whether CONN is negatively associated with debt financing costs. We impute interest rates on debt by taking the ratio of reported interest expense from firms’ income statements to average total liabilities, excluding firms with imputed interest rates greater than one. Results are robust to simply winsorising at the 1% level, rather than eliminating these observations. We control for additional standard determinants of debt financing costs, including firm size, operating profitability, leverage, and firms’ economic sectors.28 Column 1 shows that connected firms enjoy debt financing costs on average 270 basis points lower than unconnected firms, significant on a one-sided basis. When we include all control variables in column 2, we find that connected firms enjoy debt financing costs 200 basis points lower, though this result is not statistically significant. Two hundred basis points is high, but not unreasonable in the Venezuela context, where nominal lending rates averaged close to 2300 basis points over the 2000 – 2008 period (ECLAC 2009).

28 For all measure validation tests, we employ White (1980) robust standard errors, rather than clustered standard errors, as F-statistics cannot otherwise be produced for models with all control variables included. Although we rely on potentially manipulated financial reports to measure interest rates, we note that methods to reduce reported interest expense, such as utilising hybrid or off-balance-sheet debt, will reduce both reported interest expense and reported debt; imputed interest rates should therefore still accurately measure financing costs for at least on-balance-sheet debt

20 G. Batta et al.

Table 3.

Summary statistics and univariate correlations

Panel A – Dependent variables SMALLPOS Connected firms Mean 0.043 p50 Sd

SMALLDIFF

ROALOW ABS(ACC)

SMALLPOS SMALLDIFF 0.157

ROALOW

Unconnected firms 0.111

0.043

0.065

0.083 0.063 0.090

Mean p50 sd

LIABS TO ASSETS

SIZE

ROA

GROWTH TURN OVER

0.250 0.242 0.191

4.728 3.761 2.330

0.001 0.018 0.105

LIABS TO ASSETS

SIZE

ROA

0.367 0.330 0.349

4.640 4.159 1.983

0.010 0.011 0.105

0.046

ABS(ACC) 0.119 0.088 0.112

Panel B – Control variables LEV Connected firms Mean 0.076 p50 0.035 Sd 0.130 LEV Unconnected firms Mean 0.080 p50 0.030 Sd 0.129

20.041 20.016 0.764

CFO

EQUITY _ISS

DEBT _ISS

LN(A/ PPE)

0.036 0.021 0.066

0.087 0.027 0.551

0.449 20.047 2.334

2.391 0.754 3.236

CFO

EQUITY _ISS

DEBT _ISS

LN(A/ PPE)

0.621 0.499 0.542

0.066 0.032 0.102

0.003 20.055 0.851

0.368 0.037 1.480

1.807 0.657 3.208

EQUITY _ISS

DEBT _ISS

ROA

1.000 0.151

1.000

0.447 0.462 0.432

GROWTH TURN OVER 0.085 0.042 0.487

Panel C – Univariate correlations

LEV SIZE GROWTH TURNOVER CFO EQUITY_ISS DEBT_ISS

LEV

SIZE

GROWTH

TURN OVER

CFO

1.000 0.109 0.098 0.184 (0.012) (0.034) (0.007)

1.000 0.182 (0.237) 0.134 0.042 (0.031)

1.000 0.333 0.071 0.029 0.183

1.000 0.292 0.107 0.150

1.000 (0.054) (0.068)

ABS (ACC)

LN (A/ PPE)

LOCALAUD XLIST 0.087

0.109

LOCALAUD XLIST 0.120

0.130

(0.178) 0.052 (0.449)

0.076 (0.167) 0.110

0.330 0.056 (0.262)

0.281 0.241 (0.415)

0.385 0.048 (0.212)

(0.090) (0.121) (0.075)

0.050 (0.034) 0.099

1.000 (0.036) 0.034

1.000 (0.049)

1.000

Notes: Table 3, Panel A, provides summary statistics on dependent variables used in regression tests. SMALLPOS is equal to one when net incomet over assetst is between 0.00 and 0.005, and zero otherwise. SMALLDIFF is equal to one when the difference in net income between year t and year t21, scaled by total assets from year t21, is between 0.00 and 0.005. ROALOW is equal to one when the difference in net income between year t and year t21, scaled by total assets from year t21, is less than 20.20. Both net incomet21 and assetst21 are multiplied by ratio of the level of the Venezuelan CPI at the end of year t to the level of the CPI at the end of year t21. ABS(ACC) is performance-matched abnormal accruals using a cross-sectional version ofthe modified-Jones model of Dechow et al. (1995), computed from the following regression, run at the fiscal-year level among Venezuelan firms:  1 + b2 (DSalesit ) + b3 PPEit + 1it . ACCRUALSit = a + b1 ASSETSit ACCRUALSit is the difference between net income and operating cash flows from the cash flow statement in year t for firm i, scaled by prior-period total assets; ASSETSit is total assets; DSalesit is the yearly sales difference, scaled by prior-period total assets; and PPEit is net property, plant, and equipment, scaled by prior-period total assets. Abnormal accruals are computed as the difference between accruals in year t and predicted accruals based on the fiscal-year regression run in year t, after first subtracting the change in accounts receivable from the statement of cash flows (scaled by prior-period assets) from DSalesit before predicting accruals from the model; performance-matched accruals are computed by subtracting the abnormal accruals of the firm with the closest-matched return on assets among Venezuelan firms that year. ABS(ACC) is the absolute value of these performance-matched accruals. Panel B provides summary statistics on control variables used in the regression and other variables. LEV is long-term debtt divided by total assetst. LIABS TO ASSETS is total liabilitiest divided by total assetst. SIZE is the logarithm of total assetst. ROA is net incomet over total assetst21. GROWTH is salest minus salest21 divided by salest21. TURNOVER is salest divided by total assetst21. CFO is net operating cash flows from the statement of cash flows in year t divided by total assetst21. EQUITY_ISS is equal to the per cent change in total shareholders’ equity. DEBT_ISS is equal to the per cent change in total liabilities. LN(A/PPE) is the log of the ratio of total assets to net property, plant, and equipment. LOCALAUD is equal to one if the firm’s auditors in year t were not affiliated with an international accounting firm, and zero otherwise. XLIST is a dummy for cross-listing status. Panel C provides univariate correlations among ABS(ACC) and control variables. For Venezuelan firms’ SIZE and (1/ASSETS), total assets are deflated using the Venezuelan CPI for December 1997 as a base. For ACC, DSales, DAR, PPE, LEV, NOPAT, TURNOVER, CFO, EQUITY_ISS and DEBT_ISS, all t21 numbers for Venezuelan firms are multiplied by ratio of the level of the Venezuelan CPI at the end of year t to the level of the CPI at the end of year t21. All variables are winsorised at the 1% level.

Political Connections and Accounting Quality under High Expropriation Risk 21

ROA ABS(ACC) LN(A/PPE)

22

G. Batta et al. Table 4.

Political connection measure validation

INTEREST RATE CONNECTED

+

20.027 (21.56)

0.100∗∗∗ (10.05)

20.020 (21.23) 0.193∗ (1.66) 0.065 (1.23) 20.001 (20.24) 0.084∗∗∗ (3.19)

Yes

Yes

151 0.054 0.021 18.54∗∗∗

151 0.140 0.092 12.52∗∗∗

NOPAT LEV SIZE CONSTANT Sector dummies? Observations R2 Adj.-R2 F-Statistic

TAX RATE CONNECTED

+

0.188∗∗∗ (3.51)

20.101+ (21.50) 20.023+ (21.45) 20.70∗∗∗ (25.19) 20.878∗∗∗ (22.63) 0.504∗∗∗ (4.68)

Yes

Yes

20.126 (21.61)

PRETAXINC NEGPRETAX SIZE CONSTANT

Observations R2 Adj.-R2 F-Statistic

143 0.126 0.094 2.472∗∗

143 0.347 0.308 6.926∗∗∗

Notes: Table 4 gives results of validation tests for our political connection measure. INTEREST RATE is the ratio of interest expense to average total liabilities. TAXRATE is income tax expense less tax benefits divided by pretax income. NOPAT is net income plus interest expense multiplied by (120.34) over total assetst21. PRETAXINC is pretax income over total assetst21. NEGPRETAX equals one if pretax income is below zero, and equal to zero otherwise. GICS economic sector dummy variables are included where noted. All other variables are defined as in Table 3. All t21 numbers are multiplied by ratio of the level of the Venezuelan CPI at the end of fiscal year t to the level of the CPI at the end of fiscal year t21. Observations with INTEREST RATE greater than one are excluded from the regression. All other numerical variables are winsorised at the 1% level. Standard errors are heteroskedastic-robust. + Statistical significance below the 10% level on a one-sided basis. ∗∗ Statistical significance below the 5% level on a two-side basis. ∗∗∗ Statistical significance below the 1% level.

Table 4, columns 3 and 4, shows results for tax rates and political connections. We measure tax rates by taking the ratio of reported tax expense less tax benefits, divided by pretax income.29 To control for other determinants of tax rates, we include pretax profitability (given tax rate progressivity), size (since larger firms may attract tax authorities’ scrutiny), and sector (since certain sectors may attract special tax rates or benefits). We also include a dummy variable for negative pretax income, rather than exclude these firms as in Faccio (2010): Since Venezuelan GAAP allows recording of deferred tax assets and liabilities, the ratio of tax benefits to pretax loss may provide information either on tax rates or, if pretax loss firms nonetheless have a tax provision, it may reveal low expectations of future tax benefits.30 Additionally, rather than excluding firms with tax rates greater than one as in Faccio (2010), we verify that the underlying data do not reflect data errors by matching to original financial reports, as large taxes relative to income may simply indicate a strong level of expropriation. We then winsorise tax rates at the 1% level. Results absent control variables in Table 4 suggest that connected firms enjoy tax rates that are lower by around 12 and a half percentage points, with results that are significant below the 10%

29

In Venezuelan GAAP, benefits from the use of deferred tax assets are typically included as part of extraordinary items, though over a quarter of sample firm-years recorded an extraordinary tax benefit on their income statements. Tests with all control variables included are robust to excluding extraordinary tax benefits in measuring effective tax rates. 30 If pretax income is negative while tax benefits are positive (i.e. tax expense is reported as negative), then firms are recording deferred tax assets in anticipation of future tax benefits. If pretax income is negative while the provision for taxes is positive or zero, it suggests firms are not confident about recording the gross value of deferred tax assets.

Political Connections and Accounting Quality under High Expropriation Risk 23 level on a one-sided basis. When we include all control variables, results suggest connected firms enjoy tax rates lower by around 10 percentage points, with results again significant below the 10% level on a one-sided basis. Using predicted values from this regression, we find median predicted effective tax rates for unconnected firms equal to 28%, while median predicted effective tax rates for connected firms are only 19.5%. Results in Table 4 overall suggest that our connection measure is associated with potential political connection benefits, serving as a partial validation of our connection measure.31 In the next section, we assess whether this political connection measure is associated with earnings management.

5.2. Accounting Quality Test Results Table 5 provides results of earnings smoothing tests. Results for tests of differences in the standard deviation of net income changes show that connected firms have more variable earnings. Connected firms have a higher standard deviation of (residual) net income changes by 0.034, a difference that is statistically significant at below the 10% level. Differences in the ratios of the standard deviation of (residual) net income changes to (residual) cash flow changes also prove that connected firms engage in less earnings smoothing behaviour, with connected firms having a ratio difference of 0.709, statistically significant below the 5% level. Connected firms also have a higher difference between net income and cash flow changes, statistically significant below the 1% level. Finally, connected firms also have a lower correlation between accruals and cash flows, though, differences are not statistically significant. Overall, our evidence suggests that connected firms engage less in earnings smoothing behaviour than unconnected firms. Table 5 also shows results for a further partition for firm-years that are above and below the median of assets to net PPE. Firms above the median represent high expropriation risk firmyears. Among these firms, differences in earnings smoothing between connected and unconnected firms are all statistically significant; among low expropriation risk firms, no differences are statistically significant. This suggests that the effects of political connections on accounting quality are present only for high expropriation risk firms. Table 6 shows results for tests of whether Venezuelan firms engage in loss-avoiding behaviour. In an OLS regression framework, we find that the coefficient on CONN is negative and significant, suggesting that connected firms are less likely to engage in managing earnings to avoid losses. The result holds even when all control variables are included in the model. When we interact CONN with HIEXPROP (representing firms above the median of total assets to net PPE), we find it has a negative and significant coefficient estimate. This suggests 31

It is possible that, given our respondents are generally sympathetic to the anti-Chavez opposition, firms that are truly connected will have been classified as unconnected, to spare themselves or associates’ firms from embarrassment with having been associated with the regime. If our hypothesis is true on the negative relationship between connections and earnings management, we will have a more difficult time finding significant results due to this bias. If our hypothesis is false, and connectedness has no relationship or even a positive relationship with earnings management, then we would have to believe that a non-negligible number of the firms associated with our respondents who are then misclassified as unconnected have poorer accounting quality, presumably for capital market or contracting purposes. While we cannot rule out this possibility, there is no reason to believe the firms associated with the businessmen that were interviewed happen to have poorer accounting quality. Moreover, through our construct validation tests, we have indeed found that connected firms in our sample derive economic benefits (lower debt cost of capital, lower effective tax rates) that unconnected firms do not. There are certainly alternative plausible stories for how connected firms would have greater incidence of earnings management. However, it is more difficult to derive a story for connected firms enjoying a lower debt cost of capital, and also contrary to evidence in Khwaja and Mian (2005) and Faccio (2010). It is even more difficult to explain why connected firms would suffer a higher tax rate than unconnected firms.

24 G. Batta et al.

Table 5. Earnings smoothing tests Full sample Across-group differences STDEV of DNI STDEV of DNI/STDEV in DCFO STDEV of DNI 2 STDEV in DCFO Correlation of ACC and CFO

Connected N ¼ 46

High expropriation risk

Unconnected Connected N ¼ 108 Difference N ¼ 26

0.158 1.912 0.075

0.124 1.203 0.021

20.436

20.615

0.034∗ 0.709∗∗ 0.054∗∗∗ 0.179

Low expropriation risk

Unconnected Connected N ¼ 48 Difference N ¼ 20

0.186 2.179 0.100

0.122 1.003 0.000

20.359

20.704

0.064∗ 1.176∗∗ 0.100∗∗∗ 0.345+

Unconnected N ¼ 59 Difference

0.114 1.466 0.036

0.125 1.494 0.041

20.011 20.028 20.005

20.517

20.543

0.026

Notes: Table 5 gives results of tests of differences in earnings smoothing behaviour between connected and unconnected firms, both for the full sample and for firm-years above the median of the log of total assets to net PPE (High expropriation risk) and below the median (Low expropriation risk). STDEV of DNI is the standard deviation of residuals from a regression of the yearly difference in net income, scaled by prior-period assets, on a set of control variables, including (as defined in Table 3) SIZE, GROWTH, LEV, TURNOVER, CFO, LOCALAUD, EQUITY_ISS, DEBT_ISS, and XLIST. For SIZE, total assets are deflated using the Venezuelan CPI for December 1997 as a base. For TURNOVER, GROWTH, CFO, EQUITY_ISS and DEBT_ISS, all t21 numbers are multiplied by ratio of the level of the Venezuelan CPI at the end of year t to the level of the CPI at the end of year t21. All regressions also include GICS economic sector dummy variables. STDEV of DCFO is the standard deviation of residuals from a regression of the yearly difference in operating cash flow from the statement of cash flows, scaled by prior-period assets, on the same set of control variables. ACC is the residual from a regression of the difference between net income and operating cash flow (scaled by prior-period assets) on SIZE, GROWTH, LEV, TURNOVER, and LOCALAUD, while CFO is the residual from a regression of operating cash flow (scaled by prior-period assets) on the same set of control variables (excluding CFO). Correlation of ACC and CFO is the spearman correlation between ACC and CFO. Statistical significance is assessed by generating 1000 bootstrap samples of differences in STDEV of DNI, STDEV of DNI/STDEV of DCFO, STDEV of DNI 2 STDEV of DCFO, and Correlation of ACC and CFO between CONNECTED and UNCONNECTED firms; bootstrap samples are generated by sampling from blocks of two adjoining fiscal years’ cross-sections of firms’ data. All variables and residuals are winsorised at the 1% level. + Statistically significant differences below the 10% level on a one-sided basis. ∗ Statistically significant differences between Connected and Unconnected firms below the 10% level. ∗∗ Statistically significant differences between Connected and Unconnected firms below the 5% level. ∗∗∗ Statistically significant differences between Connected and Unconnected firms below the 1% level.

Political Connections and Accounting Quality under High Expropriation Risk 25 Table 6.

Small positive earnings SMALLPOS

CONN

20.114∗∗ (22.06)

20.123∗∗ (22.25)

HIEXPROP HIEXPROP × CONN SIZE

CONSTANT

0.157∗∗∗ (3.13)

0.001 (0.04) 0.017 (0.41) 20.353∗∗ (22.37) 20.030 (20.55) 20.465 (21.69) 0.288 (1.54) 20.010 (20.82) 0.004 (0.29) 20.024 (20.43) 0.195 (1.54)

Observations R2 Adj.-R2 F-Statistic

154 0.025 0.019 4.24∗∗

154 0.155 0.096 1.11

GROWTH LEV TURNOVER CFO LOCALAUD EQUITY_ISS DEBT_ISS XLIST

20.036 (20.57) 0.052 (0.51) 20.152 (21.36)+

0.136∗∗∗ (3.00) 153 0.037 0.017

20.027 (20.37) 20.022 (20.35) 20.173 (21.60)+ 20.014 (20.57) 0.012 (0.27) 20.321∗∗ (22.14) 20.048 (20.85) 20.507∗ (21.83) 0.411∗∗ (2.14) 20.006 (20.42) 0.004 (0.29) 0.000 (0.01) 0.276∗ (1.90) 153 0.194 0.125 1.83∗

Notes: Table 6 gives results of regression tests of differences in earnings benchmark-beating behaviour. SMALLPOS is equal to one when net incomet over assetst is between 0.00 and 0.005, and zero otherwise. HIEXPROP is equal to one for firm-years above the median of the log of total assets to net PPE, and zero otherwise. CONN is a dummy variable equal to one if a firm is deemed to be connected, and zero otherwise. All other variables are defined as in Table 3. All variable using prior-period numbers are multiplied by ratio of the level of the Venezuelan CPI at the end of fiscal year t to the level of the CPI at the end of fiscal year t21. All numerical variables are winsorised at the 1% level. Standard errors are clustered at the firm level. + Statistical significance below the 10% level on a one-sided basis. ∗ Statistical significance below the 10% level. ∗∗ Statistical significance below the 5% level. ∗∗∗ Statistical significance below the 1% level.

that the lower tendency for benchmark-beating behaviour among connected firms remains only for firms at greater risk of expropriation. Table 7 shows results for tests of whether Venezuelan firms engage in earnings-decreaseavoiding behaviour. In an OLS framework, absent control variables, we find that the coefficient on CONN is negative and significant on a one-sided basis and robust to the inclusion of control variables, suggesting that connected firms are less likely to experience small earnings decreases. Interacting CONN with HIEXPROP, we find again a lower tendency for benchmark-beating behaviour among connected, high expropriation risk firms. We also find a greater tendency for small earnings increases among high expropriation risk firms, which may serve to diminish risk of expropriation based on low productivity perceptions.

26

G. Batta et al. Table 7. Small earnings increases SMALLDIFF

CONN

+

20.068 (21.34)

20.057+ (21.59)

HIEXPROP HIEXPROP × CONN SIZE GROWTH LEV TURNOVER CFO LOCALAUD EQUITY_ISS DEBT_ISS XLIST CONSTANT

0.111∗∗ (2.65) 20.068

Observations R2 Adj.-R2 F-Statistic

154 0.012 0.005 1.79

0.007 (0.46) 20.022 (20.46) 20.104 (20.52) 20.025 (20.71) 0.252 (0.74) 0.259∗∗∗ (2.79) 20.009 (20.66) 20.008 (21.23) 0.046 (0.50) 0.054 (0.79) 20.057 154 0.134 0.073 3.39∗∗∗

0.016 (0.33) 0.174∗∗ (2.15) 20.186∗ (21.94)

0.034 (1.02) 0.016 153 0.075 0.057 1.61

0.058 (1.45) 0.177∗∗∗ (2.83) 20.244∗∗∗ (23.22) 0.005 (0.48) 20.016 (20.37) 20.002 (20.01) 20.027 (20.79) 0.221 (0.91) 0.266∗∗∗ (3.34) 0.001 (0.04) 20.015∗ (21.78) 0.094 (0.94) 20.026 (20.38) 0.058 153 0.202 0.134 4.70∗∗∗

Notes: Table 7 gives results of tests of differences in earnings increase benchmark-beating behaviour. SMALLDIFF is equal to one when the difference in net income between year t and year t21, scaled by total assets from year t21, is between 0.00 and 0.005. All other variables are defined as in Table 3. CONN is a dummy variable equal to one if a firm is deemed to be connected, and zero otherwise. HIEXPROP is equal to one for firm-years above the median of the log of total assets to net PPE, and zero otherwise. All variable using prior-period numbers are multiplied by ratio of the level of the Venezuelan CPI at the end of fiscal year t to the level of the CPI at the end of fiscal year t21. All numerical variables are winsorised at the 1% level. Standard errors are clustered at the firm level. + Statistical significance below the 10% level on a one-sided basis. ∗∗ Statistical significance below the 5% level. ∗∗∗ Statistical significance below the 1% level.

Table 8 provides results on timely loss recognition. In the first regression, using the Ball and Shivakumar (2005) framework, we find no relation between greater timely loss recognition and connected status. In the second regression, we include interactions with HIEXPROP. We find negative and significant coefficient estimates on HIEXPROP × NEGCFO and HIEXPROP × CFO × NEGCFO, showing that high expropriation risk firms offset negative cash flow realisations with positive accruals. However, CONN × HIEXPROP × CFO × NEGCFO remains insignificant, suggesting no greater timely loss recognition among connected, high expropriation risk firms. Similarly, we find that connected firms do not have significantly greater likelihoods of reporting extreme low earnings (ROALOW), nor is CONN × HIEXPROP significant. However,

Political Connections and Accounting Quality under High Expropriation Risk 27 Table 8.

Timely loss recognition

ACCRUALS CONN NEGCFO CFO CFO × NEGCFO CONN × NEGCFO CONN × CFO CONN × CFO × NEGCFO

0.011 (0.41) 0.015 (0.55) 20.451∗∗∗ (24.82) 21.353∗∗∗ (22.81) 20.047 (21.20) 0.014 (0.09) 0.006 (0.01)

HIEXPROP HIEXPROP × NEGCFO HIEXPROP × CFO CONN × HIEXPROP HIEXPROP × CFO × NEGCFO CONN × HIEXPROP × CFO × NEGCFO CONSTANT Observations R2 Adj.-R2 F-Statistic

20.033∗ (21.89) 159 0.378 0.350 14.96∗∗∗

0.022 (0.53) 0.071+ (1.69) 20.418∗ (21.75) 0.510 (0.46) 20.039 (20.89) 0.114 (0.43) 21.367 (20.54) 0.077∗∗ (2.57) 20.094∗∗ (22.05) 20.106 (20.45) 20.054∗ (21.85) 21.798+ (21.43) 1.041 (0.43) 20.063∗∗ (22.05) 159 0.424 0.372 48.79∗∗∗

ROALOW CONN

20.008 (20.25)

20.017∗ (21.96) 20.073∗∗ (22.37) 0.196 (0.78) 20.055+ (21.55) 20.084 (21.08) 20.006 (20.14) 0.078∗∗∗ (6.14) 0.001 (0.12) 20.063∗ (21.82)

20.018 (20.29) 20.060+ (21.47) 0.028 (0.36) 20.022∗ (21.96) 20.076∗∗ (22.34) 0.175 (0.71) 20.061∗ (21.77) 20.082 (20.82) 0.035 (0.52) 0.077∗∗∗ (5.78) 0.003 (0.36) 20.069∗ (21.97)

0.163∗∗∗ (2.86)

0.215∗∗∗ (3.06)

HIEXPROP HIEXPROP × CONN SIZE GROWTH LEV TURNOVER CFO LOCALAUD EQUITY_ISS DEBT_ISS XLIST

CONSTANT

154 0.181 0.123 8.36∗∗∗

153 0.194 0.125 49.01∗∗∗

Notes: Table 8 gives results of regression tests of differences in timely loss recognition. CONN is a dummy variable equal to one if a firm is deemed to be connected, and zero otherwise. HIEXPROP is equal to one for firm-years above the median of the log of total assets to net PPE, and zero otherwise. CFO is cash from operations from the cash flow statement, scaled by prior-period assets. NEGCFO is a dummy variable equal to one if CFO is less than zero, and equal to zero otherwise. ACCRUALS is the difference between net income and cash from operations, scaled by prior-period assets. ROALOW is equal to one when the difference in net income between year t and year t21, scaled by total assets from year t21, is less than 20.20. All other variables are defined as in Table 3. All variable using prior-period numbers are multiplied by ratio of the level of the Venezuelan CPI at the end of fiscal year t to the level of the CPI at the end of fiscal year t21. All numerical variables are winsorised at the 1% level. Standard errors clustered at the firm level. + Statistical significance below the 10% level on a one-sided basis. ∗ Statistical significance below the 10% level. ∗∗ Statistical significance below the 5% level. ∗∗∗ Statistical significance below the 1% level.

with a significant, negative loading on HIEXPROP, we again find that high expropriation risk firms are less likely to report extreme negative earnings. One possible explanation for our inability to see differences in timely loss recognition for connected and unconnected firms is that, countering expected expropriation-related motivations for less timely loss recognition, unconnected firms may have to rely more so on private lenders, who will place greater demands for more conservative accounting policies on their borrowers than government-related lenders.

28

G. Batta et al. Table 9. Absolute value of abnormal accruals ABS(ACC)

CONN

20.055∗∗ (22.39)

HIEXPROP CONN × HIEXPROP SIZE LEV CFO SD GROWTH SD TURNOVER SD GROWTH LOCALAUD EQUITY_ISS DEBT_ISS XLIST CONSTANT Observations R2 Adj.-R2 F-statistic

20.012 (21.20) 20.047 (20.57) 0.887 (0.76) 20.007 (20.44) 20.001 (20.02) 0.024 (0.92) 0.063 (1.13) 0.029 (0.97) 20.006 (20.92) 20.030 (21.62) 0.190∗∗∗ (3.28) 87 0.126 20.002 2.92∗∗

0.004 (0.07) 0.090∗ (2.07) 20.106∗ (21.79) 20.006 (20.44) 0.012 (0.14) 0.611 (0.65) 20.015 (20.82) 0.011 (0.12) 0.024 (0.91) 0.005 (0.06) 0.040 (1.14) 20.009 (21.49) 20.014 (20.81) 0.118∗ (1.88) 87 0.194 0.050 6.86∗∗∗

Notes: Table 9 gives results of regression tests of differences in the absolute value of performance-matched abnormal accruals, ABS(ACC), computed as described in Table 3. CONN is equal to one for connected firms, and zero otherwise. HIEXPROP is equal to one for firm-years above the median of the log of total assets to net PPE, and zero otherwise. CFO SD is the 5-year trailing standard deviation of cash flows over prior-period total assets; GROWTH SD is the trailing 5-year standard deviation of sales growth; TURNOVER SD is the trailing 5-year standard deviation of sales over prior-period assets. All other variables are defined as in Table 3. All numerical variables are winsorised at the 1% level. Standard errors are clustered at the firm level. ∗ Statistical significance below the 10% level. ∗∗ Statistical significance below the 5% level. ∗∗∗ Statistical significance below the 1% level.

Table 9 shows results for the absolute value of abnormal accruals. We find a negative and significant coefficient on CONN, suggesting a smaller magnitude of abnormal accruals for connected firms equal to around 5.5% of prior-period assets. Although none of the control variable coefficient estimates is significant, this is likely due to the small sample size and high univariate correlations among control variables shown in Table 2. An untabulated Ftest of the joint significance of the control variables indeed suggests they are collectively significant, at below the 1% level. When we interact CONN with HIEXPROP, results are similar to most prior tests, with HIEXPROP being positively associated with abnormal accrual magnitudes, and CONN × HIEXPROP resulting in lower accruals, suggesting connected, high

Political Connections and Accounting Quality under High Expropriation Risk 29 expropriation risk firms’ lower propensity to use accruals to mitigate earnings volatility, or to manage earnings upwards or downwards to meet targets.32 Overall, results are suggestive that connected firms have superior accounting quality relative to unconnected firms. Tests of small positive earnings and small earnings increases suggest unconnected firms’ desire to avoid appearing unproductive, while earnings smoothing and abnormal accruals tests are consistent with firms’ desires to appear neither unproductive nor excessively profitable. With the exception of small positive earnings tests, we also find that accounting quality is lower for high expropriation risk firms. Finally, in all but timely loss recognition tests, we find that accounting quality is higher among connected, high expropriation risk firms. Results suggest that, for most manifestations of earnings quality, political connections serve as a ‘shield’ against expropriation.

6.

Robustness Checks

6.1. Cluster Bootstrap Standard Errors Because of the relatively small samples we employ in our study, asymptotic-based inference using clustered standard errors may over-reject the null hypothesis. We therefore follow Cameron, Miller, and Gelbach (2008) in employing a cluster bootstrap procedure for all OLS tests employing clustered standard errors.33 We find that OLS regression results for small earnings increases, small earnings increases, timely loss recognition, and abnormal accruals are robust to using cluster bootstrap standard errors.

6.2. Family Owned Firms In their study of the effect of political connections on earnings quality, Chaney et al. (2010) account for possible confounding effect on earnings quality of family owned firms. Entrenched family members are deemed to have both the incentive and capability to manipulate earnings in order to hide expropriation from minority shareholders (Fan & Wong, 2002). The majority of firms in our sample were controlled by families or individuals.34 In untabulated tests, we assess whether including an indicator variable for family owned firms affected our accounting quality results. We find that all results hold when we include this variable. 32

Chaney et al. (2011) use industry and year-matched firms for their benchmark accruals model. However, Dopuch, Mashruwala, Seethamraju, and Zach (2011) note that the assumption of a homogeneous accrual-generating process within industries may be suspect. Ecker, Francis, Olsson, and Schipper (2013) propose instead using within-country year and lagged asset-matched firms for the accruals benchmark model, especially in contexts like ours where there is an insufficient number of within-country industry peers to compute abnormal accruals. They also note that differences in institutional features (such as accounting standards, legal codes, or culture) may affect the usefulness of cross-country, industry-matched peer models. Results are similar when we use the 10 largest lagged asset peer firms to measure ‘normal’ accruals, though only the interaction term of EXPROP and CONN × EXPROP remain significant and in the predicted direction. 33 Specifically, for asymmetric loss timeliness and abnormal accruals tests, per Cameron et al.’s (2008) optimal specification, we use wild bootstrap with +1 and 21 weights on negative versus positive residuals; we ‘impose the null’ in generating predicted values and residuals; and we use 999 bootstrap replications. For small positive earnings, small earnings increase, and large negative earnings tests, we use non-parametric bootstrap, as wild bootstrap produces simulated dependent variables no longer equal to dichotomous outcomes necessary for these tests. 34 We identified family ownership through news searches and from our interviews evaluating political connections (Section 4.1).

30

G. Batta et al.

6.3. Firm Visibility It is possible that politically connected firms receive more scrutiny due to their connected status, and that this visibility engenders heightened monitoring of accounting quality. Because political connections in Venezuela are not generally public knowledge, this is a less plausible alternative explanation for our findings. Nonetheless, we control for firm visibility by conducting a LexisNexis news search of articles in El Nacional related to each firm. Results are similar when we include the yearly count of articles as an additional regressor. 7.

Conclusion

Prior literature has found that accounting quality decreases with stronger political connections. In this paper, we assess whether the risk of government expropriation may moderate this relationship, as firms attempt to avoid the appearance of low productivity and mitigate earnings volatility in to avoid state intervention. We do so by examining the link between connections and accounting quality among Venezuelan industrial firms, which face one of the highest levels of expropriation risk worldwide. Our results suggest that politically connected Venezuelan firms have higher levels of accounting quality. Measures of firms’ propensities to smooth earnings, meet earnings benchmarks, and use accruals to manage earnings are all lower among politically connected firms. Results are generally robust to a battery of sensitivity checks, and the impact of political connections on accounting quality is strongest among high expropriation risk firms. Our results suggest that the effect of political connections on accounting quality may be moderated by important institutional factors like high expropriation risk, providing nuance to previous research on the relationship between accounting quality and political connections. References Ball, R., Robin, A., & Wu, J. (2001). Accounting standards, the institutional environment and issuer incentives: Effect of timely loss recognition in China. Asia-Pacific Journal of Accounting and Economics, 7, 71– 96. Ball, R., Robin, A., & Wu, J. (2003). Incentives versus standards: Properties of accounting income in four East Asia countries. Journal of Accounting and Economics, 36, 235– 270. Ball, R., & Shivakumar, L. (2005). Earnings quality in UK private firms: Comparative loss recognition timeliness. Journal of Accounting and Economics, 39, 83–128. Barth, M., Landsman, W., & Lang, M. (2008). International accounting standards and accounting quality. Journal of Accounting Research, 46, 467 –498. Bodnurak, A., & Massa, M. (2011). Every family has a white stripe: Between vertical and horizontal governance. Working paper, INSEAD. Boubraki, N., Guedhami, O., Mishra, D., & Saffar, W. (2012). Political connections and the cost of equity capital. Journal of Corporate Finance, 18, 541 –559. Burgstahler, D., & Dichev, L. (1997). Earnings management to avoid earnings decreases and losses. Journal of Accounting and Economics, 24, 99– 126. Bushman, R., & Piotroski, J. (2006). Financial reporting incentives for conservative accounting: The influence of legal and political institutions. Journal of Accounting and Economics, 42, 107–148. Bushman, R., Piotroski, J., & Smith, A. (2004). What determines corporate transparency? Journal of Accounting Research, 42, 207–252. Businesswire. (2010, October 22). Fitch comments on possible impacts of the nationalization of FertiNitro finance, Inc. Retrieved from http://www.businesswire.com/news/home/20101022005668/en/Fitch-Comments-ImpactsNationalization-FertiNitro-Finance Cameron, A. C., Miller, D., & Gelbach, J. (2008). Bootstrap-based improvements for inference with clustered error. Review of Economics and Statistics, 90, 414–427. Chaney, P., Faccio, M., & Parsley, D. (2011). The quality of accounting information in politically connected firms. Journal of Accounting and Economics, 51, 58–76. Chen, C., Ding, Y., & Kim, C. (2010). High-level politically connected firms, corruption, and analyst forecast accuracy around the world. Journal of International Business Studies, 41, 1505–1524.

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Appendix 1.

Political connection strength

Company

Industry

Connections

Range of data

Compania Anonima Nacional Telefonos De Venezuela (CANTV) Cemex Venezuela

Communications equipment

No known connection (nationalised 2007)

2000–2005

Construction materials

Ceramica Carabobo

Building products

Mild Nationalised Mild

Corporacion Grupo Quimico Corporacion Industrial De Energia Corimon

Commodity chemicals Electric utilities

Strongly opposed No known connection

2000–2007 2008 2000–2001, 2004–2008 2001–2004 2001–2005

Commodity chemicals

No known connection

Dominguez & Cia

Containers and packaging Electric utilities

La Electricidad De Caracas Envases Venezolanos Fabrica Nacional De Cementos Fabrica Nacional De Vidrio Grupo Zuliano H.L. Boulton & Co. Hotel Tamanaco IBH Inversiones Selva

Electric utilities Construction materials Containers and packaging Commodity chemicals Marine Hotels, restaurant, and leisure Metals and mining

Manufacturas De Papel (Manpa)

Containers and packaging Paper and forest products

Mavesa Proagro Protinal Ron Santa Teresa SVS

Food products Food products Food products Beverages Metals and mining

Mild No known connection No known connection (nationalised 2007) No known connection No known connection (nationalised 2008) Strongly opposed Mild Mild Strong no known connection No known connection Strongly opposed (partial nationalisation in 2009) No known connection No known connection (temporary government occupation in 2013) No known connection No known connection No known connection Mild No known connection Strongly opposed (partial nationalisation 2009)

2000–2001, 2005 2008 2001, 2006 2000–2005 2002–2007 2000–2007 2001 2006–2008 2000–2008 2000–2008 2000–2007 2001–2003 2004 2000, 2003– 2005 2000–2004 2000 2002–2007 2000–2008 2002–2008 2000–2003 2004–2007 (Continued)

Political Connections and Accounting Quality under High Expropriation Risk 33 Appendix 1.

Continued

Company Soldaduras Y Tuberias De Oriente Suelopetrol Telares De Palo Grand Terminales Maracaibo Tordisca Distribuidoras Torvenca Venepal Venaseta

Industry Energy equipment and services Oil, gas and consumable fuels Textiles, apparel and luxury goods Transportation infrastructure Machinery Paper and forest products Textiles, apparel and luxury goods

Connections

Range of data

No known connection

2005–2008

No known connection Strong No known connection

2004 2005–2008 2000–2006

No known connection (nationalised 2009) No known connection

2000

No known connection (nationalised 2005) No known connection

2000, 2003– 2007 2000, 2003 2000–2001

Appendix 2. Bootstrapping procedure for earnings smoothing tests In untabulated tests, we find evidence of both temporal and cross-sectional error dependence in residuals from the net income change regression: the first lag of residuals for each panel has a significant autocorrelation coefficient of 20.18 (significant below the 10% level on a onesided basis, with robust standard errors) with longer lags insignificant. Additionally, in a regression of residuals on year dummies, the joint significance of year dummies is significant below the 1% level. To adjust for these potential error correlations, we bootstrap an error distribution for the differences in standard deviations, as well as other earnings smoothing metrics described below that do not have standard tests for differences. To preserve the evident temporal series and cross-sectional error dependence structure, we adopt a panel version of the moving block bootstrap of Ku¨nsch (1989), also described in Kapetanios (2008) and, in the context of fixed effect panel data models, Goncalves (2011). In each of 1000 bootstrap replications, we sample from blocks of two adjoining years of data, with all firms with data present in each year included in each year two-year block. For each bootstrap replication, we compute the standard deviation of model residuals for connected and unconnected groups, and then compute the difference; the difference in the difference in net income and cash flow change standard deviations; and the difference in the ratio of net income and cash flow change standard deviations. We then percentile rank the original-sample difference in standard deviations against the sorted, demeaned bootstrapped differences in standard deviations, in order to assess a two-sided p-value. We follow a similar procedure in testing for accruals and cash flow correlation differences, computing in each replication the difference in correlations between the two groups.

Stochastics: An International Journal of Probability and Stochastic Processes Vol. 85, No. 1, February 2013, 111–143

Precautionary measures for credit risk management in jump models Masahiko Egamia1 and Kazutoshi Yamazakib* a

Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto 606-8501, Japan; bCenter for the Study of Finance and Insurance, Osaka University, 1-3 Machikaneyama-cho, Toyonaka City, Osaka 560-8531, Japan (Received 5 April 2010; final version received 23 December 2011) Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the trade-off between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank’s loan/credit business portfolios, we represent its net worth by Le´vy processes, and solve explicitly for the double exponential jump-diffusion process and for a general spectrally negative Le´vy process. Keywords: credit risk management; double exponential jump diffusion; spectrally negative Le´vy processes; scale functions; optimal stopping Mathematics Subject Classification (2000): Primary: 60G40; Secondary: 60J75

1.

Introduction

As an aftermath of the recent devastating financial crisis, more sophisticated risk management practices are now being required under the Basel II accord. In order to satisfy the capital adequacy requirements, a bank needs to closely monitor how much of its asset values has been damaged; it needs to examine whether it maintains sufficient equity values or needs to start enhancing its equity to asset ratio by raising more capital and/or selling its assets. Due to unexpected sharp declines in asset values as experienced in the fall of 2008, optimally determining when to undertake the action is an important and difficult problem. In this paper, we give a new framework for this problem and obtain its solutions explicitly. We propose an alarm system that determines when a bank needs to start enhancing its own capital ratio. We use Le´vy processes with jumps in order to model defaults in its loan/credit assets and sharp declines in their values under unstable market conditions. Because of their negative jumps and the necessity to allow time for completing its capital reinforcement plans, early practical action is needed to reduce the risk of violating the capital adequacy requirements. On the other hand, there is also a cost of premature undertaking. If the action is taken too quickly, it may run a risk of incurring a large amount of opportunity costs including burgeoning administrative and monitoring expenses. In other words, we need to solve this trade-off in order to implement this alarm system.

*Corresponding author. Email: [email protected] ISSN 1744-2508 print/ISSN 1744-2516 online q 2013 Taylor & Francis http://dx.doi.org/10.1080/17442508.2011.653566 http://www.tandfonline.com

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In this paper, we properly quantify the costs of delay and premature undertaking and set a well-defined objective function that models this trade-off. Our problem is to obtain a stopping time that minimizes the objective function. We expect that this precautionary measure gives a new framework in risk management. 1.1 Problem Let ðV; F ; PÞ be a complete probability space on which a Le´vy process X ¼ {X t ; t $ 0} is defined. We represent, by X, a bank’s net worth or equity capital allocated to its loan/credit business and model the defaults in its credit portfolio in terms of the negative jumps. For example, for a given standard Brownian motion B ¼ {Bt ; t $ 0} and a jump process J ¼ {J t ; t $ 0} independent of B, if it admits a decomposition X t ¼ x þ m t þ s Bt þ J t ;

0#t,1

and

X 0 ¼ x;

for some m [ R and s $ 0, then Jt models the defaults as well as rapid increase in capital whereas the non-jump terms mt and sBt represent, respectively, the growth of the capital (through the cash flows from its credit portfolio) and its fluctuations caused by non-default events (e.g. change in interest rates). Since X is spatially homogeneous, we may assume, without loss of generality, that the first time X reaches or goes below zero signifies the event that the net capital requirement is violated. We call this the violation event and denote it by

u U inf{t $ 0 : X t # 0}; where we assume inf B ¼ 1. Let F ¼ ðF t Þt$0 be the filtration generated by X. Then, u is an F-stopping time taking values on ½0; 1Š. We denote by S the set of all stopping times smaller than or equal to the violation event, namely, S U {stopping time t : t # u

a:s:}:

We only need to consider stopping times in S because the violation event is observable and the game is over once it happens. By taking advantage of this, we see that the problem can be reduced to a well-defined optimal stopping problem; see Section 2. Our goal is to obtain among S the alarm time that minimizes the two costs we describe below. The first cost we consider is the risk that the alarm will be triggered at or after the violation event: 
x 2qu e ð t Þ U E 1 t [ S: RðqÞ {t$u;u,1} ; x

Here, q [ ½0; 1Þ is a discount rate and Px is the probability measure and Ex is the expectation under which the process starts at X0 ¼ x. We call this the violation risk. In particular, when q ¼ 0, it can be reduced under a suitable condition to the probability of the event {t $ u}; see Section 2. The second cost relates to premature undertaking measured by
 ðu x 2q t t Þ U E 1 e hðX Þdt ; t [ S: ð H ðq;hÞ {t,1} t x t

We shall call this the regret, and here we assume h : ð0; 1Þ ! ½0; 1Þ to be continuous, non-decreasing and
ð u  x 2q t e hðX t Þdt , 1; x . 0: ð1:1Þ E 0

Stochastics: An International Journal of Probability and Stochastic Processes 113 The monotonicity assumption reflects the fact that if a bank has a higher capital value X, then it naturally has better access to high-quality assets and hence the opportunity cost hð·Þ becomes higher accordingly. In particular, when h ; 1 (i.e. h(x) ¼ 1 for every x . 0), we have ðtÞ H ð0;1Þ x

x

x

¼ Eu 2 Et

and

H xðq;1Þ ðtÞ

  1 x 2qt x 2qu E ½e Š 2 E ½e Š ; ¼ q

q . 0;

ð1:2Þ

where the former is well defined by (1.1). Now, using some fixed weight g . 0, we consider a linear combination of these two costs described above: ðq;hÞ ðtÞ; U xðq;hÞ ðt; gÞ U RðqÞ x ðtÞ þ gH x

t [ S:

ð1:3Þ

We solve the problem of minimizing (1.3) for the double exponential jump-diffusion process and a general spectrally negative Le´vy process. The objective function is finite thanks to the integrability assumption (1.1), and hence the problem is well defined. The form of this objective function in (1.3) has an origin from the Bayes risk in mathematical statistics. In the Bayesian formulation of change-point detection, the Bayes risk is defined as a linear combination of the expected detection delay and false alarm probability. In sequential hypothesis testing, it is a linear combination of the expected sample size and misdiagnosis probability. The optimal solutions in these problems are those stopping times that minimize the corresponding Bayes risks. Namely, the trade-off between promptness and accuracy is modelled in terms of the Bayes risk. Similarly, in our problem, we model the trade-off between the violation risk and regret by their linear combination U. We first consider the double exponential jump-diffusion process, a Le´vy process consisting of a Brownian motion and a compound Poisson process with positive and negative exponentially distributed jumps. We consider this classical model as an excellent starting point mainly due to a number of existing analytical properties and the fact that the results can potentially be extended to the hyper-exponential jump-diffusion model (HEM) and more generally to the phase-type Le´vy model. Due to the memoryless property of the exponential distribution, the distributions of the first passage times and overshoots by this process can be obtained explicitly (Kou and Wang [20]). It is this property that leads to analytical solutions in various problems that would not be possible for other jump processes. Kou and Wang [21] used this process as an underlying asset and obtained a closed-form solution to the perpetual American option and the Laplace transforms of lookback and barrier options. Sepp [35] derived explicit pricing formulae for doublebarrier and double-touch options with time-dependent rebates. See also Lipton and Sepp [28] for applications of its multidimensional version in credit risk. Some of the results for the double exponential jump-diffusion process have been extended to the HEM and phasetype models, for example, by Cai et al. [7,8] and Asmussen et al. [1]. We then consider a general spectrally negative Le´vy process, or a Le´vy process with only negative jumps. Because we are interested in defaults, the restriction to negative jumps does not lose much reality in modelling. We also see that positive jumps do not have much influence on the solutions. We shall utilize the scale function to simplify the problem and obtain analytical solutions. In order to identify the candidate optimal strategy, we shall apply continuous and smooth fit for the cases X has paths of bounded and unbounded variation, respectively. The scale function is an important tool in most spectrally negative Le´vy models and can be calculated via algorithms such as Surya [40] and Egami and Yamazaki [13].

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1.2 Literature review Our model is original, and, to the best of our knowledge, the objective function defined in (1.3) cannot be found elsewhere. It is, however, relevant to the problem, arising in the optimal capital structure framework, of determining the endogenous bankruptcy levels. The original diffusion model was first proposed by Leland [26] and Leland and Toft [27], and it was extended, via the Wiener –Hopf factorization, to the model with jumps by Hilberink and Rogers [17]. Kyprianou and Surya [24] studied the case with a general spectrally negative Le´vy process. In their problems, the continuous and smooth fit principle is a main tool in obtaining the optimal bankruptcy levels. Chen and Kou [10] and Dao and Jeanblanc [12], in particular, focus on the double exponential jump-diffusion case. In the insurance literature, as exemplified by the Cramer – Lundberg model, the compound Poisson process is commonly used to model the surplus of an insurance firm. Recently, more general forms of jump processes have been used (e.g. Huzak et al. [18] and Jang [19]). For generalizations to the spectrally negative Le´vy model, see Avram et al. [2], Kyprianou and Palmowski [23], and Loeffen [29]. The literature also includes computations of ruin probabilities and extensions to jumps with heavy-tailed distributions; see Embrechts et al. [14] and references therein. See also Schmidli [34] for a survey on stochastic control problems in insurance. Mathematical statistics problems as exemplified by sequential testing and changepoint detection have a long history. It dates back to 1948 when Wald and Wolfowitz [41,42] used the Bayesian approach and proved the optimality of the sequential probability ratio test. There are essentially two problems, the Bayesian and the variational (or the fixed-error) problems; the former minimizes the Bayes risk whereas the latter minimizes the expected detection delay (or the sample size) subject to a constraint that the error probability is smaller than some given threshold. For comprehensive surveys and references, we refer the reader to Peskir and Shiryaev [31] and Shiryaev [36]. Our problem was originally motivated by the Bayesian problem. However, it is also possible to consider its variational version where the regret needs to be minimized on constraint that the violation risk is bounded by some threshold. Optimal stopping problems involving jumps (including the discrete-time model) are, in general, analytically intractable owing to the difficulty in obtaining the overshoot distribution. This is true in our problem and in the literatures introduced above. For example, in sequential testing and change-point detection, explicit solutions can be realized only in the Wiener case. For this reason, recent research focuses on obtaining asymptotically optimal solutions by utilizing renewal theory, see, for example, Baron and Tartakovsky [3], Baum and Veeravalli [4], Lai [25] and Yamazaki [44]. Although we do not address in this paper, asymptotically optimal solutions to our problem may be pursued for a more general class of Le´vy processes via renewal theory. We refer the reader to Gut [16] for the overshoot distribution of random walks and Siegmund [37] and Woodroofe [43] for more general cases in nonlinear renewal theory. 1.3

Outline

The rest of the paper is structured as in the following. We first give an optimal stopping model for a general Le´vy process in the next section. Section 3 focuses on the double exponential jump-diffusion process and solve for the case when h ; 1. Section 4 considers the case when the process is a general spectrally negative Le´vy process; we obtain the solution explicitly in terms of the scale function for a general h. We conclude with numerical results in Section 5. Long proofs are deferred to Appendix.

Stochastics: An International Journal of Probability and Stochastic Processes 115 2. Mathematical model In this section, we first reduce our problem to an optimal stopping problem and illustrate the continuous and smooth fit approach to solve it. 2.1

Reduction to optimal stopping

Fix t [ S, q . 0 and x . 0. The violation risk is

x 2qu RðqÞ 1{t$u;u,1} Š ¼ Ex ½e2qu 1{t ¼u;u,1} Š ¼ Ex ½e2qt 1{t ¼u;u,1} Š; x ðtÞ ¼ E ½e

where the second equality follows because t # u a.s. by definition. Moreover, we have 1{t ¼u;t ,1} ¼ 1{Xt #0;t ,1}

a:s:

because {t ¼ u; t , 1} ¼ {X t # 0; t ¼ u; t , 1} and {X t # 0; t , 1} ¼ {X t # 0; t # u; t , 1} ¼ {X t # 0; t ¼ u; t , 1}; where the first equality holds because t [ S and the second equality holds by the definition of u. Hence, we have x 2qt RðqÞ 1{X t #0;t ,1} Š; x ðtÞ ¼ E ½e

t [ S:

For the regret, by the strong Markov property of X at time t, we have   ðu   H xðq;hÞ ðtÞ U Ex 1{t,1} e2qthðX t Þdt ¼ Ex e2qtQ ðq;hÞ ðX t Þ1{t,1} ; t

ð2:1Þ

ð2:2Þ

where {Q ðq;hÞ ðX t Þ; t $ 0} is an F-adapted Markov process such that ð u  Q ðq;hÞ ðxÞ U Ex e2qthðX t Þdt ; x [ R: 0

Therefore, by (2.1) and (2.2), if we let GðxÞ U 1{x#0} þ gQ ðq;hÞ ðxÞ1{x.0} ;

x[R

ð2:3Þ

denote the cost of stopping, we can rewrite the objective function (1.3) as U ðq;hÞ ðt; gÞ ¼ Ex ½e2qtGðX t Þ1{t ,1} Š; x

t [ S:

Our problem is to obtain inf Ex ½e2qtGðX t Þ1{t ,1} Š

t[S

and an optimal stopping time t * [ S that attains it if such a stopping time exists. It is easy to see that G(x) is non-decreasing on ð0; 1Þ because h(x) is. If Gð0þÞ $ 1, then clearly u is optimal. Therefore, we ignore the trivial case and assume throughout this paper that Gð0þÞ , 1:

ð2:4Þ

As we will see later, when X has paths of unbounded variation, Gð0þÞ ¼ 0 and the assumption above is automatically satisfied. The problem can be naturally extended to the undiscounted case with q ¼ 0. The integrability assumption (1.1) implies Exu , 1 (without this assumption G(x) ¼ 1 and the problem becomes trivial). This also implies u , 1 a.s. and the violation risk reduces to

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the probability x x x Rð0Þ x ðtÞ ¼ P {t $ u} ¼ P {t ¼ u} ¼ P {X t # 0};

t [ S:

We shall study the case q ¼ 0 for the double exponential jump-diffusion process in Section 3. 2.2

Obtaining optimal strategy via continuous and smooth fit

Similarly to obtaining the optimal bankruptcy levels in Leland [26], Leland and Toft [27], Hilberink and Rogers [17] and Kyprianou and Surya [24], the continuous and smooth fit principle will be a useful tool in our problem. Focusing on the set of threshold strategies defined by the first time the process reaches or goes below some fixed threshold, say A,

tA U inf{t $ 0 : X t # A};

A $ 0;

we choose the optimal threshold level that satisfies the continuous or smooth fit condition and then verify the optimality of the corresponding strategy. Let the expected value corresponding to the threshold strategy tA for fixed A . 0 be

fA ðxÞ U U xðq;hÞ ðtA ; gÞ;

x[R

and the difference between the continuation and stopping values be  ð tA  x 2qt dA ðxÞ U fA ðxÞ 2 GðxÞ ¼ RðqÞ t g E e hðX Þdt ; 0 , A , x: ð Þ 2 t A x 0

ð2:5Þ

We then have 8 GðxÞ þ dA ðxÞ; > > < fA ðxÞ ¼ GðxÞ; > > : 1;

x . A; 0 , x # A; x # 0:

ð2:6Þ

The continuous and smooth fit conditions are dA ðAþÞ ¼ 0 and d0A ðAþÞ ¼ 0, respectively. For a comprehensive account of continuous and smooth fit principle, see Peskir and Shiryaev [31 – 33]. 2.3 Extension to the geometric model It should be noted that a version of this problem with an exponential Le´vy process Y ¼ {Y t ¼ expðX t Þ; t $ 0} and a slightly modified violation time

u~ ¼ inf{t $ 0 : Y t # a}; for some a . 0 can be modelled in the same framework. Indeed, defining a shifted Le´vy process X~ U {X~ t ¼ X t 2 log a; t $ 0}; we have

u~ ¼ inf{t $ 0 : X~ t # 0}: Moreover, the regret function can be expressed in terms of X~ by replacing h(x) with ~ ¼ hðexp ðx þ log aÞÞ for every x . 0. The continuity and non-decreasing properties hðxÞ remain valid because of the property of the exponential function.

Stochastics: An International Journal of Probability and Stochastic Processes 117 3. Double exponential jump diffusion In this section, we consider the double exponential jump-diffusion model that features exponential-type jumps in both positive and negative directions. We first summarize the results from Kou and Wang [20] and obtain explicit representations of our violation risk and regret. We then find analytically the optimal strategy both when q . 0 and when q ¼ 0. We assume throughout this section that h ; 1, i.e. the regret function reduces to (1.2). 3.1

Double exponential jump diffusion The double exponential jump-diffusion process is a Le´vy process of the form X t U x þ m t þ s Bt þ

Nt X

Z i;

ð3:1Þ

t $ 0;

i¼1

where m [ R, s . 0, B ¼ {Bt ; t $ 0} is a standard Brownian motion, N ¼ {N t ; t $ 0} is a Poisson process with parameter l . 0 and Z ¼ {Z i ; i [ N} is a sequence of i.i.d. random variables having a double exponential distribution with its density f ðzÞ U ph2 eh2 z 1{z,0} þ ð1 2 pÞhþ e2hþ z 1{z.0} ;

z [ R;

ð3:2Þ

for some h2 ; hþ $ 0 and p [ ½0; 1Š. Here B, N and Z are assumed to be mutually independent. The Laplace exponent of this process is given by   1 ph2 ð1 2 pÞhþ cðbÞ U E0 ½ebX1 Š ¼ mb þ s 2b 2 þ l þ 2 1 ; b [ R: ð3:3Þ h2 þ b hþ 2 b 2 We later see that the Laplace exponent and its inverse function are useful tools in simplifying the problem and characterizing the structure of the optimal solution. Fix q . 0. There are four roots of cðbÞ ¼ q, and in particular we focus on j1;q and j2;q such that 0 , j1;q , h2 , j2;q , 1

and

cð2j1;q Þ ¼ cð2j2;q Þ ¼ q:

Suppose that the overall drift is denoted by u U E0 ½X 1 Š, then it becomes   p 12p u ¼ m þ l 2 þ h2 hþ and

j1;q ! j1;0

(

¼ 0;

u # 0

. 0;

u . 0

)

and

j2;q ! j2;0

as q ! 0;

ð3:4Þ

for some j1;0 and j2;0 satisfying 0 # j1;0 , h2 , j2;0 , 1

and

cð2j1;0 Þ ¼ cð2j2;0 Þ ¼ 0:

See Figure 1 for an illustration. When u , 0, by (3.4), l’Hoˆpital’s rule and cð2j1;q Þ ¼ q, we have

j1;q q#0 1 1 1 !2 0 ¼2 ¼ : q c ð0Þ u juj

ð3:5Þ

We will see that these roots characterize the optimal strategies; the optimal threshold levels can be expressed in terms of j1;q and j2;q when q . 0 and j2;0 and u when q ¼ 0.

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M. Egami and K. Yamazaki

(a)

(b)

Figure 1.

−ξ2,0

−ξ2,q

−ξ1,q

−η−

0

η+

ψ (β)

ψ (β)

−ξ2,q

−ξ

2,0

−ξ1,q

−η−

−ξ1,0

η+

0

β

β

u– < 0

u– > 0

Illustration of the Laplace exponent and its roots when the drift u is negative and positive.

Due to the memoryless property of its jump-size distribution, the violation risk and regret can be obtained explicitly. The following two lemmas are due to Kou and Wang [20], Theorem 3.1 and its corollary. Here, we let h2 2 j1;q j2;q 2 h2 l1;q U . 0 and l2;q U . 0; q $ 0; j2;q 2 j1;q j2;q 2 j1;q where, in particular, when q ¼ 0 and u , 0, l1;0 ¼

h2 j2;0 2 h2 . 0 and l2;0 ¼ . 0: j2;0 j2;0

Notice that l1;q þ l2;q ¼ 1 for every q $ 0. Lemma 3.1 (violation risk). For every q $ 0 and 0 , A , x, we have   e2h2 A ðqÞ 2j1;q ðx2AÞ 2j2;q ðx2AÞ Rx ðtA Þ ¼ : ðj2;q 2 h2 Þl1;q e 2 ðh2 2 j1;q Þl2;q e h2 In particular, when q ¼ 0 and u , 0, this reduces to   ð0Þ 2h2 A 2j2;0 ðx2AÞ Rx ðtA Þ ¼ l2;0 e 12e : Lemma 3.2. (functional associated with the regret when h ; 1). For every q . 0, we have   ð tA  

1 x 2j1;q ðx2AÞ 2j2;q ðx2AÞ 2qt l1;q j2;q 1 2 e E e dt ¼ ; 0 # A , x: þ l2;q j1;q 1 2 e q h2 0 Furthermore, it can be extended to the case q ¼ 0, by taking q # 0 via (3.5) and the monotone convergence theorem; i 8 h < 1 ðx 2 AÞ þ j2;0 2h2 ð1 2 e2j2;0 ðx2AÞ Þ ; if u , 0; h j j u j 2 2;0 Ex ½ t A Š ¼ : 1; if u $ 0: For a general Le´vy process, Lemma 3.2 can be alternatively achieved by obtaining Px {X e q , A} where eq is an independent exponential random variable with parameter

Stochastics: An International Journal of Probability and Stochastic Processes 119 q $ 0 and X t U inf0#u#t X u is the running infimum of X; see Bertoin [5], Kyprianou [22] or Chapter 2 of Surya [39]. In particular, Px {X e q , A} admits an analytical form when jumps are of phase-type (Asmussen et al. [1]); the above results can be seen as its special case. Remark 3.1. As in Kou and Wang [20], we assume throughout this section that X contains a diffusion component (s . 0). Here, we do not consider the case s ¼ 0 because its spectrally negative case is covered in the next section. The results for s ¼ 0 can be obtained similarly using the results by Asmussen et al. [1]. 3.2

Optimal strategy when h ; 1

We shall obtain the optimal solution for q $ 0. When q ¼ 0, we focus on the case when u , 0 because Exu ¼ 1 otherwise by Lemma 3.2 and the problem becomes trivial as we discussed in Section 2. Suppose q . 0. By Lemma 3.2, the stopping value (2.3) becomes ð u  X g X C i;q ð1 2 e2ji;q x Þ ¼ 2 e2qt dt ¼ ð3:6Þ C i;q e2ji;q x ; x . 0; GðxÞ ¼ gEx q 0 i¼1;2 i¼1;2 where C 1;q U

g l1;q j2;q q h2

and

C 2;q U

g l2;q j1;q ; q h2

ð3:7Þ

which satisfy X

i¼1;2

Ci;q ¼

g q

and

X

C i;q

i¼1;2

h2 g ¼ : h2 2 ji;q q

ð3:8Þ

The difference between the continuation and stopping values (defined in (2.5)) becomes, by Lemmas 3.1 and 3.2,   l1;q g ðj2;q 2 h2 Þe2j1;q ðx2AÞ2h2 A 2 j2;q ð1 2 e2j1;q ðx2AÞ Þ dA ðxÞ ¼ h2 q   l2;q g 2ðh2 2 j1;q Þe2j2;q ðx2AÞ2h2 A 2 j1;q ð1 2 e2j2;q ðx2AÞ Þ ; 0 , A , x: þ h2 q ð3:9Þ Suppose q ¼ 0 and u , 0. By Lemma 3.2, we have   g j2;0 2 h2 xþ ð1 2 e2j2;0 x Þ ; GðxÞ ¼ juj h2 j2;0

x.0

and taking q ! 0 in (3.9) via the monotone convergence theorem (or by Lemmas 3.1 and 3.2)   l1;0 g dA ðxÞ ¼ ðj2;0 2 h2 Þe2h2 A 2 j2;0 ðx 2 AÞ h2 juj   ð3:10Þ l2;0 g 2h2 e2j2;0 ðx2AÞ2h2 A 2 ð1 2 e2j2;0 ðx2AÞ Þ ; 0 , A , x: þ h2 juj Remark 3.2. We have dA ðAþÞ ¼ 0 for every A . 0, i.e. continuous fit holds whatever the choice of A is. This is due to the fact that X has paths of unbounded variation (s . 0). As

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we see in the next section, continuous fit is applied to identify the optimal threshold level for the bounded variation case. We shall obtain the threshold level A * such that the smooth fit condition holds, i.e. ¼ 0 if such a threshold exists. By (3.9) and (3.10), we have i 9 8 h g 2h A 1 > > = < h2 ðh2 2 j1;q Þðj2;q 2 h2 Þe 2 2 q j1;q j2;q ; q . 0 ; A . 0: d0A ðAþÞ ¼ > > ; : ðj2;0 2 h2 Þe2h2 A 2 jujgh j2;0 ; q ¼ 0 and u , 0  2

d0A * ðA * þÞ

Therefore, on condition that 8 g < ðh2 2 j1;q Þðj2;q 2 h2 Þ . q j1;q j2;q ; q . 0 g

: ðj2;0 2 h2 Þ . jujh2 j2;0 ;

q¼0

and

9 =

u , 0 ;

;

ð3:11Þ

the smooth fit condition d0A ðAþÞ ¼ 0 is satisfied if and only if A ¼ A * . 0 where 8

j1;q j2;q g 1 > > < 2 h2 log q ðh2 2j1;q Þðj2;q 2h2 Þ ; q . 0;

ð3:12Þ A* U gj2;0 1 > > q ¼ 0 and u , 0: : 2 h2 log jujh2 ðj2;0 2h2 Þ ;

If (3.11) does not hold, we have d0A ðAþÞ , 0 for every A . 0; in this case, we set A * ¼ 0. We now show that the optimal value function is f U fA * (see (2.6)). Suppose q . 0 and A * . 0. Simple algebra shows that

dðxÞ U dA * ðxÞ ¼

h i g 1 * * j2;q ðe2j1;q ðx2A Þ 2 1Þ 2 j1;q ðe2j2;q ðx2A Þ 2 1Þ ; q j2;q 2 j1;q

This together with (3.6) shows that 8 P ðLi;q 2 C i;q Þe2ji;q x ; > > > i¼1;2 > > < P g C i;q e2ji;q x ; fðxÞ ¼ q 2 > i¼1;2 > > > > : 1;

x . A *: ð3:13Þ

x . A *; 0 , x # A *; x # 0;

where C1;q and C 2;q are defined in (3.7) and L1;q U

g j2;q * ej1;q A q j2;q 2 j1;q

and

L2;q U 2

g j1;q * ej2;q A : q j2;q 2 j1;q

When q ¼ 0 and A * . 0, we have

dðxÞ ¼ 2

 *  g 1 2 e2j2;0 ðx2A Þ ðx 2 A * Þ 2 ; juj j2;0

x . A*

ð3:14Þ

Stochastics: An International Journal of Probability and Stochastic Processes 121 and consequently,

8 g * þ j2;0 2h2 ð1 2 e2j2;0 x Þ þ 1 ð1 2 e2j2;0 ðx2A * Þ Þ ; > A > h j j j u j > 2 2;0 2;0 > <

2 j h g 2;0 2 2j2;0 x fðxÞ ¼ Þ ; > juj x þ h2 j2;0 ð1 2 e > > > : 1;

x . A *; 0 , x # A *; x # 0:

Finally, it is understood for the case A * ¼ 0 (for both q . 0 and q ¼ 0) that

fðxÞ ¼ limf1 ðxÞ; 1#0

ð3:15Þ

where, by Lemma 3.1, x 2qu 1{Xu ,0;u,1} Š ¼ limRðqÞ x ðt1 Þ ¼ E ½e 1#0

 ðh2 2 j1;q Þðj2;q 2 h2 Þ  2j1;q x e 2 e2j2;q x : h2 ðj2;q 2 j1;q Þ

This is the expectation of the cost incurred only when it jumps over the level zero. When A * . 0 the value function f can be attained by tA*, and when A * ¼ 0 it can be approximated arbitrarily closely by f1 ðxÞ (which is attained by t1 ) for sufficiently small 1 . 0. We, therefore, only need to verify that fðxÞ # Ex ½e2qtGðX t Þ1{t,1} Š for any t [ S. We first show that fð·Þ is dominated from above by the stopping value Gð·Þ. Lemma 3.3. We have fðxÞ # GðxÞ for every x [ R. Proof. For every 0 , A , x, ›fA ðxÞ=›A ¼ ›dA ðxÞ=›A equals i 9 8  h 2j1;q ðx2AÞ 1 > > þ l2;q e2j2;q ðx2AÞ 2ðh2 2 j1;q Þðj2;q 2 h2 Þe2h2 A þ gq j1;q j2;q ; q . 0 > > = < h2 l1;q e h i :   > 1 2j ðx2AÞ > > q ¼ 0 and u , 0 > 2h2 ðj2;0 2 h2 Þe2h2 A þ jguj j2;0 ; ; : h2 l1;0 þ l2;0 e 2;0

Here, in both cases, the term in the first bracket is strictly positive whereas that in the second bracket is increasing in A. Therefore, when A * . 0, A * is the unique value that makes it vanish, and consequently ›fA ðxÞ=›A $ 0 if and only if A $ A * . On the other hand, if A * ¼ 0, ›fA ðxÞ=›A $ 0 for every 0 , A , x. These imply when x . A * that by Remark 3.2

fðxÞ # limfA ðxÞ ¼ GðxÞ and fðxÞ ¼ limf1 ðxÞ # limfA ðxÞ ¼ GðxÞ; A"x

1#0

A"x

when A * . 0 and when A * ¼ 0, respectively. On the other hand, when 21 , x # A * , we A have fðxÞ ¼ GðxÞ by definition and hence the proof is complete.

Remark 3.3. (1) Suppose q . 0. In view of (3.6), Gð·Þ is bounded from above by g/q uniformly on x [ ð0; 1Þ. (2) Suppose u , 0 and fix x . 0. We have GðxÞ # Exu , 1 uniformly on q $ 0; namely, G(x) is bounded by Exu , 1 uniformly on q [ ½0; 1Þ.

122

M. Egami and K. Yamazaki

(3) Suppose A * . 0. Using the same argument as in the proof of Lemma 3.3, we have fðxÞ # lim1#0 f1 ðxÞ ¼ Ex ½e2qu 1{X u ,0;u,1} Š # 1 for every x . A * . When 0 , x # A * , using the monotonicity of Gð·Þ and continuous fit (see Remark 3.2), we have fðxÞ ¼ GðxÞ # GðA * Þ ¼ fðA * Þ # lim1#0 f1 ðA * Þ # 1. When A * ¼ 0, we have fðxÞ # 1 in view of (3.15). Therefore, fð·Þ is uniformly bounded and it only takes values on [0,1]. Now, we show that LfðxÞ $ qfðxÞ on ð0; 1Þn{A * } where L is the infinitesimal generator of X such that ð1 1 2 00 0 LwðxÞ ¼ mw ðxÞ þ s w ðxÞ þ l ½wðx þ zÞ 2 wðxފf ðzÞdz; x [ R; ð3:16Þ 2 21 for any C 2-function w : R ! R. The proof for the following lemma is lengthy and technical, and therefore is relegated to Appendix. Lemma 3.4. (1) If A * . 0, then we have x . A *;

LfðxÞ 2 qfðxÞ ¼ 0; LfðxÞ 2 qfðxÞ . 0;

0,x,

A *:

ð3:17Þ ð3:18Þ

(2) If A * ¼ 0, then (3.17) holds for every x . 0. Lemmas 3.3 and 3.4 show the optimality. The proof is very similar to that of Proposition 4.1 given in the next section (see Appendix A.6) and hence we omit it. Proposition 3.1. We have

fðxÞ ¼ inf Ex ½e2qtGðX t Þ1{t,1} Š; t[S

4.

x . 0:

Spectrally negative case

In this section, we analyse the case for a general spectrally negative Le´vy process. We shall obtain the optimal strategy and the value function in terms of the scale function for a general h. We assume throughout this section that q . 0. The results obtained here can, in principle, be extended to the case q ¼ 0 on condition that Exu , 1 along the same line as in the discussion in the previous section. The proofs of all lemmas and propositions are given in Appendix. 4.1

Scale functions

Let X be a spectrally negative Le´vy process with its Laplace exponent ð i h 1 ðe2bx 2 1 þ bx1{0,x,1} ÞPðdxÞ; b [ R; cðbÞ U E0 ebX1 ¼ cb þ s 2b 2 þ 2 ð0;1Þ Ð where c [ R, s $ 0 and P is a measure on ð0; 1Þ such that ð0;1Þ ð1 ^ x 2 ÞPðdxÞ , 1. See Kyprianou [22], p. 212. In particular, when ð ð1 ^ xÞPðdxÞ , 1; ð4:1Þ ð0;1Þ

Stochastics: An International Journal of Probability and Stochastic Processes 123 we can rewrite 1 cðbÞ ¼ mb þ s 2b 2 þ 2

ð

ð0;1Þ

ðe2bx 2 1ÞPðdxÞ;

b [ R;

where

mUcþ

ð

ð4:2Þ

xPðdxÞ: ð0;1Þ

The process has paths of bounded variation if and only if s ¼ 0 and (4.1) holds. It is also assumed that X is not a negative subordinator (decreasing a.s.). In other words, we require m to be strictly positive if s ¼ 0. It is well known that c is zero at the origin, convex on Rþ and has a right-continuous inverse:

zq U sup{l $ 0 : cðlÞ ¼ q};

q $ 0:

Associated with every spectrally negative Le´vy process, there exists a (q-)scale function W ðqÞ : R ! R;

q $ 0;

which is continuous and strictly increasing on ½0; 1Þ and satisfies ð1 1 ; b . zq : e2b xW ðqÞ ðxÞdx ¼ c ð b Þ 2q 0

If tþ a is the first time the process goes above a . x . 0, we have h h i W ðqÞ ðxÞ i W ðqÞ ðxÞ þ and Ex e2qu 1{tþa .u;u,1} ¼ Z ðqÞ ðxÞ 2 Z ðqÞ ðaÞ ðqÞ ; Ex e2qta 1{tþa ,u;tþa ,1} ¼ ðqÞ W ðaÞ W ðaÞ where ðx Z ðqÞ ðxÞ U 1 þ q W ðqÞ ðyÞdy;

ð4:3Þ

x [ R:

0

Here, we have W ðqÞ ðxÞ ¼ 0 on ð21; 0Þ and

Z ðqÞ ðxÞ ¼ 1 on ð21; 0Š: (q)

ð4:4Þ

is continuously We assume that P does not have atoms; this implies that W differentiable on ð0; 1Þ. See Chan et al. [9] for the smoothness properties of the scale function. The scale function increases exponentially; indeed, we have W ðqÞ ðxÞ ,

e zq x c0 ð z q Þ

ð4:5Þ

as x ! 1:

There exists a (scaled) version of the scale function W zq ¼ {W zq ðxÞ; x [ R} that satisfies, for every fixed q $ 0, W zq ðxÞ ¼ e2zq xW ðqÞ ðxÞ;

x [ R;

and ð1 0

e2b xW zq ðxÞdx ¼

1 ; cðb þ zq Þ 2 q

b . 0:

124

M. Egami and K. Yamazaki

Moreover W zq ðxÞ is increasing and as is clear from (4.5) W zq ðxÞ b

1

c0 ðz

ð4:6Þ

as x ! 1:

qÞ

From Lemmas 4.3 and 4.4 of Kyprianou and Surya [24], we also have the following results about the behaviour in the neighbourhood of zero:

W ðqÞ ð0Þ ¼

8 < 0;

9 unbounded variation =

and : m1 ; bounded variation ; 9 8 2 s .0 > > 2 ; > > s > > > > = < 0 ðqÞ 1; s ¼ 0 and Pð0; 1Þ ¼ 1 : W ð0þÞ ¼ > > > > qþPð0;1Þ > > > > ; : m 2 ; compound Poisson

ð4:7Þ

For a comprehensive account of the scale function, see Bertoin [5,6], Kyprianou [22] and Kyprianou and Surya [24]. See Surya [40] and Egami and Yamazaki [13] for numerical methods for computing the scale function. 4.2 Rewriting the problem in terms of the scale function for a general h We now rewrite the problem in terms of the scale function. For fixed A $ 0, define a random measure M

ðA;qÞ

ðv; BÞ U

ð t A ðvÞ 0

e2qt 1{X t ðvÞ[B} dt;

B [ BðRÞ:

v [ V;

Lemma 4.1. For any v [ V, we have ð tA ð v Þ 0

2qt

e hðX t ðvÞÞdt ¼

ð

R

M ðA;qÞ ðv; dyÞhðyÞ:

ð4:8Þ

With this lemma and the property of the random measure, we have x

E

 ð tA 0

2qt



x

e hðX t Þdt ¼ E

ð

R

M

ðA;qÞ



ðv; dyÞhðyÞ ¼

ð

R

ðdyÞhðyÞ; mðA;qÞ x

ð4:9Þ

where

mxðA;qÞ ðBÞ U Ex ½M ðA;qÞ ðBފ;

B [ BðRÞ

is a version of the q-resolvent kernel that has a density owing to the Radon –Nikodym theorem, see Bertoin [6]. By using Theorem 1 of Bertoin [6] (see also Emery [15] and

Stochastics: An International Journal of Probability and Stochastic Processes 125 Suprun [38]), we have for every B [ BðRÞ and a . x "ð # ð  ðqÞ  tA ^tþ a W ðx 2 AÞW ðqÞ ða 2 yÞ x ðqÞ 2qt E e 1{Xt [B} dt ¼ 2 1{x$y} W ðx 2 yÞ dy W ðqÞ ða 2 AÞ 0 B>½A;1Þ ¼

ð

B>½A;1Þ



2zq ðy2AÞ W

e

ðqÞ

 ðx 2 AÞW zq ða 2 yÞ ðqÞ 2 1{x$y} W ðx 2 yÞ dy: W zq ða 2 AÞ

Moreover, by taking a " 1 via the dominated convergence theorem in view of (4.6), we have   ð ðqÞ ðx 2 AÞW zq ða 2 yÞ ðqÞ 2zq ðy2AÞ W 2 1 mðA;qÞ W ðx 2 yÞ dy lim e ðBÞ ¼ {x$y} x W zq ða 2 AÞ B>½A;1Þ a!1 ð  2z ðy2AÞ ðqÞ  W ðx 2 AÞ 2 1{x$y} W ðqÞ ðx 2 yÞ dy; e q ¼ B>½A;1Þ

where the second equality holds by (4.6). Hence, we have the following result. Lemma 4.2. Fix q . 0 and 0 , A , x. We have
( 2zq ðy2AÞ ðqÞ W ðx 2 AÞ 2 1{x$y} W ðqÞ ðx 2 yÞ dy; e mðA;qÞ ðdyÞ ¼ x 0;

y $ A; y , A:

By (4.9) and Lemma 4.2, we have, for any arbitrary 0 , A , x, ð tA  ð1 ðx Ex e2qthðX t Þdt ¼ W ðqÞ ðx 2 AÞ e2zq yhðy þ AÞdy 2 W ðqÞ ðx 2 yÞ hðyÞdy; 0

0

A

ð4:10Þ

and this can be used to express the regret function and Ðthe stopping value. This also implies 1 that the integrability condition (1.1) is equivalent to 0 e2zq yhðyÞdy , 1. Using the q-resolvent kernel, we can also rewrite the violation risk. Lemma 4.3. For every 0 , A , x, we have ð ð1 1 ðqÞ 1 1 2zq ðu2AÞ RðqÞ ð Þ ¼ t W ðx 2 AÞ PðduÞð1 2 e Þ 2 PðduÞðZ ðqÞ ðx 2 AÞ 2 Z ðqÞ ðx 2 uÞÞ: A x zq q A A By (4.10) and Lemma 4.3, the difference function (2.5) becomes, for all 0 , A , x, ð tA  x 2qt ð Þ 2 e hðX Þdt dA ðxÞ ¼ RðqÞ t g E A t x 0    ð1 ð 1 ðqÞ 1 1 ¼ W ðx 2 AÞ PðduÞ 1 2 e2zq ðu2AÞ 2 PðduÞ Z ðqÞ ðx 2 AÞ zq q A A    ð1 ðx 2 Z ðqÞ ðx 2 uÞ 2 g W ðqÞ ðx 2 AÞ e2zq yhðy þ AÞdy 2 W ðqÞ ðx 2 yÞhðyÞdy : 0

A

ð4:11Þ

126

M. Egami and K. Yamazaki

4.3 Continuous and smooth fit We now apply continuous and smooth fit to obtain the candidate threshold levels for the bounded and unbounded variation cases, respectively. Firstly, the continuous-fit condition dA ðAþÞ ¼ 0 requires in view of (4.11) that W ðqÞ ð0ÞFðAÞ ¼ 0;

ð4:12Þ

where FðAÞ U

1 zq

ð1 A

ð1 PðduÞð1 2 e2zq ðu2AÞ Þ 2 g e2zq yhðy þ AÞdy;

A . 0:

0

Note in this calculation we used the fact that the second term in (4.11) vanishes as x # A by (4.4). Condition (4.12) automatically holds for the unbounded variation case by (4.7), but for the bounded variation case it requires FðAÞ ¼ 0:

ð4:13Þ

For the unbounded variation case, we apply smooth fit. For the violation risk, by using (4.3) and (4.4), we have  ð1  › ðqÞ 1 ðqÞ0  R ðtA Þ ¼ W ð0þÞ PðduÞð1 2 e2zq ðu2AÞ Þ: zq ›x x A x¼Aþ

For the regret function, by using (4.7) in particular W ðqÞ ð0Þ ¼ 0, we obtain   ð  ð1 ðx › x tA 2qt ðqÞ0 2 zq y ðqÞ0 E ¼ lim W ðx 2 AÞ e hðy þ AÞdy 2 W ðx 2 yÞhðyÞdy e hðX t Þdt x#A ›x 0 A 0 x¼Aþ ð1 0 ¼ W ðqÞ ð0þÞ e2zq yhðy þ AÞdy: 0

Therefore, for the unbounded variation case, smooth fit requires 0

W ðqÞ ð0þÞFðAÞ ¼ 0: Consequently, once we get (4.13), continuous fit holds for the bounded variation case and both continuous and smooth fits hold for the unbounded variation case (see Figure 4 in Section 5 for an illustration). Because h is non-decreasing by assumption, we have ð1 ð1 2zq ðu2AÞ 0 2 g e2zq yh0 ðy þ AÞdy , 0; A . 0: F ðAÞ ¼ 2 PðduÞe A

0

Hence, there exists at most one root that satisfies (4.13). We let A * be the root if it exists and zero otherwise. Because limA"1 FðAÞ , 0, A * ¼ 0 means that FðAÞ , 0 for all A . 0. 4.4 Verification of optimality We now show as in the last section that the optimal value function is f U fA * (see (2.6)) where in particular the case A * ¼ 0 is defined by (3.15). When A * . 0, it can be attained by the strategy tA * while when A * ¼ 0 it can be approximated arbitrarily closely by t1

Stochastics: An International Journal of Probability and Stochastic Processes 127 with sufficiently small 1 . 0. Recall that fðxÞ ¼ dðxÞ þ GðxÞ (with d U dA * ) for x . A * and that G(x) in (2.3) for x . 0 can be expressed in terms of the scale function by taking the limit A # 0 in (4.10). The corresponding candidate value function becomes both for A * ¼ 0 and for A * . 0 ð1 ð 1 1 1 * fðxÞ ¼ W ðqÞ ðx 2 A * Þ PðduÞð1 2 e2zq ðu2A Þ Þ 2 PðduÞðZ ðqÞ ðx 2 A * Þ zq q A* A*  ð1 ð1 2 Z ðqÞ ðx 2 uÞÞ þ g W ðqÞ ðxÞ e2zq yhðyÞdy 2 W ðqÞ ðx 2 A * Þ e2zq yhðy þ A * Þdy 0 0 # ð * A

2

0

W ðqÞ ðx 2 yÞhðyÞdy ;

ð4:14Þ for every x . 0. By definition, fðxÞ ¼ 1 for every x # 0. In particular, when A * . 0, we can simplify by using (4.13), ð 1 1 PðduÞðZ ðqÞ ðx 2 A * Þ 2 Z ðqÞ ðx 2 uÞÞ fðxÞ ¼ 2 q A* " # ð ð * þ g W ðqÞ ðxÞ

1

A

e2zq yhðyÞdy 2

0

0

W ðqÞ ðx 2 yÞhðyÞdy ;

x . 0:

These expressions for the candidate value function f are valid not only on ðA * ; 1Þ but also on ð0; A * Š thanks to (4.4) and continuous fit. In order to verify that f is indeed optimal, we only need to show that (1) f is dominated by G and (2) LfðxÞ $ qfðxÞ on ð0; 1Þn{A * } where ð1 1 Lf ðxÞ ¼ cf 0 ðxÞ þ s 2f 00 ðxÞ þ ½f ðx 2 zÞ 2 f ðxÞ þ f 0 ðxÞz1{0,z,1} ŠPðdzÞ; 2 0 ð1 Lf ðxÞ ¼ mf 0 ðxÞ þ ½f ðx 2 zÞ 2 f ðxފPðdzÞ; 0

for the unbounded and bounded variation cases, respectively; see (4.2) for the definition of m. The former is proved in the following lemma: Lemma 4.4. We have fðxÞ # GðxÞ for every x [ R. þ

Now we recall that the processes {e2qðt^u^ta ÞW ðqÞ ðX t^u^tþa Þ; t $ 0} and þ {e2qðt^u^ta ÞZ ðqÞ ðX t^u^tþa Þ; t $ 0} are Px-martingales for any 0 , x , a; see page 229 in Kyprianou [22]. Therefore, we have LW ðqÞ ðxÞ ¼ qW ðqÞ ðxÞ

and

LZ ðqÞ ðxÞ ¼ qZ ðqÞ ðxÞ;

We take advantage of (4.15) to show the following lemma. Lemma 4.5. (1) We have LfðxÞ 2 qfðxÞ ¼ 0;

x . A *:

x . 0:

ð4:15Þ

128

M. Egami and K. Yamazaki

(2) If A * . 0, we have LfðxÞ 2 qfðxÞ . 0;

0 , x , A *:

Finally, Lemmas 4.4 and 4.5 are used to show the optimality of f. Proposition 4.1. We have   fðxÞ ¼ inf Ex e2qtGðX t Þ1{t,1} ; t[S

5.

x . 0:

Numerical results

We conclude this paper by providing numerical results on the models studied in Sections 3 and 4. We obtain optimal threshold levels A * for (1) the double exponential jumpdiffusion case with h ; 1, and for (2) the spectrally negative case with h in the form of the exponential utility function. We study how the solution depends on the process X. We then verify continuous and smooth fit conditions for the bounded and unbounded variation cases, respectively. 5.1 The double exponential jump-diffusion case with h ; 1 We evaluate the results obtained in Section 3 focusing on the case h ; 1. Here, we plot the optimal threshold level A * defined in (3.12) as a function of g. The values of j1;q and j2;q are obtained via the bisection method with error bound 1024. Figure 2 shows how the optimal threshold level changes with respect to each parameter when q ¼ 0.05. The results obtained in (i) – (iv) and (vi) are consistent with our intuition because these parameters determine the overall drift u , and A * is expected to decrease in u . We show in (v) how it changes with respect to the diffusion coefficient s; although it does not play a part in determining u , we see that A * is in fact decreasing in s. This is related to the fact that, as s increases, the probability of jumping over the level zero decreases. 5.2

The spectrally negative Le´vy case with a general h

We now consider the spectrally negative case and verify the results obtained in Section 4. For the function h, we use the exponential utility function hðxÞ ¼ 1 2 e2rx ;

x . 0:

Here, r . 0 is called the coefficient of absolute risk aversion. It is well known that r ¼ 2h00 ðxÞ=hðxÞ for every x . 0, and, in particular, h ; 1 when r ¼ 1. We consider the tempered stable (CGMY) process and the variance gamma process with only downward jumps. The former has a Laplace exponent    b a ba cðbÞ ¼ cb þ C l a Gð2aÞ 1 þ 21 2 l l

and a Le´vy density given by

e2lx 1{x.0} dx; x 1þa for some l . 0 and a , 2; see Surya [39] for the calculations. It has paths of bounded variation if and only if a , 1. When a ¼ 0, it reduces to the variance gamma process; see PðdxÞ ¼ C

Stochastics: An International Journal of Probability and Stochastic Processes 129 (i) 7

(ii) 8

µ = –5 µ = 0.5 µ=1 µ=2

6 5

6 5 A*

A*

4 3

4 3

2

2

1

1 0

0 0

0.2

0.4

γ

0.6

0.8

1

0

0.2

0.4

γ

0.6

0.8

1

(iv) 6

(iii) 70 η– = 0.1 η– = 0.5 η– = 1 η– = 2

60 50 40

η+ = 0.1 η+ = 0.5 η+ = 1 η+ = 2

5 4 A*

A*

λ = 0.1 λ = 0.5 λ=1 λ=5

7

3

30 2

20

1

10 0

0

0.2

0.4

γ

0.6

0.8

0

1

(v) 7

0

0.2

0.4

γ

0.6

0.8

1

(vi) 7 σ = 0.1 σ=1 σ=2 σ=5

6 5

p = 0.1 p = 0.3 p = 0.5 p = 0.9

6 5

4 A*

A*

4

3

3

2

2

1

1 0

0 0

0.2

0.4

0.6 γ

0.8

1

0

0.2

0.4

0.6

0.8

1

γ

Figure 2. The optimal threshold level A * with respect to various parameters: the parameters are m ¼ 2 1, s ¼ 1, h2 ¼ 1.0, hþ ¼ 2.0, p ¼ 0.5, l ¼ 1 unless they are specified.

Proposition 5.7.1 of Surya [39] for the form of its Laplace exponent. We consider the case when s ¼ 0. The optimal threshold levels are computed by the bisection method using (4.13) with error bound 1026. Figure 3 shows the optimal threshold level A * as a function of g with various values of r. We see that it is indeed monotonically decreasing in r. This can be also analytically verified in view of the definition of FðAÞ because h is decreasing in r for every fixed A, and consequently the root A * must be decreasing in r. This is also clear because the regret function monotonically decreases in r.

130

M. Egami and K. Yamazaki (b) 2.5

(a) 1.8 ρ = 0.01 ρ = 0.02 ρ = 0.05 ρ = 0.1

1.6 1.4

1.5

1

A*

A*

1.2

ρ = 0.01 ρ = 0.02 ρ = 0.05 ρ = 0.1

2

0.8

1

0.6 0.4

0.5

0.2 0

0

0.1

0.2

0.3

0.4

0

0.5

0

0.1

0.2

γ

0.3

0.4

0.5

γ

(c) 2.5 2 ρ = 0.01 ρ = 0.02 ρ = 0.05 ρ = 0.1

A*

1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

γ

Figure 3. The optimal threshold level A * with various values of coefficients of absolute risk aversion r: (a) tempered stable (unbounded variation) with a ¼ 1.5, l ¼ 2, C ¼ 0.05 and c ¼ 0.05; (b) tempered stable (bounded variation) with a ¼ 0.8, l ¼ 2, C ¼ 0.075 and c ¼ 0.05; and (c) variance gamma with l ¼ 2, C ¼ 0.075 and c ¼ 0.05.

5.3

Continuous and smooth fit

We conclude this section by numerically verifying the continuous and smooth fit conditions. Unlike the optimal threshold level A * , the computation of the value function f involves that of the scale function. Here, we consider the spectrally negative Le´vy process with exponential jumps in the form (3.1) with p ¼ 1 and s $ 0. We consider the bounded variation case (s ¼ 0) and the unbounded variation case (s . 0). We also set h ; 1. This is a special case of the spectrally negative Le´vy process with phase-type jumps, and its scale function can be obtained analytically as in Egami and Yamazaki [13]. In general, scale functions can be approximated by Laplace inversion algorithm by Surya [40] or the phase-type fitting approach by Egami and Yamazaki [13]. One drawback of the approximation methods of the scale function is that the error tends to explode as x gets large (see (4.5)). Because our objective here is to accurately verify the continuous and smooth fit conditions, we use an example where an explicit form is known. Notice that the threshold level A * can be computed independently of the scale function, and hence one can alternatively approximate the value function f by simulation. Figure 4 draws the stopping value G as well as the value function f for both the bounded and unbounded variation cases. The value function f is indeed continuous at A * for the bounded variation case and differentiable for the unbounded variation case. It can

Stochastics: An International Journal of Probability and Stochastic Processes 131

0.3

0.3

0.2

0.2 φ and G

(b) 0.4

φ and G

(a) 0.4

0.1

0.1

0

0

–0.1

–0.1 –0.2

–0.2 0

0.5

1

1.5

2 X

2.5

3

3.5

4

0

0.5

1

1.5

2 X

2.5

3

3.5

4

Figure 4. Illustration of f (solid) and G (dotted) for (a) the bounded variation case with s ¼ 0 and m ¼ 0.3 and (b) the unbounded variation case with s ¼ 0.5 and m ¼ 0.175. Other parameters are q ¼ 0.05, h ; 1, l ¼ 0.5, h2 ¼ 1 and g ¼ 0.04.

be seen that G indeed dominates f. While G is monotonically increasing on ð0; 1Þ, f is decreasing for large x and is expected to converge to zero as x ! 1. Acknowledgements M. Egami is, in part, supported by Grant-in-Aid for Scientific Research (B) No. 22330098 and (C) No. 20530340, Japan Society for the Promotion of Science. K. Yamazaki is, in part, supported by Grant-in-Aid for Young Scientists (B) No. 22710143, the Ministry of Education, Culture, Sports, Science and Technology, and by Grant-in-Aid for Scientific Research (B) No. 2271014, Japan Society for the Promotion of Science. The authors thank Ning Cai, Masaaki Kijima, Michael Ludkovski, Goran Peskir and the anonymous referee for helpful suggestions and remarks.

Note 1.

Email: [email protected]

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[11] E. Cinlar and R. Vanderbei, Mathematical methods of engineering analysis, Available at http:// www.princeton.edu/rvdb/506book/book.pdf (2000). [12] B. Dao and M. Jeanblanc, Double exponential jump diffusion process: A structural model of endogenous default barrier with roll-over debt structure, The Universite´ d’E´vry, preprint (2006). [13] M. Egami and K. Yamazaki, On scale functions of spectrally negative Le´vy processes with phase-type jumps, arXiv:1005.0064 (2010). [14] P. Embrechts, C. Klu¨ppelberg, and T. Mikosch, Modelling Extremal Events, Vol. 33, Applications of Mathematics (New York), Springer-Verlag, Berlin, 1997. [15] D.J. Emery, Exit problem for a spectrally positive process, Adv. Appl. Probab. 5 (1973), pp. 498 –520. [16] A. Gut, Stopped Random Walks, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2009. [17] B. Hilberink and L.C.G. Rogers, Optimal capital structure and endogenous default, Finance Stoch. 6(2) (2002), pp. 237 – 263. [18] M. Huzak, M. Perman, H. Sˇikic´, and Z. Vondracˇek, Ruin probabilities and decompositions for general perturbed risk processes, Ann. Appl. Probab. 14(3) (2004), pp. 1378– 1397. [19] J. Jang, Jump diffusion processes and their applications in insurance and finance, Insurance Math. Econom. 41(1) (2007), pp. 62– 70. [20] S.G. Kou and H. Wang, First passage times of a jump diffusion process, Adv. Appl. Probab. 35(2) (2003), pp. 504 – 531. [21] S.G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Manag. Sci. 50(9) (2004), pp. 1178– 1192. [22] A.E. Kyprianou, Introductory Lectures on Fluctuations of Le´vy Processes with Applications, Universitext, Springer-Verlag, Berlin, 2006. [23] A.E. Kyprianou and Z. Palmowski, Distributional study of de Finetti’s dividend problem for a general Le´vy insurance risk process, J. Appl. Probab. 44(2) (2007), pp. 428 – 443. [24] A.E. Kyprianou and B.A. Surya, Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels, Finance Stoch. 11(1) (2007), pp. 131 – 152. [25] T.L. Lai, On uniform integrability in renewal theory, Bull. Inst. Math. Acad. Sinica 3(1) (1975), pp. 99– 105. [26] H.E. Leland, Corporate debt value, bond covenants, and optimal capital structure, J. Finance 49(4) (1994), pp. 1213– 1252. [27] H.E. Leland and K.B. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, J. Finance 51(3) (1996), pp. 987 – 1019. [28] A. Lipton and A. Sepp, Credit value adjustment for credit default swaps via the structural default model, J. Credit Risk 5(2) (2009), pp. 123 – 146. [29] R.L. Loeffen, On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Le´vy processes, Ann. Appl. Probab. 18(5) (2008), pp. 1669– 1680. [30] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, New York, 2005. [31] G. Peskir and A.N. Shiryaev, Sequential testing problems for Poisson processes, Ann. Statist. 28(3) (2000), pp. 837 – 859. [32] G. Peskir and A.N. Shiryaev, Solving the Poisson disorder problem, in Advances in Finance and Stochastics, Springer, Berlin, 2002, pp. 295 – 312. [33] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems., Lectures in Mathematics ETH Zu¨rich, Birkha¨user Verlag, Basel, 2006. [34] H. Schmidli, Stochastic Control in Insurance, Probability and its Applications (New York), Springer-Verlag London Ltd, London, 2008. [35] A. Sepp, Analytical pricing of double-barrier options under a double-exponential jump diffusion process: Applications of Laplace transform, Int. J. Theoret. Appl. Finance 7(2) (2004), pp. 151 – 175. [36] A.N. Shiryaev, Optimal Stopping Rules, Vol. 8, Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2008. [37] D. Siegmund, Sequential Analysis., Springer Series in Statistics, Springer-Verlag, New York, 1985. [38] V. Suprun, Problem of destruction and resolvent of terminating process with independent increments, Ukrainian Math. J. 28 (1976), pp. 39– 45.

Stochastics: An International Journal of Probability and Stochastic Processes 133 [39] B.A. Surya, Optimal stopping problems driven by Le´vy processes and pasting principles, Ph.D. diss., Universiteit Utrecht (2007). [40] B.A. Surya, Evaluating scale functions of spectrally negative Le´vy processes, J. Appl. Probab. 45(1) (2008), pp. 135 – 149. [41] A. Wald and J. Wolfowitz, Optimum character of the sequential probability ratio test, Ann. Math. Statist. 19 (1948), pp. 326 – 339. [42] A. Wald and J. Wolfowitz, Bayes solutions of sequential decision problems, Ann. Math. Statist. 21 (1950), pp. 82– 99. [43] M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis, Vol. 39, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1982. [44] K. Yamazaki, Essays on sequential analysis: Bandit problems with availability constraints and sequential change detection and identification, Ph.D. diss., Princeton University (2009).

Appendix: Proofs A.1 Proof of Lemma 3.4 We shall prove for the case q.0 and then extend it to the case q ¼ 0 and u , 0. We first prove the following for the proof of Lemma 3.4. Lemma A.1. Fix q . 0 and x [ R. Suppose that a function w : R ! R in a neighbourhood of x . 0 is given by wðxÞ ¼ k þ

X

ki e2ji;q x ;

i¼1;2

for some k, k1 and k2 in R. Then, we have "ð

qðwðxÞ2 kÞ ¼ LwðxÞ 2 l

!#  X  ph2 ð12 pÞhþ 2ji;q x : ki wðxþ zÞf ðzÞdz2 k þ þ e h2 2 ji;q hþ þ ji;q 21 i¼1;2 1

Proof. Because cð2j1;q Þ ¼ cð2j2;q Þ ¼ q, we have qðwðxÞ 2 kÞ ¼

X

i¼1;2

ki cð2ji;q Þe2ji;q x :

Moreover, the right-hand side equals by (3.3)   X  1 2 ph2 ð1 2 pÞhþ 2 ki 2mji;q þ s ðji;q Þ þ l þ 2 1 e2ji;q x h j h j 2 þ 2 2 i;q þ i;q i¼1;2   X  1 2 ph2 ð1 2 pÞhþ 2 e2ji;q x ¼ 2lðwðxÞ 2 kÞ þ ki 2mji;q þ s ðji;q Þ þ l þ h j h j 2 þ 2 2 i;q þ i;q i¼1;2  X  ph2 1 2 00 ð1 2 pÞhþ 2ji;q x 0 e ki þ ¼ s w ðxÞ þ mw ðxÞ 2 lwðxÞ þ lk þ l h2 2 ji;q hþ þ ji;q 2 i¼1;2 "ð !#  1 X  ph 2 ð1 2 pÞhþ 2ji;q x e ¼ LwðxÞ 2 l þ ki wðx þ zÞf ðzÞdz 2 k þ ; h2 2 ji;q hþ þ ji;q 21 i¼1;2 as desired.

A

134

M. Egami and K. Yamazaki

Proof of Lemma 3.4 when q . 0. (i) Suppose A * . 0. By Lemma A.1 above, we have LfðxÞ 2 qfðxÞ equals, for every x . A *,

l

"ð

#  ph 2 ð1 2 pÞhþ 2ji;q x e ðLi;q 2 C i;q Þ fðx þ zÞf ðzÞdz 2 þ h2 2 ji;q hþ þ ji;q 21 i¼1;2 1



X

ðA:1Þ

and, for every 0 , x , A * , "ð #   1 X g p h2 ð1 2 pÞhþ 2ji;q x 2ðq þ lÞ þ l e Ci;q fðx þ zÞf ðzÞdz þ þ ; ðA:2Þ q h2 2 ji;q hþ þ ji;q 21 i¼1;2 by using (3.6) and (3.14). Proof of (3.17). We only need to show (A.1) equals zero. Notice that the integral can be split into four parts, and, by using (3.7) and (3.8), we have ð1 0

ð0

2ðx2A * Þ

fðx þ zÞf ðzÞdz ¼

fðx þ zÞf ðzÞdz ¼

X

ð 2ðx2A * Þ

fðx þ zÞf ðzÞdz ¼

2x

i¼1;2

ðLi;q 2 Ci;q Þ

ðLi;q 2 C i;q Þ

i¼1;2



X

ð1 2 pÞhþ 2ji;q x ; e hþ þ ji;q

X ph 2 ðLi;q 2 C i;q Þ e2ji;q x 2 h2 2 ji;q i¼1;2

ph2 * * e2h2 ðx2A Þ2ji;q A ; h2 2 ji;q

pg 2h2 ðx2A * Þ X ph 2 * * e C i;q 2 e2h2 ðx2A Þ2ji;q A ; 2 q h j 2 i;q i¼1;2

ð 2x

fðx þ zÞf ðzÞdz ¼ pe2h2 x :

21

Putting altogether, (A,1) equals 2

X

Li;q

i¼1;2

ph 2 pg * * * e2h2 ðx2A Þ2ji;q A þ e2h2 ðx2A Þ þ pe2h2 x h2 2 ji;q q

  g j1;q j2;q * eh2 A ¼ pe2h2 x 1 2 q ðh2 2 j1;q Þðj2;q 2 h2 Þ and this vanishes because of the way A * is chosen in (3.12).

Stochastics: An International Journal of Probability and Stochastic Processes 135 Ð1 Proof of (3.18). We shall show that (A.2) is decreasing in x. Note that 21 fðx þ zÞf ðzÞdz can be split into four parts with ð1 X * * hþ ðLi;q 2 Ci;q Þ fðx þ zÞf ðzÞdz ¼ ð1 2 pÞ e2hþ ðA 2xÞ2ji;q A ; * j h þ i;q þ A 2x i¼1;2 ð A * 2x 0

   * g 2hþ ðA 2xÞ 12e fðx þ zÞf ðzÞdz ¼ ð1 2 pÞ q #  X * * hþ  2xji;q 2 2 e2hþ ðA 2xÞ2ji;q A C i;q e ; ji;q þ hþ i¼1;2 ð0

fðx þ zÞf ðzÞdz ¼

2x

ð 2x

X pg h2 2p C i;q e2ji;q x ; q h j 2 2 i;q i¼1;2

fðx þ zÞf ðzÞdz ¼ pe2h2 x

21

and after some algebra, we have ð1

X

fðx þ zÞf ðzÞdz þ

21

C i;q

i¼1;2



 p h2 ð1 2 pÞhþ 2ji;q x þ e h2 2 ji;q hþ þ ji;q

! g g X hþ 2ji;q A * hþ ðx2A * Þ 2h2 x ¼ þ pe e Li;q þ ð1 2 pÞe 2 þ ; q q i¼1;2 ji;q þ hþ which by (3.14) equals to   g g j1;q j2;q 1 1 * : þ pe2h2 x þ ð1 2 pÞe2hþ ðA 2xÞ 2 q j2;q þ hþ j1;q þ hþ j2;q 2 j1;q q Hence, we see that (A.2) or LfðxÞ 2 qfðxÞ equals   g 2h2 x 2hþ ðA * 2xÞ j1;q j2;q 2g þ l pe 2 ð1 2 pÞe q ðj2;q þ hþ Þðj1;q þ hþ Þ and is therefore decreasing in x on ð0; A * Þ. We now only need to show that limx"A * ðLfðxÞ 2 qfðxÞÞ . 0. For every x . A * ,

d00 ðxÞ ¼

i g j1;q j2;q h * * j1;q e2j1;q ðx2A Þ 2 j2;q e2j2;q ðx2A Þ q j2;q 2 j1;q

and hence, after taking x # A * ,

g d00 ðA * þÞ ¼ 2 j1;q j2;q , 0: q Consequently, by the continuous and smooth fit conditions dðA * þÞ ¼ d0 ðA * þÞ ¼ 0 and

136

M. Egami and K. Yamazaki

the definition of L in (3.16), we have limðLfðxÞ 2 qfðxÞÞ . limðLfðxÞ 2 qfðxÞÞ ¼ 0;

x"A *

x#A *

as desired. (ii) Suppose A * ¼ 0. By (3.15), we have   fðxÞ ¼ C e2j1;q x 2 e2j2;q x ;

where

CU

x . 0;

ðh2 2 j1;q Þðj2;q 2 h2 Þ : h2 ðj2;q 2 j1;q Þ

By Lemma A.1 above, we have LfðxÞ 2 qfðxÞ equals, for every x . 0, "ð #  1 X  ph2 ð1 2 pÞhþ 2ji;q x e C l fðx þ zÞf ðzÞdz 2 þ h2 2 ji;q hþ þ ji;q 21 i¼1;2 and this vanishes after some algebra.

A

Proof of Lemma 3.4 when q ¼ 0. We extend the results above to the case q ¼ 0 and u , 0. Solely in this proof, let us emphasize the dependence on q and use A*q , fq ð·Þ, Gq ð·Þ, dq ð·Þ and dA;q ð·Þ with a specified discount rate q $ 0. We shall show that Lf0 ðxÞ ¼ limq!0 Lfq ðxÞ for all x . 0. First notice that A*q ! A*0 as q ! 0. Clearly, Gq ðxÞ ! G0 ðxÞ by the monotone convergence theorem. Furthermore, we have   X g j2;0 2 h2 2j2;0 x q#0 g ¼ G00 ðxÞ; 1þ j1;q j2;q e li;q e2ji;q x ! G0q ðxÞ ¼ qh2 j u j h 2 i¼1;2 G00q ðxÞ ¼ 2

  X g g ðj2;0 2 h2 Þj2;0 2j2;0 x q#0 ¼ G000 ðxÞ: j1;q j2;q ji;q li;q e2ji;q x ! 2 e qh2 j u j h 2 i¼1;2

Fix x . A*0 . We suppose A*0 ¼ 0 and focus on q [ ½0; q0 Š with q0 . 0 sufficiently small such that A*q ¼ 0 for all 0 # q # q0 . We have, by applying the monotone convergence theorem on (3.15), fq ðxÞ ! f0 ðxÞ as q ! 0. Furthermore,

f0q ðxÞ ¼ f00q ðxÞ ¼


q#0 ðh2 2 j1;q Þðj2;q 2 h2 Þ 2j1;q e2j1;q x þ j2;q e2j2;q x ! j2;0 e2j2;0 x ¼ f00 ðxÞ; h2 ðj2;q 2 j1;q Þ ðh2 2 j1;q Þðj2;q 2 h2 Þ 2 2j1;q x q#0 ðj1;q e 2 j22;q e2j2;q x Þ ! 2 j22;0 e2j2;0 x ¼ f000 ðxÞ: h2 ðj2;q 2 j1;q Þ

Suppose A*0 . 0 and focus on q [ ½0; q0 Š with q0 . 0 sufficiently small such that x . A*q for all 0 # q # q0 . Note that jdq ðxÞ 2 d0 ðxÞj # jdq ðxÞ 2 dA*q ;0 ðxÞj þ jdA*q ;0 ðxÞ 2 d0 ðxÞj;

Stochastics: An International Journal of Probability and Stochastic Processes 137 where on the right-hand side the former vanishes as q ! 0 by the monotone convergence theorem in view of (2.5) and the latter vanishes because A*q ! A*0 ; hence, dq ðxÞ ! d0 ðxÞ as q ! 0. Furthermore,

d0q ðxÞ ¼ d00q ðxÞ ¼

iq#0 g j1;q j2;q h 2j1;q ðx2A*q Þ g * * þ e2j2;q ðx2Aq Þ ! 2 ½1 2 e2j2;0 ðx2A0 Þ Š ¼ d00 ðxÞ; 2e q j2;q 2 j1;q juj

iq#0 g j1;q j2;q h g * * * j1;q e2j1;q ðx2Aq Þ 2 j2;q e2j2;q ðx2Aq Þ ! 2 j2;0 e2j2;0 ðx2A0 Þ ¼ d000 ðxÞ: q j2;q 2 j1;q juj

In summary, we have fq ðxÞ ! f0 ðxÞ, f0q ðxÞ ! f00 ðxÞ and f00q ðxÞ ! f000 ðxÞ as q ! 0 for every x . 0. Moreover, by Remark 3.3-(3), via the dominated convergence theorem, ð1

f0 ðx þ zÞf ðzÞdz ¼ lim

ð1

q!1 21

21

fq ðx þ zÞf ðzÞdz;

x $ 0:

Consequently, limq!0 ðLfq ðxÞ 2 qfq ðxÞÞ ¼ Lf0 ðxÞ. This together with the result for q . 0 shows the claim. A A.2 Proof of Lemma 4.1 Because h is continuous on ð0; 1Þ, it is Borel measurable. Hence, there exists a converging sequence of simple functions ðh ðnÞ Þn[N increasing to h in the form h ðnÞ ðyÞ U

lðnÞ X

bðnÞ i 1{y[Bn;i } ;

n $ 1;

i¼1

for some l : N ! N, {bðnÞ i ; n $ 1; 0 # i # lðnÞ} and Borel measurable {Bn;i ; n $ 1; 0 # i # lðnÞ}; see page 99 of Cinlar and Vanderbei [11]. Then the right-hand side of (4.8) is, by the monotone convergence theorem, ð

M

ðA;qÞ

ðv; dyÞhðyÞ ¼

R

ð

M ðA;qÞ ðv; dyÞ lim

n!1

R

¼ lim

n!1

lðnÞ X

biðnÞ

i¼1

ð

lðnÞ X

sets

bðnÞ i 1{y[Bn;i }

i¼1

M ðA;qÞ ðv; dyÞ1{y[Bn;i } ¼ lim

n!1

R

lðnÞ X

ðA;qÞ bðnÞ ðv; Bn;i Þ: i M

i¼1

This is indeed equal to the left-hand side of (4.8) because, by the monotone convergence theorem, ð tA ð v Þ

e2qthðX t ðvÞÞdt ¼

0

ð tA ð v Þ 0

¼ lim

n!1

as desired.

e2qt lim

lðnÞ X i¼1

n!1

biðnÞ

lðnÞ X i¼1

ð t A ð vÞ 0

bðnÞ i 1{X t ðvÞ[Bn;i } dt

e2qt 1{X t ðvÞ[Bn;i } dt ¼ lim

n!1

lðnÞ X i¼1

ðA;qÞ bðnÞ ðv; Bn;i Þ; i M

138

M. Egami and K. Yamazaki

A.3 Proof of Lemma 4.3 Let Nð·; ·Þ be the Poisson random measure for 2X and X t U min0#s#t X s for all t $ 0. By the compensation formula (see, e.g. Theorem 4.4 in Kyprianou [22]), we have x RðqÞ x ðtA Þ ¼ E

¼E

x

ð 1 ð 1 0

0

ð 1

e

Nðdt; duÞe2qt 1{X t2 2u#0;X t2 .A}

2qt

dt

0

0

¼

ð1

PðduÞ

0

¼

ð1

ð1

ð1 0

PðduÞ

ð1 0

0

PðduÞ1{X t2 2u#0;Xt2 .A}





  dt e2qt Px {X t2 # u; X t2 . A}

  dt e2qt Px {X t2 # u; tA $ t} :

By using the q-resolvent kernel that appeared in Lemma 4.2, we have for u . A ð1 0



dt e

2qt

x



P {X t2 # u; tA $ t} ¼

ðu

A

¼

ð u2A 0

¼

  dy e2zq ðy2AÞ W ðqÞ ðx 2 AÞ 2 W ðqÞ ðx 2 yÞ   dz e2zq z W ðqÞ ðx 2 AÞ 2 W ðqÞ ðx 2 z 2 AÞ

1 ðqÞ W ðx 2 AÞð1 2 e2zq ðu2AÞ Þ 2 zq

ð u2A

dzW ðqÞ ðx 2 z 2 AÞ

0

and it is zero on 0 # u # A. Substituting this, we have RðqÞ x ðtA Þ

ð1



1 ðqÞ W ðx 2 AÞð1 2 e2zq ðu2AÞ Þ 2 ¼ PðduÞ z q A

ð u2A 0

dzW

ðqÞ



ðx 2 A 2 zÞ :

By (4.4), we have RðqÞ x ðtA Þ ¼

ð1
1 ðqÞ W ðx 2 AÞ PðduÞ 1 2 e2zq ðu2AÞ zq A ðx ð u2A ð1 ð x2A 2 PðduÞ dzW ðqÞ ðx 2 A 2 zÞ 2 PðduÞ dzW ðqÞ ðx 2 A 2 zÞ A

0

x

ð1

0

1 ðqÞ W ðx 2 AÞ PðduÞð1 2 e2zq ðu2AÞ Þ zq A ðx ð 1 1 1 2 PðduÞðZ ðqÞ ðx 2 AÞ 2 Z ðqÞ ðx 2 uÞÞ 2 PðduÞðZ ðqÞ ðx 2 AÞ 2 1Þ q A q x ð1 1 ¼ W ðqÞ ðx 2 AÞ PðduÞð1 2 e2zq ðu2AÞ Þ zq A ð1 1 2 PðduÞðZ ðqÞ ðx 2 AÞ 2 Z ðqÞ ðx 2 uÞÞ; q A

¼

as desired.

Stochastics: An International Journal of Probability and Stochastic Processes 139 A.4 Proof of Lemma 4.4 Fix 0 , A , x. We have  ð 1 › ðqÞ 1 0 Rx ðtA Þ ¼ W ðqÞ ðx 2 AÞ 2 W ðqÞ ðx 2 AÞ PðduÞð1 2 e2zq ðu2AÞ Þ zq ›A A ð1
1 ¼ 2 W z0 q ðx 2 AÞezq ðx2AÞ PðduÞ 1 2 e2zq ðu2AÞ zq A

and because we can write ð t A  ð1 ðx Ex e2qthðX t Þdt ¼ ezq xW zq ðx 2 AÞ e2zq yhðyÞdy 2 W ðqÞ ðx 2 yÞhðyÞdy; 0

A

A

we have

› x E ›A

ð tA 0

2qt



e hðX t Þdt ¼

2W z0 q ðx

2 AÞe

zq x

ð1

e2zq yhðyÞdy 2 W zq ðx 2 AÞezq ðx2AÞhðAÞ

A

þ W ðqÞ ðx 2 AÞhðAÞ ¼

2W z0 q ðx

2 AÞe

zq ðx2AÞ

ð1

e2zq yhðy þ AÞdy:

0

Summing up these, we obtain

› dA ðxÞ ¼ 2W z0 q ðx 2 AÞezq ðx2AÞ FðAÞ: ›A Here, because W z0 q ðx 2 AÞ . 0 and FðAÞ is decreasing in A and attains zero at A * , we have

› dA ðxÞ . 0 () FðAÞ , 0 () A . A * : ›A

ðA:3Þ

Now suppose A * . 0, we have

fðxÞ ¼ GðxÞ þ dA * ðxÞ # GðxÞ þ limdA ðxÞ # GðxÞ; A"x

x . A *;

where the last inequality holds because continuous fit holds everywhere for the unbounded variation case and because limA"x dA ðxÞ ¼ W ðqÞ ð0ÞFðxÞ , 0 by (A.3) for the bounded variation case (by noting that x . A * ). The case with A * ¼ 0 holds in the same way by the definition that fðxÞ ¼ lim1#0 f1 ðxÞ. Finally, because f ¼ G on ð21; A * Š, the proof is complete. A.5

Proof of Lemma 4.5 (1) When x . A * , f is defined in (4.14). Let f~ be defined such that f~ðxÞ ¼ fðxÞ for all x . 0 and f~ðxÞ ¼ 0 for all x # 0. We obtain ð  ð1 1 1 PðduÞLZ ðqÞ ðx 2 uÞ 2 q PðduÞZ ðqÞ ðx 2 uÞ Lf~ðxÞ 2 qf~ðxÞ ¼ q x x ¼ 2Pðx; 1Þ:

140

M. Egami and K. Yamazaki

Here, the first equality holds by (4.15) and because the operator L can go into the integrals thanks to the fact that Z ðqÞ is C 1 everywhere and C 2 on Rn{0} for the unbounded variation case and it is C 0 everywhere and C 1 on Rn{0} for the bounded variation case, and also to the fact that x . A * . The second equality holds by (4.4). Because f~ðxÞ ¼ fðxÞ and LfðxÞ 2 Lf~ðxÞ ¼ Pðx; 1Þ for every x . 0, we have the claim. (2) Suppose 0 , x , A * . We can write ðx ð1 fðxÞ ¼ gW ðqÞ ðxÞ e2zq yhðyÞdy 2 g W ðqÞ ðx 2 yÞhðyÞdy þ LðxÞ; ðA:4Þ 0 0 x [ ð21; A * Þn{0}; where LðxÞ ¼ 1{x#0} for every x [ R. After applying ðL 2 qÞ, the first term vanishes thanks to (4.15). For the second term, by integration by parts,  y¼x ðx ðqÞ ðqÞ ðqÞ q W ðx 2 yÞhðyÞdy ¼ hðyÞðZ ðxÞ 2 Z ðx 2 yÞÞ 0

y¼0

ðx

h0 ðyÞðZ ðqÞ ðxÞ 2 Z ðqÞ ðx 2 yÞÞdy ðx ¼ hðxÞðZ ðqÞ ðxÞ 2 1Þ 2 h0 ðyÞðZ ðqÞ ðxÞ 2 Z ðqÞ ðx 2 yÞÞdy 2

0

0

¼ hðxÞðZ ðqÞ ðxÞ 2 1Þ 2 ðhðxÞ 2 hð0ÞÞZ ðqÞ ðxÞ ðx þ h0 ðyÞZ ðqÞ ðx 2 yÞdy 0 ðx ¼ 2hðxÞ þ hð0ÞZ ðqÞ ðxÞ þ h0 ðyÞZ ðqÞ ðx 2 yÞdy 0 ðM ðqÞ ¼ 2hðMÞ þ hð0ÞZ ðxÞ þ h0 ðyÞZ ðqÞ ðx 2 yÞdy; 0

for any M . x where the last equality holds because Z ðqÞ ðxÞ ¼ 1 on ð21; 0Š. The operator ðL 2 qÞ can again go into the integral thanks to the smoothness of Z ðqÞ as discussed in (1) and we obtain ð x    ðM 1 ðqÞ 0 ðqÞ W ðx 2 yÞhðyÞdy ¼ ðL 2 qÞ 2hðMÞ þ h ðyÞZ ðx 2 yÞdy ðL 2 qÞ q 0 0 ðM 1 h0 ðyÞðL 2 qÞZ ðqÞ ðx 2 yÞdy ¼ hðMÞ þ q 0 ðM ¼ hðMÞ 2 h0 ðyÞdy ¼ hðxÞ; x

where the second to last equality holds by (4.15). For the last term of (A.4), we have ð1 ðL 2 qÞLðxÞ ¼ ðLðx 2 uÞ 2 LðxÞÞPðduÞ ¼ Pðx; 1Þ: 0

Stochastics: An International Journal of Probability and Stochastic Processes 141 Putting altogether, we have noting that x , A * and h is increasing, ðL 2 qÞfðxÞ ¼ Pðx; 1Þ 2 ghðxÞ $

ð1

A*

  2zq ðu2A * Þ 2 ghðxÞ PðduÞ 1 2 e

¼

ð1

  ð1 * PðduÞ 1 2 e2zq ðu2A Þ 2 gzq e2zq yhðxÞdy

$

ð1

  ð1 2zq ðu2A * Þ 2 gzq e2zq yhðA * þ yÞdy; PðduÞ 1 2 e

A*

A*

0

0

which is zero because A * satisfies (4.13). This completes the proof. A.6 Proof of Proposition 4.1 Due to the discontinuity of the value function at zero, we need to proceed carefully. By (2.4) and Lemma 4.4, we must have 1 ¼ fð0Þ . fð0þÞ. We first construct a sequence of functions fn ð·Þ such that (1) it is C 2 (resp. C 1) everywhere except at A * when s . 0 (s ¼ 0), (2) fn ðxÞ ¼ fðxÞ on x [ ð0; 1Þ and (3) fn ðxÞ " fðxÞ pointwise for every fixed x [ ð21; 0Þ (with limn!1 fn ð0Þ ¼ fð0þÞ , fð0Þ). It can be shown along the same line as Remark 3.3-(3) that fð·Þ is uniformly bounded. Hence, we can choose so that fn is also uniformly bounded for every n $ 1. Because f0 ðxÞ ¼ f0n ðxÞ and f00 ðxÞ ¼ f00n ðxÞ on ð0; 1Þn{A * }, we have ðL 2 qÞ ðfn 2 fÞðxÞ # 0 for every fixed x [ ð0; 1Þn{A * }. Furthermore, by Lemma 4.5 Ex

ð t

0

ð t   e2qs ððL 2 qÞfn ðX s2 ÞÞds $ Ex e2qs ððL 2 qÞðfn 2 fÞðX s2 ÞÞds . 21; t [ S: 0

ðA:5Þ

Here, the last lower bound is obtained because x

E

ð t

e

2qs

0

 ð u  x 2qs ððL 2 qÞðfn 2 fÞðX s2 ÞÞds $ 2KE e PðX s2 ; 1Þds ; 0

where K , 1 is the maximum difference between f and fn. Using N as the Poisson random measure for 2X and X as the running infimum of X as in the proof of Lemma 4.3, we have by the compensation formula

E

x

ð u

0

e

2qs



PðX s2 ; 1Þds ¼ E

x

ð 1 ð 1 0 0 ð 1 ð1

2qs

e

1{u$s;u.Xs2 } PðduÞds



 e 1{X s2 .0;u.X s2 } PðduÞds ¼E ð 01 ð 01  x 2qs e 1{Xs2 .0;u.Xs2 } Nðds; duÞ ¼E   0 0 x 2qu ¼ E e 1{X u ,0;u,1} , 1: x

2qs

142

M. Egami and K. Yamazaki

By (A.5), we have uniformly in n Ex Ðt

0e

Ð t

2qs

0e

2qs

 jðL 2 qÞðfn 2 fÞðX s2 Þjds , 1;

jðL 2 qÞðfn 2 fÞðX s2 Þjds , 1;

ðA:6Þ

Px 2 a:s:

We remark here for the proof of Proposition 3.1 that, in the double exponential case with q ¼ 0, there also exists a finite bound because the Le´vy measure is a finite measure and Exu , 1 by assumption. Notice that, although fn is not C 2 (resp. C 1) at A * for the case s . 0 (the case of bounded variation), the Lebesgue measure of the set where fn at which X ¼ A * is zero and hence f00n ðA * Þ (f0n ðA * Þ) can be chosen arbitrarily; see also Theorem 2.1 of [30]. By applying Ito’s formula to {e2qðt^uÞfn ðX t^u Þ; t $ 0}, we see that 

e

2qðt^uÞ

fn ðX t^u Þ 2

ð t^u

e

2qs

ððL 2 qÞfn ðX s2 ÞÞds;

t$0

0



ðA:7Þ

is a local martingale. Suppose {s k ; k $ 1} is the corresponding localizing sequence, namely, x



E e

2qðt^u^s k Þ

x



fn ðX t^u^s k Þ ¼ fn ðxÞ þ E

ð t^u^s k

2qs

e



ððL 2 qÞfn ðX s2 ÞÞds ;

0

k $ 1:

Now by applying the dominated convergence theorem on the left-hand side and Fatou’s lemma on the right-hand side via (A.5), we obtain   Ex e2qðt^uÞfn ðX t^u Þ $ fn ðxÞ þ Ex

ð t^u 0

 e2qs ððL 2 qÞfn ðX s2 ÞÞds :

Hence (A.7) is in fact a submartingale. Now fix t [ S. By the optional sampling theorem, for any M $ 0, x



E e

2qðt^MÞ

ð t^M

x



fn ðX t^M Þ $ fn ðxÞ þ E

0

x

¼ fn ðxÞ þ E

ð t^M 0

þ Ex

ð t^M 0

e

2qs

e

2qs

 ððL 2 qÞfn ðX s2 ÞÞds ððL 2 qÞfðX s2 ÞÞds



 e2qs ððL 2 qÞðfn 2 fÞðX s2 ÞÞds ;

where the last equality holds because the expectation can be split by (A.5). Applying the dominated convergence theorem on the left-hand side and the monotone convergence theorem on the right-hand side (here the integrands in the two expectations are positive and negative, respectively) along with Lemma 4.5, we obtain x



2qt

E e



fn ðX t Þ1{t,1} $ fn ðxÞ þ E

x

ð t

0

e

2qs



ððL 2 qÞðfn 2 fÞðX s2 ÞÞds :

ðA:8Þ

Stochastics: An International Journal of Probability and Stochastic Processes 143 For the left-hand side, the dominated convergence theorem implies h i   lim Ex e2qtfn ðX t Þ1{t,1} ¼ Ex e2qt lim fn ðX t Þ1{t,1} n!1

n!1

  ¼ Ex e2qt ðfðX t Þ1{X t –0} þ fð0þÞ1{Xt ¼0} Þ1{t,1}   # Ex e2qtfðX t Þ1{t,1} ;

ðA:9Þ

which holds by equality when s ¼ 0 because X creeps the level zero only when s . 0. For the right-hand side, by (A.6), ð t  x 2qs e ððL 2 qÞðfn 2 fÞðX s2 ÞÞds lim E n!1

¼ Ex

0

ð t

0

 e2qs lim ððL 2 qÞðfn 2 fÞðX s2 ÞÞds : n!1

ðA:10Þ

Here, for every Px -a.e. v [ V, because X s2 ðvÞ . 0 for Lebesgue-a.e. s on ð0; tðvÞÞ ð1 PðduÞ lim ðfn 2 fÞðX s2 ðvÞ 2 uÞ ¼ 0: lim ððL 2 qÞðfn 2 fÞðX s2 ÞðvÞÞ ¼ n!1

X s2 ðvÞ

n!1

Hence, (A.10) vanishes. Therefore, by taking n ! 1 on both sides of (A.8) (note fðxÞ ¼ fn ðxÞ for any x . 0), we have     fðxÞ # Ex e2qtfðX t Þ1{t,1} # Ex e2qtGðX t Þ1{t,1} ; t [ S;

where the last inequality follows from Lemma 4.4. Finally, the stopping time tA * attains the value function f when A * . 0 while t1 and f1 approximate f by taking 1 sufficiently small when A * ¼ 0. This completes the proof.

European Accounting Review, 2014 http://dx.doi.org/10.1080/09638180.2014.918518

Real Earnings Management Uncertainty and Corporate Credit Risk TSUNG-KANG CHEN∗ , YIJIE TSENG∗∗ and YU-TING HSIEH† ∗

Department of Finance and International Business, Fu Jen Catholic University, New Taipei City, Taiwan, ROC, Department of Accounting, Fu Jen Catholic University, New Taipei City, Taiwan, ROC and †Department of Accounting, National Taiwan University, Taipei City, Taiwan, ROC ∗∗

(Received: October 2012; accepted: April 2014)

A BSTRACT This study examines the accounting information uncertainty effects on corporate credit risk from the perspective of real earnings management (RM) activities by investigating 9565 American bond observations from year 2001 to 2008. The main results show that the volatilities of RM activities significantly and positively affect corporate bond yield spreads when well-known bond spread determinant variables are controlled. In addition, the results are robust to alternative model specifications, including the suspect firm analyses, another less ambiguous measure of abnormal cash flows from operations, and abnormal production cost analyses in manufacturing industry or with control of the input price variation. This research also finds that the positive effects of RM volatilities become weaker if a firm has a lower credit rating. Finally, our results remain hold with considering endogeneity issues and analyst characteristic variables and for another estimation period of RM volatilities.

1.

Introduction

Many studies have documented that a firm’s idiosyncratic risks have enormous impacts on its credit risk (Campbell & Taksler, 2003; Chen, Chen, & Liao, 2011; Duffie & Lando, 2001). Incomplete accounting information, a major idiosyncratic risk, plays a critical role in corporate credit risk, as described in the influential work of Duffie and Lando (2001). The effects of incomplete accounting information on corporate credit risk have been explored from the perspectives of disclosure ranking (Yu, 2005), analyst forecasts (Gu¨ntay & Hackbarth, 2010), and tradingbased information asymmetry (Lu, Chen, & Liao, 2010).1 Moreover, Duffie and Lando’s (2001) model reveals that not only does the incomplete accounting information level (degree) have a significant impact on a firm’s real asset value distributions, and therefore its credit risk, but so do its variations. However, how incomplete accounting information variations affect credit risk is rarely discussed in empirical studies. To address this issue, this work employs the information content variations embedded within a firm’s real earnings management (RM) activities to explore the effects of incomplete accounting information uncertainty on corporate credit risk. Correspondence Address: Yijie Tseng, Department of Accounting, Fu Jen Catholic University, 259 Loyola Building, No. 510, Zhongzheng Rd., Xinzhuang Dist., New Taipei City 24205, Taiwan, ROC. Email: [email protected] Paper accepted by Guochang Zhang. 1 Yu (2005), Gu¨ntay and Hackbarth (2010), and Lu et al. (2010), respectively, employ the corporate disclosure ranking assessed by financial analysts, analyst forecasts, and information asymmetry between informed and uninformed traders to explain the variations of bond yield spreads. # 2014 European Accounting Association

2

T.-K. Chen et al.

Figure 1. Structural credit model frameworks of Merton (1974) and Duffie and Lando (2001). Here D is the default threshold, f (v) is the probability density function (p.d.f.) of a firm’s asset value, and m measures a firm’s expected value. The probability of default is the area of asset value p.d.f. less than default threshold. For the effects of RM levels on a firm’s credit risk, due to that RM is associated with firm operating performance shown in the previous RM-related studies (e.g. Cohen & Zarowin, 2010; Gunny, 2010; Leggett et al., 2009), we hypothesise that the RM levels affect a firm’s asset value, which influences the firm’s credit risk (Merton, 1974). For the effects of RM variations on a firm’s credit risk, Lambert et al. (2007) show that the assessed variance of a firm’s asset value increases with its poor quality accounting information (low precision or high variation). Since RM cannot be fully detected by outside investors, the assessed effects of RM on the accounting information present measurement errors. Hence, the assessed asset value distribution (the dotted line) given RM fluctuations has a higher variation relative to the real asset value distribution (the solid line). That is, a firm with more volatile RM activities has greater uncertain and ambiguous asset value distributions, which increase the firm’s credit risk (Duffie & Lando, 2001).

Since the decisive work of Roychowdhury (2006), RM has become more important than accrual manipulations, especially for the management of operational activities. The perspective that RM has greater consequences compared with accrual-based earnings management is consistent with Graham, Harvey, and Rajgopal (2005), Gunny (2005), Zang (2006), and Cohen and Zarowin (2010). In general, RM makes a firm’s reported earnings deviate from underlying real economic substance, and therefore, a firm’s assessed asset value distributions diverge from the actual ones. As a result, RM seems to constitute a major portion of a firm’s incomplete accounting information. According to the structural credit model frameworks of Merton (1974) and Duffie and Lando (2001) shown in Figure 1,2 credit risk is defined as the probability of a firm’s asset value being less than a given default threshold. The factors influencing credit risk include the drivers leading to changes of a firm’s asset value distribution. For the effects of RM variations on a firm’s credit risk,3 Lambert, Leuz, and Verrecchia (2007) indicate that better quality accounting information (higher precision or less variation) reduces the assessed variance of the firm’s asset value. This perspective is consistent with Duffie and Lando’s (2001) model, which proposes that uncertainties due to incomplete accounting information have significant impacts on its asset value distribution and therefore its credit risk. To make this argument more concrete, we apply an accounting information 2

In Figure 1, Merton’s model (1974) shows that a firm’s credit risk is determined by its asset value distribution (including asset return and asset volatility) and default threshold. Duffie and Lando (2001) show that the lack of precise accounting information can lead to different predictions on the shape of a firm’s value distribution. Hence, asset return, asset volatility, default threshold, and incomplete accounting information all contribute to the essence of corporate credit risk. 3 We focus on addressing the effects of RM variations, but still empirically examining the effects of both RM levels and RM variations on the credit risk in Section 5. Gunny (2010) suggests RM is associated with better performance, while other studies reach the opposite conclusions (Cohen & Zarowin, 2010; Gunny, 2005; Gupta, Pevzner, & Seethamraju, 2010; Leggett et al., 2009; Roychowdhury, 2006). Therefore, we hypothesise that the RM levels affect the operating performance, which influences the credit risk, but the direction is either positive or negative, depending on the costs of RM and the incentives to manage earnings. In addition, Achleitner, Guenther, Kaserer, and Siciliano (2014) find that nonfamily firms engage more RM activities than family firms.

Real Earnings Management Uncertainty and Corporate Credit Risk 3 model based on Lambert et al. (2007) to show the effects of RM variations on a firm’s credit risk. Since RM cannot be fully detected by outside investors, the assessed effects of RM on the accounting information present measurement errors. Increasing RM fluctuation limits investors’ ability to undo the earnings management, implying that the accounting information is less precise. Therefore, the assessed asset value distribution given RM fluctuations has a higher variation. The finding of Leggett, Parsons, and Reitenga (2009) that a firm’s RM activities change the firm’s future cash flow streams could also lead to a similar inference. That is, a firm with more volatile RM activities has greater uncertainty about future cash flows and larger asset value variations, which increase the firm’s credit risk based on structural credit models. In addition, in the application model of Lambert et al. (2007), the posterior mean of asset value under RM manipulations is estimated by the precision (namely inverse-variance) weighted sum of all available information.4 The inflated accounting information embedded with incomeincreasing RM activities causes investors to overestimate a firm’s asset mean value and then underestimate its credit risk when controlling for the costs of RM activities. However, increasing RM fluctuations lead to investors having less precise knowledge of the inflated accounting information and assigning less weight to it, which suggests that the underestimation of the credit risk is alleviated. Therefore, the credit risk assessed by investors is increased. Accordingly, this research leads to the hypothesis that a firm’s RM fluctuations positively relate to its credit risk (bond yield spreads). To measure the levels of RM, this study follows Roychowdhury (2006), Cohen, Dey, and Lys (2008), and Cohen and Zarowin (2010) to employ abnormal levels of cash flow from operations (R_CFO), the production costs (R_PROD), and discretionary expenses (R_DISX) to detect three manipulation activities. Moreover, three comprehensive metrics of RM (RM1, RM2, RM3) are developed according to Zang (2006), Cohen et al. (2008), and Cohen and Zarowin (2010) to capture the total effects of RM. For the measurement of a firm’s RM fluctuations, this work employs the RM’s standard deviation (RMV) calculated by the past 4-year RM data as the main proxies of RM volatility risks. The estimation period is based on Baber, Kang, and Li (2011) and Chan, Jegadeesh, and Sougiannis (2004). Hence, in this study, we employ three measures of RM variations relating to three specific types of RM activities (R_CFOV, R_PRODV, and R_DISXV) and three comprehensive RM variations (RM1_V, RM2_V, and RM3_V). Higher standard deviations of the RM proxies denote greater variations in a firm’s RM activities. To properly evaluate a firm’s credit risk, this study employs the firm’s bond yield spreads as the main measure of credit risk. In contrast to previous studies, this work uses American bond market data to examine the effect of a firm’s incomplete accounting information uncertainty on bond yield spreads from an RM perspective. This study uses a sample of 9565 annual bond observations from 2001 through 2008 to empirically investigate whether RM fluctuations significantly affect bond yield spreads when wellknown yield spread determinant variables related to firm characteristics and bond features are controlled. Our results show that RMV is significantly and positively related to bond yield spreads. The yield spreads increase 37.69, 30.75, and 32.70 bps (basic points) per standard deviation increase in R_CFOV, R_PRODV, and R_DISXV, respectively. The comprehensive RMV measures also lead to a similar conclusion. Our findings also show that the positive effects of R_CFOV and R_DISXV on a firm’s credit risk become weaker when a firm has a lower credit rating. The implication is that bond investors are more concerned about RMV for the firms with higher credit ratings. Furthermore, this study also shows that absolute discretionary accrual variations significantly and positively relate to bond yield spreads.

4

The available information could be classified into two categories, including the prior information and the current accounting information implied by RM-managed earnings.

4

T.-K. Chen et al.

Our results are robust to alternative empirical specifications, such as employing alternative RM volatility measure (R_CFOCV), restricting the analysis to sample firms with incentives to meet earnings benchmark (suspect firm analyses) or manufacturing firms, replacing the estimation period of RMV, and additionally controlling for analyst-related variables and endogeneity issues. Therefore, this study supports that the accounting information uncertainty effects resulting from the RM have a substantial impact on corporate credit risks. This paper contributes to the literature on both RM and credit risks. Most RM studies focus on the impacts of levels of RM activities. This study extends the extant research by developing an application model, which provides a theoretic framework for the effects of RM variations on credit risks and empirically shows that the RM variations are positively associated with the bond yield spreads. Although the levels of RM methods have different and ambiguous effects on abnormal cash flows, the volatilities of RM methods consistently and definitely influence the direction of abnormal cash flow volatility.5 Moreover, RM variations affect the precision of investors’ knowledge of accounting information embedded within RM activities, cash flow patterns, and firm’s asset value distributions. Hence, RM variation is an important determinant of the cost of debt. Furthermore, idiosyncratic risk is also a hot issue in credit risk literature. This study contributes to the credit risk literature by showing that idiosyncratic risk effects resulting from RM variations have substantial impacts on firm credit risk. The remainder of this paper is organised as follows. Section 2 introduces the methodology of measuring RMV. Section 3 presents the hypotheses. Section 4 summarises major variables used in the empirical examinations. Section 5 presents and analyses empirical results. Finally, Section 6 offers concluding remarks.

2.

Measuring Information Risks of RM Activities

This section demonstrates the methods for measuring the information content and risks of RM activities. Based on Roychowdhury (2006), three types of RM activities are identified, including sales manipulations, overproduction, and reduction of discretionary expenditures. We employ R_CFO, R_PROD, and R_DISX to measure the levels of RM activities suggested by Roychowdhury (2006), Cohen et al. (2008), and Cohen and Zarowin (2010). The detailed estimation methods of R_CFO, R_PROD, and R_DISX are shown in the following equations:       CFOit 1 Salesit DSalesit = a1t + a3t + 1it , + a2t Assetit−1 Assetit−1 Assetit−1 Assetit−1

(1)

        Prodit 1 Salesit DSalesit DSalesit−1 + a2t + a3t + a4t = a1t Assetit−1 Assetit−1 Assetit−1 Assetit−1 Assetit−1 + 1it ,     DiscExpit 1 Salesit + a2t + 1it , = a1t Assetit−1 Assetit−1 Assetit−1

(2) (3)

where CFO represents operating cash flows from continuing operations; Prod shows production costs, which are the sum of COGS (cost of goods sold) and changes in inventories; and DiscExp 5

Traditionally, the influencing direction of R_DISX reduction on abnormal cash flows is opposed to other two RM methods so that the effects on abnormal cash flows of all RM methods used by managers are ambiguous.

Real Earnings Management Uncertainty and Corporate Credit Risk 5 signifies discretionary expenses, which are the sum of advertising expenses, R&D (research and development) expenses, and SG&A (sales, general, and administration). It should be noted that advertising expenses and R&D expenses are set as zero if they are missing but SG&A is available, consistent with Cohen et al. (2008) and Cohen and Zarowin (2010). Asset is total assets, and Sales and DSales stand for net sales and change in net sales, respectively. The residuals of Equations (1) – (3) represent the levels of real activity manipulations, R_CFO, R_PROD, and R_DISX, respectively. Firms with lower R_DISX or higher R_PROD are more likely to engage in real earnings manipulations.6 However, both sales manipulations and overproduction have a negative effect on R_CFO, while reduction of discretionary expenditures has a positive effect on R_CFO. Consequently, the effects of RM on R_CFO are ambiguous. This study tries to mitigate this concern by employing an alternative RM volatility measure, R_CFOC, which is the estimated residual of Equation (4) for each industry given each year. Because R_CFOC reflects the abnormal cash flows that are uncorrelated to R_PROD and R_DISX, firms that engage in greater RM activities, especially for sales manipulation, are expected to have lower R_CFOC: R CFOit = a0t + a1t R PRODit + a2t R DISXit + 1it .

(4)

In addition, following Zang (2006) and Cohen and Zarowin (2010), this study also employs three types of aggregate RM measures (RM1, RM2, and RM3) to capture the total effects of real manipulation activities. The functional form of RM1 is R_PROD 2 R_CFO 2 R_DISX, and the functional forms of RM2 and RM3 are R_PROD 2 R_DISX and 2R_CFO 2 R_DISX, respectively. The higher the amount of these aggregate measures (RM1, RM2, and RM3), the more likely the firm engaged in RM activities. To further demonstrate a firm’s RM fluctuation risks (RMV), this study employs the standard deviations of the RM proxies (R_CFOV, R_PRODV, R_DISXV, and R_CFOCV) and those of three comprehensive RM measures (RM1_V, RM2_V, and RM3_V), calculated for 4 years prior to the end of each year (Baber et al., 2011; Chan et al., 2004), as the main proxies. This is because most of the current net discretionary accruals have reversals in the future 4 years. Despite the opposite effects of different RM activities on the RM proxies of R_CFO, all kinds of RM variations have positive effects on RMV. Hence, higher standard deviations of RM proxies signify greater RM variations and accounting information uncertainties. In addition, this work also discusses how the variations of absolute discretionary accruals affect corporate credit risks. Following Kothari, Andrew, and Charles (2005), this study employs a performance-matched discretionary accruals model to estimate discretionary accruals in Equation (5). This research also uses the earnings before interest expense and tax (EBIT)based definition of total accruals (TA_EBIT), suggested by Cohen et al. (2008), in Equation (6) for the robustness purpose7:       TAit 1 DSalesit − DARit PPEit + a2t + a3t = a0t + a1t Assetit−1 Assetit−1 Assetit−1 Assetit−1 + a4t ROAit + 1it ,

(5)

6 The reduction of discretionary expenditures leads to lower R_DISX. Excessive price discounts and overproduction lead to higher R_PROD. Since overproduction as an RM strategy is only available to firms in the manufacturing industry, this study also restricts sample firms in the manufacturing industry to examine the effects of overproduction on credit risks. 7 In Equations (5) and (6), TA represents total accruals (balance sheet perspective), which are equal to change in noncash current assets minus the change in current liabilities excluding the current portion of long-term debt,

6

T.-K. Chen et al.       TA EBITit 1 DSalesit − DARit PPEit + a3t + a2t = a0t + a1t Assetit−1 Assetit−1 Assetit−1 Assetit−1 + a4t ROAit + 1it .

(6)

The residuals of Equations (5) and (6) represent the levels of discretionary accruals, DA and DA_EBXI, respectively. Similarly, this study uses the standard deviations of absolute values of discretionary accruals (A_DAV and A_DAV_EBXI), calculated for 4 years prior to the end of each year, to describe the information risks of accrual-based earnings management. 3.

Main Hypothesis

This section proposes a hypothesis for the effects of RM volatilities on corporate credit risk (bond yield spreads). We review the related literature and develop an application model based on the accounting information model of Lambert et al. (2007) to provide the theoretical foundations for the hypothesis development. Previous RM-related studies suggest that a firm’s RM activities affect its future operating performance, either positively (Gunny, 2010) or negatively (Cohen & Zarowin, 2010; Gunny, 2005; Gupta et al., 2010; Leggett et al., 2009; Roychowdhury, 2006). As a result, this study could reasonably conjecture that the fluctuations of RM activities have impacts on a firm’s operating performance uncertainty, namely, its operating risk. Leggett et al. (2009) also show that a firm’s RM activities change the firm’s future cash flow streams. Therefore, variations of RM activities increase the firm’s cash flow uncertainty, operating risk, and asset value distribution fluctuations, which increase firm credit risk based on structural credit models (Duffie & Lando, 2001; Merton, 1974). In addition, Lambert et al. (2007) demonstrate a cash flow-based equity pricing model, in which the available information includes the prior information and the current accounting information. Since RM might make outside investors’ knowledge of a firm’s observed asset value distributions diverge from the actual ones,8 RM may make the firm’s accounting information ambiguous. Therefore, this study employs RM activities to present the current accounting information in the Lambert et al. (2007) model and discusses how RM variations affect a firm’s cash flow patterns, asset value distributions, and credit risk. The following application model based on Lambert et al. (2007) facilitates clarification of this issue. Consider the impact on a firm’s asset value distribution if the new accounting information becomes available. Let the firm’s liabilities consist of one zero-coupon bond with principal D maturing in T. The default probability is defined as the probability that the asset value is below D (the default threshold). Assume that the firm’s asset value V at maturity T follows a normal distribution with the following parameters: 

 1 , V T ≏ N m, 4 minus depreciation and amortisation. TA_EBIT is total accruals (income statement perspective), which are equal to earnings before extraordinary items minus the operating cash flows from continuing operations. DAR and PPE stand for change in account receivables and net property, plant, and equipment, respectively. ROA is returns on assets, defined by net income scaled by the lagged total assets. The definitions of Asset and DSales are same as those in Equations (1) –(3). 8 RM may make a firm’s operating and investing activities deviate from normal business practices to meet certain earnings targets. At this time, a firm’s reported earnings are different from the underlying real economic substance.

Real Earnings Management Uncertainty and Corporate Credit Risk 7 where m and 4 represent the (ex ante) expected value of assets and precision (inverse of variance) of assets, respectively. Suppose investors receive new accounting information, Z, about the ultimate realisation of the firm’s asset value at maturity T. Z is assumed to be normally distributed with precision g. Without RM manipulations, the assessed posterior distribution for the asset value at maturity follows a normal distribution with the following parameters: 

 4 g 1 , m+ Z, VT |Z ≏ N 4+g 4+g 4+g where the posterior mean is a weight average of the prior expected value of assets m and the new information Z, where the weights of m and Z are 4/(4 + g) and g/(4 + g), respectively. The posterior precision, 4 + g, is the sum of the precision of prior information (4) and that of new accounting information (g). When a firm engages in RM manipulations to boost the accounting information, it is assumed that the observed accounting information Z increases by m. At this time, outside investors cannot see the precise effects of RM manipulations on the accounting information but can anticipate the effects with measurement error 1. That is, when outside investors revise their belief about the asset value, the new accounting information with RM manipulations, Z + m, is adjusted to Z + 1. Suppose the measurement error is normally distributed with zero mean and precision of t. For simplicity, we also assume that Z and e are independent. As a result, the assessed posterior distribution for the asset value at maturity given RM manipulations is normally distributed, with the following parameters: 

 4 gt/(g + t) 1 . (Z + 1), m+ VT |(Z + 1) ≏ N (4 + gt)/(g + t) (4 + gt)/(g + t) (4 + gt)/(g + t) √ Let DDR = (E(V|Z + 1) − D)/ Var(V|Z + 1) be the distance to default with RM manipulations defined by the number of standard deviations that the expected value of assets with RM manipulations, E(V|Z + 1), is away from the default D. Accordingly, the degree of a firm’s credit risk is expressed as F( − DDR ), where F denotes the cumulative standard normal distribution. Similarly, DDNR and F( − DDNR ) are denoted as the distance to default and the degree of credit risk without RM manipulations, respectively. Given default threshold D and the posterior distribution for asset value with or without RM in T, the difference between F( − DDR ) and F( − DDNR ) can be expressed as the following formula:     D − E(V|Z + 1) D − E(V|Z) √  √   −F F Var(V|Z + 1) Var(V|Z)    D − 4m/(4 + gt (g + t)) − (Z + 1)(gt/(g + t))/(4 + gt/(g + t)) √  =F 1/(4 + gt/(g + t))   D − 4m/(4 + g) − gZ/(4 + g) √ −F 1/(4 + g) The analysis above formulises the notion that the precision of the measurement error, t, is one of the components of DDR . Since increasing RM variations constrains investors’ ability to undo

8

T.-K. Chen et al.

the impacts of RM manipulations, higher RM variations imply lower precision of the measurement error.9 For a going-concern firm with RM activities, the situation holds that the firm’s posterior expected asset value with RM manipulations is higher than the default threshold, namely the distance to default is positive (i.e. DDR . 0). All active firms are assumed to satisfy this condition in this work. As a result, the RM variations affect a going-concern firm’s credit risk in two ways. First, a firm with higher RM variations has a lower measurement error’s precision, t, which suggests that the variance of accounting information with RM manipulations is increased. There√ fore, the denominator of DDR , Var(V|Z + 1), goes up, which shortens the distance to default with RM manipulations (DDR ) and then leads to a higher credit risk (F( − DDR )). Second, when outside investors underestimate the firm’s income-increasing RM manipulations, it is more probable that the adjusted RM-managed accounting information, Z + 1, is higher than the prior expected asset value, m (i.e. Z + 1 . m). Increasing RM variations lead to a greater amount of 4/(4 + gt/(g + t)), which represents the weight investors assign to the prior expected asset value. As a result, given the condition that Z + 1 . m, as RM variations go up, the distance to default with RM manipulations, DDR , narrows, suggesting an increase of credit risk, F( − DDR ). In sum, this application model predicts that RM variations are positively related to a firm’s credit risk. According to the preceding discussion, the results of Leggett et al. (2009) and the application of Lambert et al. (2007) model both provide foundations for our hypothesis development. A firm’s future cash flow uncertainty and asset value distributions seem to be affected by RM variations. Therefore, this research hypothesises that a firm’s RM volatilities positively relate to credit risk (bond yield spreads)10: Main hypothesis: A firm’s RM volatilities positively relate to bond yield spreads.

4.

Data and Methodology

Given that bond yield spread is the main measure of firm credit risk, this study uses American corporate bonds as our sample observations. These bond issues must meet four requirements: (1) be fixed rate coupons; (2) be unsecured or unguaranteed by others; (3) be issued by nonfinancial industry or low regulated firms (nonfinancial or government unrelated firms); and (4) have no special clauses or embedded options (nonconvertible or noncallable bonds). This restricts the sample to US straight corporate bonds with fixed coupon payment and collateralised by firm assets. The bond issue characteristic variables, such as yield spreads, coupon rate, maturity, bond age, issued amount, and bond credit rating, are acquired from Datastream. The characteristics data of bond issuer are obtained from TAQ (e.g. information asymmetry), COMPUSTAT (e.g. leverage ratio, returns of assets, accrual quality (Francis, LaFond, Olsson, & Schipper, 2005), and cash flow volatility), CRSP (e.g. equity volatility) and I/B/E/S (e.g. analyst variables) databases. Finally, the RM-related data are available on COMPUSTAT database. 9

Note that when outside investors infer the RM activities perfectly and rationally undo the RM effects (i.e. t  1), the posterior distribution for the asset value in T turns out to be the one without RM activities. 10 The positive association between RM variations and the credit risk may not hold when Z + 1 is smaller than m. Under this scenario, high RM variations have both positive and negative effects on firm credit risk. The positive effect is attrib√ uted to the increase of the posterior asset value variations, Var(V|Z + 1). On the other hand, the negative effect occurs when investors assign higher weight on m, implying that E(V|Z + 1)is increased given the condition that m . Z + 1. Fortunately, the manager’s incentive to engage in RM activities is usually to meet certain positive earnings targets in practice (namely, Z + 1 . m). As a result, an increase of RM variations leads to higher credit risk.

Real Earnings Management Uncertainty and Corporate Credit Risk 9 Table 1. Sample distribution Year

Above Aa

A

Baa

Noninvestment grade

Total

2001 2002 2003 2004 2005 2006 2007 2008 Total

32 33 41 46 48 52 64 82 398

128 156 224 209 228 237 334 413 1929

251 298 362 415 465 488 638 698 3615

206 250 268 446 551 586 707 609 3623

617 737 895 1116 1292 1363 1743 1802 9565

Note: the sample period is yearly between 2001 and 2008. During the sample period, the sample includes 9565 annual bond observations for all firms. The numbers of pooled observations in the given years are reported. The rating subsamples are sorted by Moody’s credit ratings.

With data from 2001 to 2008, the sample period covers the pre- and post-subprime periods. The final sample is 9565 annual bond observations (excluding samples with invalid and missing data). Table 1 shows the sample distribution categorised by Moody’s credit ratings. As shown in Table 1, approximately 62% of the bonds are investment-grade bonds. The main dependent variable, yield spread (YS), is defined as the difference in yield between corporate bond yield and Treasury bond yield of the same maturity (Lu et al., 2010; Yu, 2005). The test variables are RMV proxies, which are detailed in Section 2. The control variables can be divided as bond issue features and firm characteristics. Bond issue features include five variables: amount issued (Lnamt; Yu, 2005), bond age (Bage; Warga, 1992), coupon rate (Coupon; Elton, Gruber, Agrawal, & Mann, 2001), bond credit rating (RAT), and bond maturity (LFFL; Helwege & Turner, 1999). Lnamt is the logarithm of the originally issued dollar amount, which is positively associated with external liquidity (Yu, 2005). Opposed to the Lnamt variable, Bage, defined as the time interval during the issuing date and the settlement date, has a negative effect on external liquidity. Coupon is the annualised interest rate stated on a bond and is expected to positively relate to YS because more taxes are charged for higher coupon bonds. RAT is the variable that converts Moody’s bond credit ratings into a nominal categorical variable: 1 is Aaa, 2 is Aa1, 3 is Aa2, 4 is Aa3, and so on. LFFL, which constitutes a major portion of term structure risks, is defined as time to the bond maturity (in years). Among the five bond feature variables, the first one is conjectured to negatively relate to YS, while the rest have the opposite effect. Firm characteristics include six variables: returns of assets (ROA), leverage ratio (LEV; CollinDufresne, Goldstein, & Martin, 2001), equity volatility (VOL; Campbell & Taksler, 2003), operating cash flow volatility (OCFV; Molina, 2005; Tang & Yan, 2010), trading-based information asymmetry (ADJPIN; Lu et al., 2010), and accrual quality (AQ; Lu et al., 2010). ROA refers to the ratio of EBIT to total asset value. LEV refers to the ratio of debt book value to debt book value plus equity market value (Yu, 2005). VOL is estimated as the annualised standard deviation of daily stock returns over the preceding 150 days. OCFV represents the past 12-quarter standard deviation of the operating cash flows before interest payments scaled by total asset. ADJPIN is the adjusted probability of information-based trading developed by Duarte and Young (2009). It represents a firm’s pure information asymmetry level, estimated by Equation (7) with the EM algorithm of Chen and Chung (2007).11 The data required to estimate a firm’s ADJPIN are the number of 11

The definitions of symbols in Equation (7) are as follows. Db and Ds are the symmetric order flow shocks for buys and sells, respectively. ub and us represent the number of buyer-initiated informed trades and that of seller-initiated trades, respectively, and the distributions of ub and us are different. a is the probability of the occurrence of a private information

10

T.-K. Chen et al.

buyer- and seller-initiated trades for each firm-day that could be collected from the TAQ database. The selection criteria of trades and quotes are based on Lee and Ready (1991), Chordia, Roll, and Subrahmanyam (2002), and Duarte and Young (2009). Finally, AQ presents the past 5-year standard deviation of residuals of Equation (8), namely the modified Dechow and Dichev model (Francis et al., 2005).12 All firm characteristic variables are conjectured to positively relate to YS except the ROA variable:

ADJPIN =

a × (d × ub + (1 − d) × us ) , a × (d × ub + (1 − d) × us ) + (Db + Ds ) × (a × u′ + (1 − a) × u) + 1s + 1b

        TCAit CFOit−1 CFOit CFOit+1 DSalesit = a0t + a1t + a2t + a3t + a4t Assetit Assetit Assetit Assetit Assetit   PPEit + 1it . + a5t Assetit

(7)

(8)

Table 2 presents the descriptive statistics of the major variables. The average of YS is 322.74 bps; the averages of R_CFOV, R_PRODV, and R_DISX are 0.2285, 0.4501, and 0.5759, respectively;13 the averages of RM1_V, RM2_V, and RM3_V are 0.9871, 0.9254, and 0.6025, respectively; the average of LEV is 30.55%; the average of VOL is 30.21%; the average of coupon rate (Coupon) is 7.00%; the average of LFFL is 12.05 years; the average of Bage is 5.47 years; the probability of pure information asymmetry (ADJPIN) is 0.0343; and the average numerical bond rating (RAT) is 10.83 (between Baa3 and Ba1 in Moody’s rating system).

5.

Empirical Analyses

5.1. The Effects of RM Volatilities on Corporate Bond Yield Spreads To investigate the RM volatility effects on bond yield spreads, this study employs panel regressions with controlling for firm and year fixed effects, shown as Equation (9). In addition, the current research discusses how accrual-based earnings management variations affect bond yield spreads, shown as Equation (10): YSit = a + b1 RMit + b2 RMVit + 1it ,

(9)

event in a given day. When there is no private information, buys arrive at a rate of 1b and sells arrive at a rate of 1s. d (1 2 d ) indicates the conditional probability with the occurrence of a positive (negative) private information event, respectively. In addition, these days happen with probability u ′ when private information arrives and with probability u in the absence of private information. 12 TCA represents total current accruals, which are equal to change in noncash current assets minus the change in current liabilities excluding the current portion of long-term debt. The definitions of other notations are same as those described in Equations (1)– (3). 13 We employ all COMPUSTAT firms belong to the same industry and year to estimate RM proxies (R_CFO, R_PROD, and R_DISX), and then limit our sample firms with bond issuers when examining the association between RM variations and bond yield spreads. As a result, the means of residuals of R_CFO, R_PROD, and R_DISX for firms issuing bonds in Table 2 may not equal to zero.

Real Earnings Management Uncertainty and Corporate Credit Risk 11 Table 2. Variable YS LEV VOL Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ R_CFO R_PROD R_DISX A_DA A_DA_EBXI A_DAV R_CFOV R_PRODV R_DISXV A_DAV_EBXI RM1 RM1_V RM2 RM2_V RM3 RM3_V

Summary statistics of major variables

Mean

Median

Stdev.

Min.

Max.

322.7358 0.3055 0.3021 7.0031 5.4679 12.0504 5.1253 10.8277 0.0343 0.0451 0.0145 0.1260 0.0997 20.1866 20.1346 0.1461 0.6063 0.1945 0.2285 0.4501 0.5759 1.0741 20.1778 0.9871 20.0719 0.9254 0.0283 0.6025

196.6000 0.2683 0.2473 7.0000 4.2658 8.9528 5.3194 10.0000 0.0199 0.0509 0.0102 0.0549 0.0423 20.0439 20.0186 0.0423 0.0698 0.0621 0.0698 0.0644 0.1664 0.1238 20.0738 0.2641 20.0369 0.2272 20.0394 0.2076

357.1936 0.1944 0.1851 1.4023 4.4726 7.8767 0.7481 4.8977 0.0434 0.0790 0.0139 0.2046 0.4329 1.2978 1.0176 0.4431 3.2049 0.4476 0.3848 1.4031 0.8142 3.8772 1.9393 1.9282 1.7971 1.7778 1.0285 0.8227

0.2000 0.0000 0.0529 0.0000 0.0274 0.5361 0.0000 1.0000 0.0000 21.0744 0.0015 0.0000 22.7426 223.8256 29.9847 0.0000 0.0000 0.0003 0.0016 0.0011 0.0016 0.0016 228.2411 0.0013 224.8698 0.0009 210.8038 0.0044

2876.0000 0.9861 1.6099 13.7500 52.8712 29.9861 6.6990 23.0000 0.3351 0.3908 0.1686 4.2635 8.8215 2.1154 8.1475 7.0757 76.8108 11.3399 4.3923 12.1828 5.5902 130.4471 11.4618 17.1344 12.1001 15.6619 9.3464 5.9106

Notes: this table presents the mean, median, and standard deviation of major variables used in empirical analyses. Yield spread (YS) is the difference in yield to maturity between a corporate bond and a US Treasury bond with the same maturity. LEV refers to firm leverage ratio. The equity volatility (VOL) measures the annualised daily volatility of previous 150-day stock returns. LFFL, Coupon, and Lnamt stand for the time to maturity, annual coupon rate, and natural log of amount issued, respectively. Bond age (Bage) is defined as the difference between the settlement date and the issuing date. Bond rating (RAT) is the numerical scores bond rating from Datastream system, where Aaa is 1, Aa1 is 2, Aa2 is 3, etc. ADJPIN is the proxy for information asymmetry according to Duarte and Young (2009). ROA represents the returns of assets. OCFV stands for cash flow (the sum of operating cash flows and interest expense is divided by total asset value) volatility estimated by the standard deviations of previous 12-quarter cash flow data. AQ represents accrual quality, measured by the standard deviation of firm i’s residuals over year t24 through t of the modified Dechow and Dichev model. R_CFO, R_PROD, and R_DISX represent three real activity manipulation (RM) variables of abnormal levels of cash flow from operations, production costs, and discretionary expenses, respectively. R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. A_DA and A_DA_EBXI represent the absolute values of asset- and EBIT-based discretionary accruals (DA). A_DAV and A_DAV_EBXI are the standard deviations of absolute values of two DA variables, respectively. RM1, RM2, RM3 are aggregate RM measures. RM1 is calculated by the sum of R_PROD, the product of minus one and the R_CFO, and the product of minus one and R_DISX. RM2 is the value of that R_PROD minus R_DISX. RM3 is the value of that product of minus one and the sum of R_CFO and R_DISX. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively.

where RM ¼ R_CFO, R_PROD, R_DISX, RM1, RM2, RM3; RMV ¼ R_CFOV, R_PRODV, R_DISXV, RM1_V, RM2_V, RM3_V: YSit = a + b1 DAit + b2 DAVit + 1it , where DA ¼ A_DA, A_DA_EBXI; DAV ¼ A_DAV, A_DAV_EBXI.

(10)

12

T.-K. Chen et al.

Table 3 provides the results of Equations (9) and (10) for the entire sample period from 2001 to 2008. The results from columns (2), (3), and (4) reveal that RMV (R_CFOV, R_PRODV, and R_DISXV) are all significantly and positively related to bond yield spreads when the RM-level variables are considered. The coefficients (97.9364, 21.9127, and 40.1657) show that yield spreads increase 37.69 bps (0.3848 × 97.9364), 30.75 bps (1.4031 × 21.9127), and 32.70 bps (0.8142 × 40.1657) for one standard deviation increase in R_CFOV, R_PRODV, and R_DISXV, respectively. Besides, the results from columns (6), (7), and (8) suggest that RMV is positively associated with bond yield spreads using the comprehensive measures (RM1_V, RM2_V, RM3_V). Finally, the results from columns (1) and (5) show that the volatilities of absolute discretionary accruals (A_DAV and A_DAV_EBXI) have significant and positive effects on bond yield spreads. From the results, the main hypothesis is preliminarily supported. Other interesting findings are that R_CFO significantly and negatively relates to bond yield spreads, while R_PROD has positive effects on bond yield spreads. It is consistent with Gupta et al. (2010), suggesting that a higher degree of RM may negatively relate to operating performance in subsequent periods. To examine the robustness of RM volatility effects on bond yield spreads, this study includes other well-known yield spread explanatory variables mentioned in the literature in Equation (11). Also, this study discusses whether the bond yield spread determinants distort the effects of accrual-based earnings management variations on yield spreads, shown as Equation (12): YSit = a + b1 RMit + b2 RMVit + b3 LEVit + b4 VOLit + b5 Couponit + b6 Bageit + b7 LFFLit + b8 Lnamtit + b9 RATit + b10 ADJPINit + b11 ROAit

(11)

+ b12 OCFVit + b13 AQit + 1it ,

where RM ¼ R_CFO, R_PROD, R_DISX, RM1, RM2, RM3; RMV ¼ R_CFOV, R_PRODV, R_DISXV, RM1_V, RM2_V, RM3_V, YSit = a + b1 DAit + b2 DAVit + b3 LEVit + b4 VOLit + b5 Couponit + b6 Bageit + b7 LFFLit + b8 Lnamtit + b9 RATit + b10 ADJPINit + b11 ROAit + b12 OCFVit (12) + b13 AQit + 1it ,

where DA ¼ A_DA, A_DA_EBXI; DAV ¼ A_DAV, A_DAV_EBXI. Table 4 shows the results of Equations (11) and (12) for the entire sample period. The results from columns (2), (3), and (4) show that RMV variables still significantly and positively relate to bond yield spreads (the coefficients are 77.5171, 13.1385, and 27.4028 for R_CFOV, R_PRODV, and R_DISXV, respectively) when RM levels, accrual quality, and other wellknown variables are controlled. The coefficients reveal that the yield spreads increase 29.83 bps (0.3848 × 77.5171), 18.43 bps (1.4031 × 13.1385), and 22.31 bps (0.8142 × 27.4028) per standard deviation increase in R_CFOV, R_PRODV, and R_DISXV, respectively. The comprehensive measures of RM1_V, RM2_V, and RM3_V present the similar results, as shown in columns (6), (7), and (8). In addition, untabulated results suggest that the effects of RMV are similar if the RM-level variables are not included in the regressions. According to the preceding discussions, the main hypothesis is empirically supported. Our findings also show that, among the six proxies of RM activity level, only RM1 and RM2 have a significantly positive effect on bond yield spreads. These results support the argument that when controlling for other firm characteristic variables, managers engage in RM with either positive or negative

Table 3. (1) A_DA A_DAV

R_CFOV R_PROD R_PRODV R_DISX R_DISXV A_DA_EBXI A_DAV_EBXI RM1 RM1_V RM2 RM2_V RM3 RM3_V

(2)

(3)

(4)

(5)

(6)

(7)

(8)

22.4554 (20.51) 53.5425∗∗∗ (4.08) 214.3952∗ (21.94) 97.9364∗∗∗ (3.49) 3.4172∗∗ (2.33) 21.9127∗∗∗ (4.99) 23.3629 (21.03) 40.1657∗∗∗ (3.63) 20.1435 (20.29) 6.3091∗∗∗ (3.46) 6.3636∗∗∗ (4.28) 25.4773∗∗∗ (5.75) 6.5243∗∗∗ (4.24) 28.5240∗∗∗ (6.21) 4.7277 (1.56) 41.4158∗∗∗ (3.32) (Continued)

Real Earnings Management Uncertainty and Corporate Credit Risk 13

R_CFO

The effects of RM level and volatility on corporate bond yield spreads

14 T.-K. Chen et al.

Table 3. (1) Constant Observations R2

(2) ∗∗∗

269.0068 (18.57) 9021 0.5645

(3) ∗∗∗

280.1554 (14.59) 9565 0.5583

Continued (4)

∗∗∗

270.4696 (18.74) 9443 0.5525

(5) ∗∗∗

282.4307 (24.46) 8706 0.5666

(6) ∗∗∗

268.6164 (18.93) 9414 0.5634

(7) ∗∗∗

277.1336 (26.24) 8584 0.5658

(8) ∗∗∗

279.5071 (26.34) 8585 0.5663

273.3672∗∗∗ (26.59) 8705 0.5666

Notes: this table shows the results of eight different panel regressions with the yield spreads (YS) as the dependent variable against various explanatory variables of RM level and volatility using data of 9565 annual bond observations for all firms. The fixed effects (firm and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFO, R_PROD, and R_DISX represent three real activity manipulation (RM) variables of abnormal levels of cash flow from operations, production costs, and discretionary expenses, respectively. A_DA and A_DA_EBXI represent the absolute values of asset- and EBIT-based discretionary accruals (DA). R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. A_DAV and A_DAV_EBXI are the standard deviations of absolute values of two DA variables, respectively. RM1, RM2, RM3 are aggregate RM measures. RM1 is calculated by the sum of R_PROD, the product of minus one and the R_CFO, and the product of minus one and R_DISX. RM2 is the value of that R_PROD minus R_DISX. RM3 is the value of the product of minus one and the sum of R_CFO and R_DISX. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively. This table presents the regression coefficients and adjusted R2. The t-statistics calculated by firm-level clustered standard errors for each coefficient appears immediately underneath. Bold values are used to highlight the results of the main variables this study focus on. ∗ Significance of 10%. ∗∗ Significance of 5%. ∗∗∗ Significance of 1%.

Table 4. Regressions of yield spreads against RM level and volatility controlled by characteristic variables (1) A_DA A_DAV

(2)

(3)

(4)

(5)

(6)

(7)

25.8417 (21.17) 55.4300∗∗∗ (4.52) 210.9477 (21.64) 77.5171∗∗∗ (4.84)

R_CFOV R_PROD

0.5337 (0.40) 13.1385∗∗ (2.51)

R_PRODV R_DISX

20.6223 (20.24) 27.4028∗∗∗ (2.77)

R_DISXV A_DA_EBXI

0.2790 (0.51) 5.1044∗∗ (2.25)

A_DAV_EBXI

2.3892∗ (1.88) 15.8574∗∗∗ (3.29)

RM1 RM1_V

2.5000∗ (1.81) 18.6031∗∗∗ (3.57)

RM2 RM2_V RM3 RM3_V 443.6634∗∗∗ (5.46)

451.0451∗∗∗ (5.57)

437.2870∗∗∗ (5.26)

450.0312∗∗∗ (5.06)

446.4170∗∗∗ (5.44)

438.7010∗∗∗ (4.91)

439.7138∗∗∗ (4.95)

1.3237 (0.50) 28.8305∗∗∗ (3.03) 450.4648∗∗∗ (5.06) (Continued)

Real Earnings Management Uncertainty and Corporate Credit Risk 15

R_CFO

LEV

(8)

Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ Constant Observations R2

(2)

(3)

(4)

(5)

(6)

(7)

(8)

781.2744∗∗∗ (5.29) 6.7601∗∗ (2.31) 1.5352∗ (1.94) 0.1213 (0.34) 18.4560∗∗∗ (4.10) 2.2378∗∗ (2.30) 27.5519 (20.03) 2491.0338∗∗∗ (22.85) 3236.6405∗∗∗ (4.15) 278.3223 (21.34) 2322.8176∗∗∗ (25.10) 7957 0.6811

776.2150∗∗∗ (5.26) 6.7671∗∗ (2.35) 1.5265∗ (1.95) 0.1033 (0.29) 18.1150∗∗∗ (4.06) 1.9504∗∗ (1.98) 227.9014 (20.13) 2499.6453∗∗∗ (22.91) 3049.7137∗∗∗ (4.02) 26.5360 (20.17) 2374.7148∗∗∗ (26.00) 7967 0.6833

796.2972∗∗∗ (5.19) 6.9130∗∗ (2.33) 1.4565∗ (1.82) 0.0975 (0.28) 18.7104∗∗∗ (4.14) 2.3175∗∗ (2.41) 254.6157 (20.24) 2487.6977∗∗∗ (22.82) 3205.2606∗∗∗ (4.15) 271.7269 (21.03) 2363.9647∗∗∗ (25.77) 7887 0.6790

714.9126∗∗∗ (5.70) 7.9951∗∗ (2.47) 1.5835∗ (1.90) 0.0182 (0.05) 18.8221∗∗∗ (4.14) 2.0810∗∗ (2.02) 237.9837 (20.16) 2500.4968∗∗∗ (22.85) 2959.7879∗∗∗ (4.19) 246.6667 (20.90) 2303.1408∗∗∗ (25.38) 7296 0.6824

780.8297∗∗∗ (5.26) 7.0126∗∗ (2.38) 1.4831∗ (1.86) 0.1036 (0.29) 18.7697∗∗∗ (4.15) 2.2616∗∗ (2.34) 230.5097 (20.14) 2487.8642∗∗∗ (22.83) 3189.5430∗∗∗ (4.05) 266.0639 (21.01) 2324.3468∗∗∗ (25.12) 7957 0.6801

721.7968∗∗∗ (5.60) 7.8647∗∗ (2.39) 1.6130∗ (1.94) 0.0092 (0.02) 19.1985∗∗∗ (4.22) 2.1510∗∗ (2.08) 246.0268 (20.19) 2495.4623∗∗∗ (22.83) 2964.0806∗∗∗ (4.21) 292.7996 (21.31) 2292.4368∗∗∗ (25.62) 7219 0.6813

721.0583∗∗∗ (5.59) 7.8731∗∗ (2.40) 1.5843∗ (1.90) 0.0119 (0.03) 18.9759∗∗∗ (4.19) 2.1338∗∗ (2.07) 241.9062 (20.17) 2492.6706∗∗∗ (22.82) 2961.4642∗∗∗ (4.19) 297.1937 (21.38) 2293.7043∗∗∗ (25.64) 7219 0.6818

711.4445∗∗∗ (5.68) 8.0158∗∗ (2.47) 1.6143∗ (1.96) 0.0205 (0.05) 19.0277∗∗∗ (4.18) 2.0409∗ (1.96) 253.9616 (20.23) 2505.8699∗∗∗ (22.88) 2963.0196∗∗∗ (4.22) 248.0945 (20.92) 2298.2040∗∗∗ (25.74) 7296 0.6825

Notes: this table shows the results of eight different panel regressions with the yield spread (YS) as the dependent variable against various explanatory variable combinations using data of all bond observations with firms in the sample period. The fixed effects (firm and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFO, R_PROD, and R_DISX represent three real activity manipulation (RM) variables, respectively. A_DA and A_DA_EBXI represent the absolute values of asset- and EBIT-based discretionary accruals (DA). R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. A_DAV and A_DAV_EBXI are the standard deviations of absolute values of two DA variables, respectively. RM1, RM2, RM3 are aggregate RM measures. RM1 is calculated by the sum of R_PROD, the product of minus one and the R_CFO, and the product of minus one and R_DISX. RM2 is the value of that R_PROD minus R_DISX. RM3 is the value of that product of minus one and the sum of R_CFO and R_DISX. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively. The control variables include leverage ratio (LEV), equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT) and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. This table presents the regression coefficients and adjusted R2. The t-statistics calculated by firm-level clustered standard errors for each coefficient appears immediately underneath. Bold values are used to highlight the results of the main variables this study focus on. ∗ Significance of 10%. ∗∗ Significance of 5%. ∗∗∗ Significance of 1%.

T.-K. Chen et al.

VOL

Continued

16

Table 4. (1)

Real Earnings Management Uncertainty and Corporate Credit Risk 17 effects on future performance. The mixed effects offset one another on average, and therefore, the effects of RM levels on bond yield spreads are less likely to be significant. Additionally, the effects of accrual-based earnings management variations on yield spreads are significant when well-known bond yield spread determinants are controlled. 5.2. Additional Analysis 5.2.1. The RMV effect distortions from a credit rating perspective This section investigates whether credit rating weakens the RMV effects on bond yield spreads by adding the interaction term between credit rating and RMV variables as shown in Equation (13). Similarly, this work explores whether credit rating twists DAV effects on bond yield spreads as shown in Equation (14): Y Sit = a + b1 RMVit + b2 LEVit + b3 VOLit + b4 Couponit + b5 Bageit + b6 LFFLit + b7 Lnamtit + b8 RATit + b9 ADJPINit + b10 ROAit + b11 OCFVit + b12 AQit + b13 RATit × RMVit + 1it ,

(13)

where RMV ¼ R_CFOV, R_PRODV, R_DISXV, RM1_V, RM2_V, RM3_V, Y Sit = a + b1 DAVit + b2 LEVit + b3 VOLit + b4 Couponit + b5 Bageit + b6 LFFLit + b7 Lnamtit + b8 RATit + b9 ADJPINit + b10 ROAit + b11 OCFVit + b12 AQit + b13 RATit × DAVit + 1it ,

(14)

where DAV ¼ A_DAV, A_DAV_EBXI. Table 5 demonstrates the results of Equations (13) and (14) and shows that the interaction term variables of R_CFOV × RAT, R_DISXV × RAT, RM1_V × RAT, RM3_V × RAT, A_DAV × RAT, and A_DAV_EBXI × RAT are significantly and negatively related to bond yield spreads. The results indicate that an issuer with lower credit quality weakens the RMV and DAV effects on bond yield spreads. That is, bond investors might neglect the RMV and DAV effects for firms with lower credit ratings, while they are more concerned with the RMV and DAV effects for those with higher credit ratings. 5.2.2. Alternative measure of abnormal cash flow volatility To address the issue that the different RM activities may have opposite effects on R_CFO, this study employs R_CFOC (described in Section 2) as an additional proxy for abnormal cash flow from operations (abnormal CFO). R_CFOCV and R_CFOCV3 are estimated from the standard deviations of R_CFOC for 4 years and 3 years prior to the end of each year, respectively. The results using R_CFOCV and R_CFOCV3 as the alternative measures of abnormal CFO volatility are shown in Table 6. Both R_CFOCV and R_CFOCV3 are positively related to bond yield spreads. The coefficients of columns (2) and (4) reveal that the yield spreads increase 12.83 bps (0.4524 × 28.3616) and 13.64 bps (0.4596 × 29.6816) per standard deviation increase in R_CFOCV and R_CFOCV3, respectively. It suggests that RMV effect holds under alternative abnormal CFO specifications. 5.2.3. The RMV effects on manufacturing firms Abnormally high production costs for a firm indicate either sales manipulation or overproduction. Since overproduction is only available to manufacturing firms, this study re-estimates

18

Table 5. The RM volatility effects on yield spreads: credit rating perspective

A_DAV A_DAV × RAT R_CFOV

(2)

(3)

(4)

(5)

(6)

(8)

94.7952 (4.60) 24.1819∗∗∗ (22.66)

123.6345∗∗∗ (4.31) 24.8660∗∗ (22.50)

R_CFOV 3 RAT R_PRODV

15.2522∗∗∗ (2.62) 20.3266 (20.79)

R_PRODV × RAT R_DISXV

48.9749∗∗∗ (4.48) 22.3887∗ (21.92)

R_DISXV 3 RAT A_DAV_EBXI

10.5173∗∗∗ (3.54) 20.5658∗∗∗ (23.16)

A_DAV_EBXI × RAT RM1_V

19.0215∗∗∗ (3.58) 20.6304∗ (21.71)

RM1_V 3 RAT RM2_V

21.3716∗∗∗ (3.65) 20.5604 (21.36)

RM2_V × RAT RM3_V RM3_V 3 RAT LEV

(7)

∗∗∗

445.1296∗∗∗ (5.50)

447.9902∗∗∗ (5.55)

437.2679∗∗∗ (5.28)

449.0005∗∗∗ (5.05)

446.5969∗∗∗ (5.45)

438.6912∗∗∗ (4.90)

440.2507∗∗∗ (4.95)

57.9037∗∗∗ (4.93) 23.2871∗∗ (22.55) 446.4608∗∗∗ (5.02)

T.-K. Chen et al.

(1)

VOL Coupon Bage LFFL

RAT ADJPIN ROA OCFV AQ Constant Observations R2

773.2351∗∗∗ (5.22) 6.7102∗∗ (2.34) 1.4946∗ (1.90) 0.0847 (0.24) 18.0556∗∗∗ (4.01) 3.4623∗∗∗ (2.90) 214.7198 (20.07) 2502.1687∗∗∗ (22.93) 3055.4076∗∗∗ (3.99) 214.4505 (20.38) 2386.5354∗∗∗ (26.15) 7967 0.6836

796.3927∗∗∗ (5.20) 6.9170∗∗ (2.34) 1.4700∗ (1.84) 0.0951 (0.27) 18.7875∗∗∗ (4.15) 2.5404∗∗ (2.37) 256.9339 (20.25) 2487.8363∗∗∗ (22.82) 3207.1045∗∗∗ (4.15) 270.4071 (21.01) 2366.7205∗∗∗ (25.80) 7887 0.6791

711.0923∗∗∗ (5.69) 7.9988∗∗ (2.48) 1.6137∗ (1.95) 0.0167 (0.04) 19.0017∗∗∗ (4.19) 3.7747∗∗∗ (2.67) 242.4263 (20.18) 2501.8408∗∗∗ (22.86) 2935.7248∗∗∗ (4.16) 249.9910 (20.99) 2313.6019∗∗∗ (25.96) 7296 0.6830

779.8525∗∗∗ (5.25) 6.8622∗∗ (2.33) 1.5348∗ (1.93) 0.1243 (0.35) 19.1575∗∗∗ (4.23) 2.8230∗∗∗ (2.86) 246.0834 (20.21) 2485.3003∗∗∗ (22.82) 3234.7298∗∗∗ (4.13) 271.5563 (21.11) 2330.2034∗∗∗ (25.21) 7957 0.6806

722.2993∗∗∗ (5.60) 7.7968∗∗ (2.38) 1.6654∗∗ (2.01) 0.0088 (0.02) 19.4322∗∗∗ (4.25) 3.0918∗∗ (2.40) 245.4521 (20.19) 2495.4936∗∗∗ (22.83) 2963.3413∗∗∗ (4.19) 294.8656 (21.32) 2299.3040∗∗∗ (25.68) 7219 0.6813

721.0685∗∗∗ (5.59) 7.8703∗∗ (2.40) 1.6211∗ (1.95) 0.0114 (0.03) 19.2554∗∗∗ (4.23) 2.9412∗∗ (2.32) 243.1998 (20.18) 2493.0592∗∗∗ (22.82) 2958.1471∗∗∗ (4.17) 2100.3354 (21.41) 2299.8119∗∗∗ (25.68) 7219 0.6817

706.1260∗∗∗ (5.65) 7.8240∗∗ (2.44) 1.6847∗∗ (2.06) 0.0076 (0.02) 19.2284∗∗∗ (4.21) 4.4836∗∗∗ (2.92) 246.5768 (20.20) 2505.6724∗∗∗ (22.89) 2928.2918∗∗∗ (4.16) 253.2606 (21.06) 2319.8806∗∗∗ (26.10) 7296 0.6835

Notes: this table shows the results of eight different panel regressions with the yield spread (YS) as the dependent variable against various explanatory variable combinations using data of all bond observations with firms in the sample period. The fixed effects (firm and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. A_DAV and A_DAV_EBXI are the standard deviations of absolute values of two DA variables, respectively. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively. The control variables include leverage ratio (LEV), equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT), and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. A_DAV × RAT (R_CFOV × RAT, R_PRODV × RAT, R_DISXV × RAT, A_DAV_EBXI × RAT, RM1_V × RAT, RM2_V × RAT, RM3_V × RAT) are interaction terms. This table presents the regression coefficients and adjusted R2. The t-statistics calculated by firm-level clustered standard errors for each coefficient appears immediately underneath. Bold values are used to highlight the results of the main variables this study focus on. ∗ Significance of 10%. ∗∗ Significance of 5%. ∗∗∗ Significance of 1%.

Real Earnings Management Uncertainty and Corporate Credit Risk 19

Lnamt

779.6832∗∗∗ (5.28) 6.8765∗∗ (2.34) 1.4919∗ (1.88) 0.1368 (0.39) 18.5779∗∗∗ (4.09) 2.9806∗∗∗ (2.92) 220.0458 (20.09) 2489.2314∗∗∗ (22.84) 3214.3874∗∗∗ (4.11) 285.2727 (21.46) 2330.6461∗∗∗ (25.23) 7957 0.6815

20

T.-K. Chen et al.

Table 6. The effects of alternative measures of abnormal cash flow volatility on bond yield spreads (1) R_CFOCV

35.2375∗∗ (2.19)

(2)

VOL Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ Constant Observations R2

276.1133∗∗∗ (27.32) 8584 0.5597

(4)

32.7053∗∗∗ (3.61)

29.6816∗∗∗ (3.35) 441.4962∗∗∗ (4.85) 750.7635∗∗∗ (5.84) 7.2692∗∗ (2.24) 1.7068∗∗ (2.04) 0.0389 (0.10) 18.9792∗∗∗ (4.11) 2.0557∗∗ (2.00) 233.9876 (20.14) 2496.0098∗∗∗ (22.79) 2988.0221∗∗∗ (4.35) 218.7205 (20.43) 2311.6640∗∗∗ (25.11) 7219 0.6799

28.3616∗∗ (2.36)

R_CFOCV3 LEV

(3)

443.9269∗∗∗ (4.88) 743.4915∗∗∗ (5.76) 7.2335∗∗ (2.23) 1.7447∗∗ (2.10) 0.0192 (0.05) 18.8110∗∗∗ (4.11) 1.9740∗ (1.91) 218.5358 (20.08) 2499.5686∗∗∗ (22.82) 2984.3653∗∗∗ (4.33) 216.6942 (20.39) 2304.7435∗∗∗ (25.00) 7219 0.6796

277.8045∗∗∗ (29.58) 8584 0.5598

Notes: this table shows the results of four different panel regressions with the yield spread (YS) as the dependent variable against various explanatory variable combinations using data of all bond observations with firms in the sample period. The fixed effects (firm and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFOC is estimated by the residual of the regression model with R_CFO (the abnormal levels of cash flow from operations) as the dependent variable against other two RM variables of R_PROD and R_DISX. R_CFOCV and R_CFOCV3 are estimated by the standard deviations of previous 4-year and 3-year R_CFOC historical data, respectively. The control variables include leverage ratio (LEV) equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT), and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. This table presents the regression coefficients and adjusted R2. The t-statistics calculated by firm-level clustered standard errors for each coefficient appears immediately underneath. Bold values are used to highlight the results of the main variables this study focus on. ∗ Significance of 10% . ∗∗ Significance of 5%. ∗∗∗ Significance of 1%.

R_PRODV by using bond observations that the issuer firms are in the manufacturing industry. Results are summarised in Table 7. The coefficient of R_PRODV is significantly positive; the yield spreads increase 22.75 bps (1.7935 × 12.6839) per standard deviation increase in R_PRODV. Compared with the full sample result (18.43 bps per standard deviation increase in R_PRODV), the magnitude of the R_PRODV impact on bond yield spreads is greater for manufacturing industry firms. In addition, the volatility of a firm’s input price may induce greater fluctuations of the firm’s production costs. As a result, the proxy of R_PRODV may capture not only the firm’s RM

Real Earnings Management Uncertainty and Corporate Credit Risk 21 Table 7. The effects of R_PROD volatility on bond yield spreads: manufacturing industry and controlling suppliers’ operating risk perspectives Manufacturing industry R_PRODV

12.6839∗∗ (2.36)

S_OCFV LEV VOL Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ Constant Observations R2

429.1784∗∗∗ (3.99) 514.4483∗∗∗ (4.53) 10.3007∗∗ (2.27) 20.1529 (20.15) 0.0948 (0.19) 16.8438∗∗∗ (2.90) 2.4977∗∗ (2.04) 280.2491 (20.26) 2480.8877∗∗ (22.12) 2176.1884∗∗∗ (3.62) 297.2671 (21.41) 2268.0317∗∗∗ (24.84) 4579 0.6470

With Suppliers’ OCFV 9.8450∗∗ (2.39) 2151.1762 (20.79) 623.6621∗∗∗ (5.46) 429.9799∗∗∗ (3.57) 3.1869 (0.68) 3.0344∗∗∗ (2.76) 20.0423 (20.06) 14.4501∗∗ (2.50) 0.4981 (0.42) 186.3742 (0.63) 2548.5895∗∗ (22.07) 22148.0147 (20.98) 238.5720 (20.58) 2153.9997∗ (21.95) 2912 0.6973

Notes: this table shows the results of four different panel regressions with the yield spread (YS) as the dependent variable against various explanatory variable combinations using data of all bond observations with manufacturing firms in the sample period. The fixed effects (firm and year) and cluster issues (Petersen, 2009) are considered in these results. This study follows Kale and Shuhrur (2007) and Chen et al. (2013) to estimate the proxy of suppliers’ operating risk. S_OCFV represents suppliers’ operating cash flow (S_OCF) volatility, estimated by S_OCF data for 4 years prior to the end of each year. The control variables include leverage ratio (LEV), equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT), and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. This table presents the regression coefficients and adjusted R2. The t-statistics calculated by firm-level clustered standard errors for each coefficient appears immediately underneath. Bold values are used to highlight the results of the main variables this study focus on. ∗ Significance of 10%. ∗∗ Significance of 5%. ∗∗∗ Significance of 1%.

variations, but the operating risks of its suppliers. To mitigate this concern, this study employs suppliers’ operating cash flow volatilities (S_OCFV) as an additional control variable to proxy the operating risks of the suppliers (namely the firm’s input price risk). To calculate a firm’s S_OCFV, this study follows Fee and Thomas (2004), Kale and Shahrur (2007), and Chen, Liao, and Kuo (2013) to identify a firm’s suppliers at the firm level by using the ‘COMPUSTAT industry segment files’ database. With the identifications of a firm’s suppliers, we can construct a firm-specific measure of suppliers’ operating cash flow volatilities

Regression of RM volatility against bond yield spreads: with earnings management incentives (measured by earnings per asset value ranging from 0 to 0.01)

R_CFOV

(2)

(3)

(4)

(5)

842.0470∗∗∗ (9.32) 865.0704∗∗∗ (9.78)

R_PRODV

166.6802∗∗∗ (9.02)

R_DISXV

87.9103∗∗∗ (9.44)

RM1_V

94.3950∗∗∗ (9.44)

RM2_V RM3_V LEV VOL Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ

(6)

2767.9670∗∗∗ (25.30) 3484.9877∗∗∗ (10.71) 6.3284 (0.26) 20.6092 (20.19) 27.3696 (21.23) 38.9364 (1.36) 3.6771 (0.59) 27890.5340∗∗∗ (27.19) 28073.9576∗∗∗ (25.19) 223,025.8488∗∗∗ (23.92) 1834.9105∗∗∗ (8.16)

2586.1205∗∗∗ (23.88) 3466.6156∗∗∗ (10.58) 2.8334 (0.12) 20.2986 (20.09) 27.3993 (21.22) 38.0651 (1.34) 2.9384 (0.47) 27063.7949∗∗∗ (26.26) 28028.1254∗∗∗ (25.15) 229,074.1837∗∗∗ (25.06) 2048.1865∗∗∗ (8.68)

2891.1260∗∗∗ (26.19) 3421.7993∗∗∗ (10.65) 6.1097 (0.25) 20.4202 (20.13) 27.7043 (21.24) 37.8069 (1.29) 4.4556 (0.65) 26822.3242∗∗∗ (26.15) 28356.8905∗∗∗ (25.09) 229,447.4722∗∗∗ (24.97) 1549.2448∗∗∗ (6.66)

2818.9927∗∗∗ (25.65) 3458.3903∗∗∗ (10.74) 2.5447 (0.11) 20.1141 (20.04) 27.7323 (21.23) 37.0604 (1.27) 3.6127 (0.53) 26894.9119∗∗∗ (26.19) 28205.9471∗∗∗ (24.99) 228,658.8447∗∗∗ (24.82) 1639.8405∗∗∗ (6.95)

2812.1477∗∗∗ (25.60) 3462.1993∗∗∗ (10.76) 2.5447 (0.11) 20.1141 (20.04) 27.7323 (21.23) 37.0604 (1.27) 3.6127 (0.53) 26936.6282∗∗∗ (26.24) 28204.2497∗∗∗ (24.99) 228,362.1164∗∗∗ (24.77) 1646.5047∗∗∗ (6.96)

115.8392∗∗∗ (9.02) 2866.5634∗∗∗ (26.01) 3437.2249∗∗∗ (10.71) 6.1097 (0.25) 20.4202 (20.13) 27.7043 (21.24) 37.8069 (1.29) 4.4556 (0.65) 26827.5151∗∗∗ (26.15) 28268.2963∗∗∗ (25.04) 229,012.7107∗∗∗ (24.89) 1574.5763∗∗∗ (6.72)

T.-K. Chen et al.

(1)

22

Table 8.

Constant Observations R2

860.7994∗∗∗ (3.07) 276 0.4597

83.5251 (0.38) 274 0.4798

372.9107 (1.56) 267 0.4626

1037.0436∗∗∗ (3.66) 265 0.4827

318.2562 (1.38) 265 0.4827

347.1893 (1.46) 267 0.4626

Real Earnings Management Uncertainty and Corporate Credit Risk 23

Notes: this table shows the results of six regressions with bond yield spread (YS) as the dependent variable against various variables of RM volatility using observations restricted to firms with earnings per asset value ranging from 0 to 0.01 (following Gunny, 2010). The fixed effects (industry and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively. The control variables include leverage ratio (LEV), equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT), and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. The t-statistics are calculated by firm-level clustered standard errors for each coefficient and appears immediately underneath. ∗ Significance at the 10% level. ∗∗ Significance at the 5% level. ∗∗∗ Significance at the 1% level.

R_CFOV

(2)

(3)

(4)

(5)

29.8285∗ (1.66) 11.6067∗∗ (2.32)

R_PRODV

15.5913∗ (1.89)

R_DISXV

11.7740∗∗∗ (3.38)

RM1_V

12.8360∗∗∗ (3.27)

RM2_V RM3_V LEV VOL Coupon Bage LFFL Lnamt RAT ADJPIN ROA OCFV AQ

(6)

238.6449∗∗∗ (3.04) 731.7044∗∗∗ (6.13) 26.6574∗∗∗ (4.77) 23.6254∗∗ (22.09) 20.3386 (20.87) 22.6754 (20.35) 9.2362∗∗∗ (5.34) 203.5605 (1.03) 217.7334 (1.21) 968.8991∗∗ (2.52) 274.4999 (21.34)

207.9818∗∗∗ (2.67) 758.5458∗∗∗ (6.21) 26.2033∗∗∗ (4.69) 23.4295∗∗ (21.98) 20.3481 (20.88) 22.6676 (20.34) 9.3033∗∗∗ (5.39) 181.8535 (0.89) 123.1341 (0.67) 971.2036∗∗ (2.53) 2101.8905 (21.60)

210.1739∗∗∗ (2.63) 728.6216∗∗∗ (5.79) 27.3656∗∗∗ (4.31) 23.4281∗ (21.76) 20.2965 (20.69) 22.7838 (20.33) 10.7629∗∗∗ (5.15) 175.5208 (0.83) 183.2183 (1.03) 832.7494∗∗ (2.15) 2104.3599∗ (21.82)

192.9266∗∗ (2.40) 755.0982∗∗∗ (5.94) 26.8459∗∗∗ (4.19) 23.1644 (21.62) 20.2708 (20.62) 21.5909 (20.19) 10.7887∗∗∗ (5.15) 159.1738 (0.74) 115.4747 (0.63) 824.6070∗∗ (2.18) 2145.4861∗∗ (22.43)

191.5546∗∗ (2.37) 753.0503∗∗∗ (5.92) 26.9745∗∗∗ (4.21) 23.2208 (21.65) 20.2777 (20.63) 21.9144 (20.22) 10.8392∗∗∗ (5.17) 154.5514 (0.72) 109.6592 (0.60) 826.2217∗∗ (2.18) 2148.3177∗∗ (22.48)

16.1517∗∗ (2.10) 211.7736∗∗∗ (2.66) 731.7022∗∗∗ (5.81) 27.1997∗∗∗ (4.26) 23.3713∗ (21.72) 20.2823 (20.66) 22.3695 (20.28) 10.7394∗∗∗ (5.14) 178.3733 (0.84) 187.0439 (1.05) 830.4923∗∗ (2.14) 2104.9835∗ (21.86)

T.-K. Chen et al.

(1)

24

Table 9. Regression of RM volatility against bond yield spreads: with earnings management incentives (measured by the change of earnings per asset value ranging from 0 to 0.01)

Constant Observations R2

2270.3113∗∗∗ (24.57) 1455 0.7721

2256.5158∗∗∗ (24.39) 1449 0.7725

2272.5793∗∗∗ (24.41) 1254 0.7714

2271.9543∗∗∗ (24.37) 1249 0.7730

2269.8116∗∗∗ (24.35) 1249 0.7729

2276.1616∗∗∗ (24.46) 1254 0.7717

Real Earnings Management Uncertainty and Corporate Credit Risk 25

Notes: this table shows the results of six regressions with bond yield spread (YS) as the dependent variable against various variables of RM volatility using observations restricted to firms with the change of earnings per asset value ranging from 0 to 0.01 (following Gunny, 2010). The fixed effects (industry and year) and cluster issues (Petersen, 2009) are considered in these results. R_CFOV, R_PRODV, and R_DISXV are these three RM volatility variables, respectively. RM1_V, RM2_V, and RM3_V are these aggregate RM volatility variables, respectively. The control variables include leverage ratio (LEV), equity volatility (VOL), bond age (Bage), time to maturity (LFFL), the natural log of amount issued (Lnamt), credit rating (RAT), and annualised coupon rate (Coupon). ADJPIN, ROA, OCFV, and AQ separately show the information asymmetry, returns of assets, cash flow volatility, and accrual quality of firms. The t-statistics are calculated by firm-level clustered standard errors for each coefficient and appears immediately underneath. ∗ Significance at the 10% level. ∗∗ Significance at the 5% level. ∗∗∗ Significance at the 1% level.

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(S_OCFV). Because a firm may have multiple suppliers, the current research computes the firm’s S_OCFV as Equation (15). In Equation (15), C ICj (Customer Input Coefficientj) represents the ratio of the firm’s purchases from the jth supplier to the firm’s total sales, S OCFVj (Supplier OCFVj) is the jth supplier’s operating cash flow volatility, and n is the number of suppliers: S OCFV =

n

j=1

S OCFVj × C ICj .

(15)

Table 7 demonstrates the results of the R_PROD volatility effects on bond yield spreads when controlling for suppliers’ operating risks (measured by the S_OCFV variable). The empirical result shows that R_PRODV is significantly and positively related to bond yield spreads when the S_OCFV variable is controlled. Therefore, the result is robust using this alternative specification. 5.2.4. Incentives to engage in earnings management (suspect firm analyses) Most previous RM studies also consider the incentives to engage in earnings management. To increase the confidence that the RM proxies capture RM activities, this study further restricts the sample to firms with earnings management incentives. Following Gunny (2010), this study identifies firms that just meet earnings benchmarks (zero or last year’s earnings) as their earnings scaled by total assets fall into the interval of [0, 0.01]. The empirical results using sample firms suspected to meet or just exceed zero earnings and last year’s earnings are shown in Tables 8 and 9, respectively.14 The results show that RMV variables are significantly and positively related to bond yield spreads, consistent with the main results. 5.3. Robustness Checks To show the robustness of the results, this study conducts three robustness examinations. First, the simultaneous equations and two-stage regressions are applied to address the endogeneity issue. Second, this study changes the time interval of estimating a firm’s RMV or DAV from 4 to 3 years. Third, this study additionally controls for analyst-related information uncertainty variables, Follow and Hetro. Follow equals the number of analysts following a stock in the I/ B/E/S database, while Hetro is calculated by the standard deviation in analysts’ fiscal year 1 earnings per share forecasts made 1 month prior to fiscal year end.15 The untabulated results are all robust and are available upon request. 6.

Concluding Remarks

In contrast to previous studies, this research used 9565 American bond observations from 2001 to 2008 to investigate the effect of a firm’s accounting information uncertainty on bond yield spreads from the RM’s idiosyncratic risk perspective. Empirical results show that RM volatilities are significantly and positively related to bond yield spreads when well-known bond spread determinant 14

Gunny (2010) indicates that RM activities mostly take place before analysts’ forecasts prior to the earnings announcement. In addition, managers tend to use forecast guidance as a mechanism to avoid missing analysts’ forecasts. Therefore, following Gunny (2010), the current work does not focus on analysts’ forecasts. 15 The Hetro variable is also scaled by the absolute value of the mean forecast. This study uses FY1 EPS forecast data to measure Hetro, which is reported in the I/B/E/S detail history database. Zhang (2006) shows that Follow negatively relates to the degree of information uncertainty, while DISP positively relates to it. Hence, the Follow variable negatively relate to yield spreads, while the Hetro variable positively relates to it based on structural credit model frameworks.

Real Earnings Management Uncertainty and Corporate Credit Risk 27 variables are controlled. In addition, the results are robust to alternative model specifications, including suspect firm analyses, another less ambiguous measure of abnormal cash flows from operations, and abnormal production cost analyses in the manufacturing industry or with control of input price variation. The results also show that the positive effects of RM volatilities on bond yield spreads weaken when a firm has a lower bond credit rating. Furthermore, the results still hold when endogeneity issues are considered, the estimation period of 4-year RMV is replaced by the previous 3-year RMV, or additional analyst-related variables are controlled. Therefore, this study contributes to the literature by developing an application model, which provides a theoretical framework for the effects of RM variations on credit risks, and empirically shows that the RM variations are positively associated with the bond yield spreads. It also supports that the idiosyncratic risk effects resulting from RM volatilities have a substantial impact on firm credit risk.

Acknowledgements We are indebted to anonymous referees and especially the editor for their insightful comments and suggestions on earlier drafts of the paper. We thank Prof. Nancy Beneda, Ting-Pin Wu, Konan Chan, Yanzhi Wang, and the participants at 2012 FMA Annual Meeting, Fu Jen Catholic University seminar (March 2013), National Taipei University seminar (March 2013), 2013 CTFA Annual Meeting (April 2013), and NTU/NCCU joint brown bag seminar (May 2013) for helpful comments and suggestions. We also appreciate Prof. Hsien-Hsing Liao for providing the ADJPIN data estimated by the EM-algorithm (Chen & Chung, 2007).

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The European Journal of Finance, 2014 Vol. 20, No. 5, 446–462, http://dx.doi.org/10.1080/1351847X.2012.714792

Risk aversion vs. individualism: what drives risk taking in household finance? Wolfgang Breuer, Michael Riesener and Astrid Juliane Salzmann∗ Department of Finance, RWTH Aachen University, Templergraben 64, 52056 Aachen, Germany (Received 24 June 2010; final version received 19 July 2012) Despite a considerable premium on equity with respect to risk-free assets, many households do not own stocks. We ask why the prevalence of stockholding is so limited. We focus on individuals’ attitudes toward risk and identify relevant factors that affect the willingness to take financial risks. Our empirical evidence contradicts standard portfolio theory, as it does not indicate a significant relationship between risk aversion and financial risk taking. However, our analysis supports the behavioral view that psychological factors rooted in national culture affect portfolio choice. Individualism, which is linked to overconfidence and overoptimism, has a significantly positive effect on financial risk taking. In microdata from Germany and Singapore, as well as in cross-country data, we find evidence consistent with low levels of individualism being an important factor in explaining the limited participation puzzle.

Keywords: household finance; individualism; risk aversion; risk taking JEL Classification: A13; D14; G11; Z13

1. Introduction Despite a considerable premium on equity with respect to risk-free assets, many households do not hold stocks. In most countries, the majority of households do not own stocks even indirectly through mutual funds or retirement accounts. Exceptions are Sweden, the UK, and the US, where stockholding participation amounts to over 40% of all households, but participation rates are below 20% in Germany, France, Italy, and Japan (Haliassos 2008). We ask why the prevalence of stockholding is so limited. We focus on individuals’ attitudes toward risk and identify relevant factors that affect the willingness to take financial risks. We test directly whether the individual’s subjectively measured rate of risk aversion influences the holding of risky assets. We further examine whether behavioral biases arising from cultural influences are helpful to explain attitudes toward risk taking in financial matters. In microdata from Germany and Singapore, as well as in cross-country data, we find evidence consistent with low levels of individualism being an important factor in explaining the limited participation puzzle. Existing research has explored a variety of factors to explain risk-taking behavior, and we are not the first to analyze the determinants of household willingness to take financial risks. However, the novel approach in this article is to decompose the variation in risk attitudes across individuals into separate effects from risk aversion and cultural values. Standard finance theory describes the choices that maximize household welfare. Investment in risky assets is rewarded by higher expected portfolio returns, and risk averse households determine their best trade-off between risk and expected return. Under the standard axioms on decisions under

∗ Corresponding

author. Email: [email protected]

© 2012 Taylor & Francis

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uncertainty, any household will select that portfolio that maximizes the expected utility of their final consumption. An increase in risk aversion reduces the demand for risky assets (Gollier 2002). In the following, we directly measure risk preferences of individuals through lottery questions and examine how these preferences correlate with the willingness to invest in risky assets. Behavioral finance theory describes the choices that households actually make. Some households make decisions that are hard to reconcile with any standard model, and observed portfolio composition often differs from predictions of the standard model. In fact, several studies reveal that non-standard models with behavioral factors explain why many individuals do not invest in stocks or other risky financial assets. Transaction and information costs, broadly interpreted, are suggested as the main reason for variations in stockholdings or the lack of stockholdings across individuals. The exact nature of these costs, however, is still not well understood (Christelis, Jappelli, and Padula 2010). In this article, we focus on cultural influences on human preferences as a potential explanation for limited willingness to take financial risks. The study of household finance is challenging because household behavior is difficult to measure (Campbell 2006). Our analysis uses detailed microeconomic data sets allowing us to control for a wide range of individual characteristics that may impact the willingness to take financial risks. Most importantly, we can directly infer a degree of risk aversion from hypothetical survey questions asking to compare different lotteries. Our data also contain information on cultural values that may be relevant for attitudes toward risk taking. The contribution of our research is threefold. First, we are able to examine the immediate relationship between risk aversion and individual-specific attitudes toward risk taking in financial matters. Though risk aversion has been recognized as a crucial determinant for financial risk taking, empirical studies seldom use information on an individual’s rate of risk aversion, as such information is rarely available. Instead, most research relies on variables that proxy risk aversion through self-reported risk attitudes (see, for example, Haliassos and Bertraut 1995; Keller and Siegrist 2006; Barsky et al. 1997). We measure risk aversion directly through a set of hypothetical lottery questions. Second, we examine the relationship between cultural values and attitudes toward risk taking in financial matters. National culture has recently emerged as a powerful determinant in economic studies (Guiso, Sapienza, and Zingales 2006). Hofstede (1983) defines culture as the ‘collective programming of the mind’, indicating that culture is composed of certain values that shape attitudes and behavior. For that reason looking further and deeper into the individual, to the very things that define an individual’s psychology, seems a fruitful approach to understand economic conduct (Durand, Newby, and Sanghani 2008). Our data set provides information on cultural values of the individual, so that we are able to test the immediate influence of culture on risk attitudes. Though the integration of cultural values into financial risk taking would certainly allow for interesting conclusions regarding participation in the stock market, this relationship remains unexplored in the current literature. Third, we extend our analysis to the country level. We not only examine whether culture exerts a direct influence on individual risk attitudes, but also investigate whether culture has an impact on actual economic outcomes. We therefore use cross-country data on cultural values and portfolio shares in equity. This analysis sheds light on the broader influence of culture, and provides an approach to identify a causal effect from national culture to economic outcomes. Our models explain financial risk taking from subjectively measured variables capturing risk aversion and cultural values as well as socio-demographic characteristics. We do not address the issue of endogeneity of the latter variables. The underlying conceptual model of our analysis is rather simple: individual characteristics (gender, age, income, and wealth) are given; risk aversion and cultural values may vary within these characteristics. In particular, we do not investigate

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possible relationships between preference parameters on the one side and income and wealth on the other side. Instead, we focus on how risk taking in financial matters is driven by risk aversion or cultural values as well as individual characteristics. Yet, we will address the endogeneity issue regarding cultural values on the one side and economic outcomes on the other side, as this relationship describes one of our major research questions. The article proceeds as follows. In Section 2, we link our study to the previous literature. Section 3 details the survey design. Section 4 contains the results. Section 5 provides evidence at the country level. Section 6 concludes. 2. Literature review The relationship between risk attitudes and the participation in the stock market is straightforward and well established in the literature. Several studies show that the probability of stockholding is smaller the larger the degree of risk tolerance (Shum and Faig 2006; Puri and Robinson 2007; Donkers and van Soest 1999; Barsky et al. 1997). However, most of these analyses employ respondents’ self-assessed risk attitudes as a proxy for risk aversion, and there is little research on the extent to which actual measures of risk aversion influence the willingness to take financial risks. Dimmock and Kouwenberg (2010) obtain direct measures of risk aversion through survey data involving hypothetical payoffs. Their evidence indicates that risk aversion is highly significant for the decision to participate in risky markets. Wärneryd (1996) measures risk aversion through a set of hypothetical lottery questions and examines the relationship to actual investments in risky assets. He finds that the fact that a household invests in more risky assets is quite well explained by risk aversion, but contrary to standard portfolio theory, the proportion of risky assets is less well explained. Similarly, Guiso, Sapienza, and Zingales (2008) consider a set of risky lotteries to derive a measure of risk aversion and discover that risk aversion has little predictive power for financial risk taking. To this effect, Fellner and Maciejovsky (2007) note that observations of how people deal with risks in real life cast some doubts on the occurrence of risk aversion. They document that studies of decision-making frequently uncover inconclusive evidence regarding the relation of risk attitudes and individual behavior. Empirical studies often find that actual risky choices deviate substantially from what maximization of expected utility presumed. Economic models of standard portfolio theory suggest that generally all households should own stocks. A household should be willing to participate in the stock market because of the equity premium (Haliassos and Bertraut 1995). Theoretical models typically derive household portfolio choice by maximizing an expected utility function conditional on household preferences. Among these, the household’s rate of risk aversion is a crucial parameter. For instance, in the standard two-period Markowitz model of portfolio theory, the choice between risky and riskless assets depends on the individual’s risk aversion parameter (Markowitz 1952). Risk aversion is defined as a preference for a sure outcome over a prospect with an equal or greater expected value (Tversky and Kahnemann 1981). Risk aversion is usually assumed as a stable personal trait. As theory predicts, households that are more risk averse should be less inclined to invest in risky assets (Guiso, Haliassos, and Jappelli 2003). Hypothesis 1: Risk aversion is negatively related to the willingness to take financial risks Empirical studies document that equity market participation is much lower than what is implied by standard portfolio theory. It follows that limited participation in the stock market must be due to the inadequacy of the standard assumptions (Campbell 2006). The literature suggests several

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explanations for the reluctance of investment in markets for risky assets. The view that seems to have gained most support is that households face some actual or perceived fixed entry or participation costs that discourage investment in the stock market (Haliassos 2008). The costs are eclectic and can be interpreted in distinct ways. One concept is to understand them broadly as transaction costs, ranging from trading costs to monitoring costs (Guiso, Haliassos, and Jappelli 2003). An alternative approach is to describe them as psychological factors that make equity ownership uncomfortable for some households. The exact nature of these costs is however not well understood and a matter of ongoing research (Campbell 2006). The empirical literature provides various findings consistent with the presence of psychological individual-specific factors that influence investment in risky assets. Puri and Robinson (2007) show that more optimistic individuals are more likely to participate in the equity market. Guiso, Sapienza, and Zingales (2008) find that an individual’s level of trust toward others is an effective measure to predict the level of stock market participation. Hong, Kubik, and Stein (2004) contend that social households are more likely to invest in the stock market than non-social households. Georgarakos and Pasini (2011) demonstrate that trust and sociability have distinct and sizeable effects on stock market participation. Behavioral finance provides several explanations for the apparent irrationality of investors. Behavioral finance is concerned with psychological influences on individual investor behavior (Charness and Gneezy 2010). Several studies acknowledge that investor portfolio diversity can be attributed to psychological factors (Shum and Faig 2006). Intrinsic differences in how to view the world may lead to heterogeneity in beliefs (He and Shi 2010). As financial decisions are often made in situations of high complexity and high uncertainty that lack formal rules for decisionmaking, many conclusions rely on intuition (Kahneman and Riepe 1998). Intuitions play a crucial role in most decisions and may cause systematic errors in judgment, so called biases. Two of the most prominent biases discussed in the investment literature are overconfidence and optimism. The interplay of overconfidence and optimism causes individuals to overestimate their knowledge, to underestimate risks, and to exaggerate their ability to control events (Giordani and Söderlind 2006). Both biases are substantially related to attitudes toward risk taking. We refer to the concept of national culture as a way of capturing and measuring information about the psychology of investors. Culture is defined as customary beliefs and values that social groups transmit fairly unchanged from one generation to another. Cultural beliefs and values reflect a person’s sense of what is good, right, fair, and just. Restricting the potential channel of cultural influence to values and beliefs provides an approach to identify a causal link from culture to economic behavior (Guiso, Sapienza, and Zingales 2006). Virtually, all decision theorists agree that values and beliefs jointly influence the willingness to invest under uncertainty (Campbell 2006). We focus on what psychologists refer to as ‘individualism’. Individualism describes the relationship between the individual and the collectivity that prevails in a given society. In individualist societies, the ties between individuals are loose, and everyone is expected to look after him- or herself. In the polar type, collectivist societies, people are integrated into strong groups that they are unquestioningly loyal to (Hofstede 2001). Although individualism does not directly measure the behavioral biases of overoptimism and overconfidence, the psychology literature suggests a link between individualism and overconfidence as well as overoptimism. In more individualistic societies, more decisions are made by the individual and these decisions are more likely to be driven by overconfidence (Chui, Titman, and Wei 2010). Markus and Kitayama (1991) indicate that people in individualistic cultures think positively about their abilities. Van den Steen (2004) argues that when individuals are overoptimistic about

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their abilities, they tend to overestimate the precision of their predictions, whereas in collectivist cultures, people are concerned with behaving properly and exert high self-monitoring. Church et al. (2006) discuss that high self-monitoring helps to reduce the cognitive bias caused by overoptimism. Odean (1998) demonstrates that overoptimism leads to a miscalibration in beliefs. Puri and Robinson (2007) find that optimism is significantly related to attitudes toward risk. Grinblatt and Keloharju (2009) contend that overconfidence might result in a miscalibration of beliefs, what implies a tendency to be excessively confident in one’s estimate of a parameter, such as the future return of a stock. Glaser and Weber (2007) evidence that overconfident investors tend to believe that one has the skill to pick winning stocks with above-average returns. Pan and Statman (2009) maintain that highly overconfident people exhibit indeed more risk tolerance than less overconfident people. Hypothesis 2: Individualism is positively related to the willingness to take financial risks Recent strands of the literature explore the link between financial risk taking and household socio-demographic characteristics. Gender, age, income, and wealth are probably those factors that affect household financial decisions the most. In general, women are found to be making more risk averse choices than men (Barnea, Cronqvist, and Siegel 2010; Renneboog and Spaenjers, 2012). There is no consensus on the relationship with age. While the majority of empirical work observe no age differences in risk propensity (Guiso, Sapienza, and Zingales 2008; Dimmock and Kouwenberg 2010), some studies find that risk taking rises with age (Shum and Faig 2006), whereas others suggest a declining risk taking (Campbell 2006). Almost all studies consistently support a positive link between wealth and income and investment in risky assets (Guiso, Sapienza, and Zingales 2008; Barnea, Cronqvist, and Siegel 2010; Campbell 2006). We control for these variables in our regression analysis. Risk and time preferences are underresearched in the context of household finance. Dimmock and Kouwenberg (2010) examine the link between such parameters and equity market participation. They reveal that, among an individual’s risk preferences, loss aversion has a significantly negative impact. Time discount rates seem to be unrelated. We include variables for these preference parameters in our regression models as well. 3. Survey design and data collection We use a specifically designed survey to obtain the different types of individual data we need to test our hypotheses. Altogether, the questionnaire for the experiment consists of five main components. The main components of the survey are questions concerning the risk-taking behavior of the individual following Puri and Robinson (2007) and different lotteries concerning risk preferences as well as questions on cultural values. In addition, the survey comprises demographic variables, a section covering the economic background of the respondent, and questions deriving time preferences. As our article seeks to explain financial risk taking, the willingness to invest in risky assets is our focal point of interest. The literature suggests many different measures for the riskiness of an individual’s portfolios, and until today there is no consensus on the most appropriate question. We employ a question developed in Puri and Robinson (2007): Which of the following statements comes closest to the amount of financial risk that you are willing to take when you save or make investments?

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Respondents were allowed to choose between the following four answers: (4) ‘Take substantial financial risks expecting to earn above average returns’ (3) ‘Take above average financial risks expecting to earn above average returns’ (2) ‘Take average financial risks expecting to earn average returns’ (1) ‘Not willing to take any financial risks’ We define the numerical answer to this question as the variable risky, implicating that the higher the value of this variable, the riskier is the individual’s behavior regarding portfolio choice. Concerning risk preferences, we use the main ideas and questions of Prospect Theory that has been developed by Kahneman and Tversky in the late 1970s. In Prospect Theory, risk preferences can be explained by three different parameters, risk aversion α, probability bias γ , and loss aversion λ. These parameters are part of the commonly used value function by Kahneman and Tversky, which has been verified empirically.  + x ≥ 0, xα (1) v(x) := α− x < 0. −λ · (−x) Prospect Theory expresses outcomes as positive and negative deviations from a neutral reference outcome. In the above formula, the reference point is assigned a value of zero. Furthermore, v is the subjective value of an amount of money x, which can be positive or negative. In the positive part, the individual gains x, while he loses x in the negative part. Risk aversion toward gains as well as risk seeking toward losses are both presented by the parameter α, since many empirical studies found the two parameters α + and α − in formula (1) to be almost equal. As another characteristic of the value function, individuals respond to losses more extremely than to gains. Therefore, Kahneman and Tversky introduce the loss aversion parameter λ, which is usually greater than 1. Time preferences are described by two parameters which are called the present bias parameter β and the long-term discount factor δ. While the former one stands for an individual’s time preference between the present period and the next period, the latter one characterizes the time preference between any two future periods. Both parameters are derived assuming a quasihyperbolic discount-model. The appendix details the exact derivation of the variables for risk and time preferences. The parameters are derived through a set of hypothetical lottery questions. Clearly, a key issue for our article is the reliability of the measures for individual preferences. Some economists are skeptical about the use of subjective survey questions in general. As survey questions are not incentive compatible, individuals might respond randomly to survey questions, which would distort the induced measures (Dominitz and Manski 1997). Yet, we are eager that in particular our measure of risk aversion captures risk preferences accurately. First, the validity of the surveybased measures is examined and confirmed in laboratory experiments using paid lottery choices (Dohmen et al. 2011). Second, for several questions, there are simple, logical relationships that must hold between the responses. We find that only very few questionnaires of our sample violate these relations (Dimmock and Kouwenberg 2010). Third, the questions used in our survey have been asked in numerous prior studies and are well accepted in the behavioral literature (Thaler 1981). Finally, the responses and derived preference parameters are quantitatively similar to a large number of other empirical studies (Tversky and Kahneman 1992). We classify culture according to Hofstede by using the four original cultural dimensions uncertainty avoidance, power distance, individualism, and masculinity. Despite the criticism regarding

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Table 1. Variables and sources. Variable

Description

Variables from the individual data set (collected in surveys) risky

Risk aversion, α Loss aversion, λ Probability bias, γ Present bias, β Long-term discount factor, δ Uncertainty avoidance Power distance Individualism Masculinity Sex Age Ln(month income) Ln(wealth)

Respondent’s answer to the question: ‘Which of the statements on this page comes closest to the amount of financial risk that you are willing to take when you save or make investments?’ [(4) Take substantial financial risks expecting to earn above-average returns; (3) Take above-average financial risks expecting to earn above-average returns; (2) Take average financial risks expecting to earn average returns; (1) Not willing to take any financial risks] Risk preferences parameter from Prospect Theory according to Kahneman and Tversky (1981). See appendix for a detailed derivation. Risk preferences parameter from Prospect Theory according to Kahneman and Tversky (1981). See appendix for a detailed derivation. Risk preferences parameter from Prospect Theory according to Kahneman and Tversky (1981). See appendix for a detailed derivation. Time preferences parameter according to the Quasi-hyperbolic discount-model. See appendix for a detailed derivation. Time preferences parameter according to the Quasi-hyperbolic discount-model. See appendix for a detailed derivation. Hofstede Uncertainty Avoidance Index following Hofstede (2001). Power Distance Index following Hofstede (2001). Hofstede Individualism Index following Hofstede (2001). Hofstede Masculinity Index following Hofstede (2001). Gender. [(1) male, (0) female]. Age of the respondent. Logarithm of the respondent’s monthly disposal in ¤. Logarithm of the respondent’s actual total wealth in ¤.

Variables from the cross-country data set (source in parentheses) Equities

Uncertainty avoidance Power distance Individualism Masculinity Market capitalization Median income Trust Dependency ratio

The ratio of equity assets to total assets held by the household sector. Equity assets consist of claims to residual value of incorporated enterprises, after claims of all creditors, and include mutual fund holdings. Data for 2008. (EIU WorldData) Hofstede Uncertainty Avoidance Index (Hofstede 2001). Hofstede Power Distance Index (Hofstede 2001). Hofstede Individualism Index (Hofstede 2001). Hofstede Masculinity Index (Hofstede 2001). Value of listed shares to GDP. Data is averaged over 2000 to 2008 (Beck and Demirgüç-Kunt 2010). Median nominal disposable income earned by households per annum. Data for 2008 (EIU WorldData). Average level of trust in a country. (World Values Survey) The dependency ratio is the sum of the ratio of the population under age 15 to the population ages 15–64 and the ratio of the population over age 64 to the population ages 15–64 (EIU WorldData).

Note: The table lists the descriptions of variables for the regressions.

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Hofstede’s work (Sivakumar and Nakata 2001), academic research has relied extensively on his framework to analyze the impact of culture (Kirkman, Lowe, and Gibson 2006). • Uncertainty avoidance deals with a society’s tolerance for uncertainty and ambiguity and refers to its search for truth. • Power distance is the extent to which different societies handle human inequality differently. • Individualism describes the relationship between the individual and the collectivity that prevails in a given society. • Masculinity refers to the distribution of roles between genders. The questions used to compute the respective indices can be found in Hofstede (2001). Each cultural dimension is calculated using the numerical answer to four different questions or statements. We follow the calculation formulas suggested in Hofstede (2001) to derive the specific values for each cultural dimension from the survey questions. Since the economic background of a respondent may have an influence on his financial decisions, we ask questions concerning wealth and monthly income. A description of these and other demographic control variables can be found in Table 1. Table 2. Summary statistics. Variable

N

Mean

Median

SD

Min.

Max.

2.09 0.59 1.66 0.61 0.65 0.85 65.50 −8.06 48.71 76.94 0.71 22.43 6.26 8.72

2.00 0.55 1.00 0.54 0.69 0.86 65.00 −5.00 50.00 80.00 1.00 22.00 6.21 8.85

0.78 0.27 1.83 0.35 0.33 0.09 59.15 41.47 57.20 91.27 0.46 2.16 0.67 1.40

1.00 0.06 1.00 0.00 0.00 0.30 −105.00 −140.00 −400.00 −370.00 0.00 19.00 3.00 3.91

4.00 1.00 22.99 3.25 1.66 1.05 215.00 130.00 205.00 370.00 1.00 33.00 9.21 12.90

Summary statistics for cross-country data set Equities 34 0.31 Power distance 34 57.56 Individualism 34 49.71 Masculinity 34 49.62 Uncertainty avoidance 34 64.41 Market capitalization 34 0.93 Median income 34 32,228 Trust 34 0.24 Dependency ratio 34 0.48

0.26 59.00 47.00 51.00 64.50 0.68 24,465 0.19 0.49

0.18 20.40 25.01 19.09 21.34 0.78 24,842 0.14 0.06

0.00 22.00 13.00 5.00 29.00 0.13 2570 0.04 0.34

0.66 104.00 91.00 95.00 95.00 4.01 83,730 0.58 0.58

Summary statistics for individual data set risky 444 Risk aversion, α 318 Loss aversion, λ 250 Probability bias, γ 318 Present bias, β 434 Long-term discount factor, δ 434 Uncertainty avoidance 420 Power distance 405 Individualism 443 Masculinity 435 Sex 447 Age 431 Ln(month income) 352 Ln(wealth) 251

Note: The first part of this table presents descriptive statistics for the variables at the individual level. The second part of this table reports descriptive statistics for the variables at the country level. SD indicates the standard deviation.

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A total of 449 economic students participated in the surveys, which were conducted in Germany and Singapore. We use data from two culturally distinct but equally developed countries, to ensure sufficient cultural variance in the sample without introducing too much heterogeneity in the economic background variables. The respondents answered the questions during the first or the last part of university lectures. The German questionnaire has been translated into English for the Singaporean survey and checked by re-translation for translation mistakes that could influence the results. In order to adjust the questionnaire for differences in the currency, the cash flows have been converted using the Purchasing Power Parity between Germany and Singapore. We used students with an economic background, because this group of individuals is easy to reach and is able to understand the tasks involving lotteries. Additionally, a group like this is homogenous and therefore comparable across countries, as requested in Hofstede (2001). For the final regressions, we only included those respondents who answered more than 50% of the questions without inconsistencies. Table 2 illustrates the main summary statistics of our data. One might doubt the validity of the use of student samples in our analysis, objecting that the value ratings obtained from a student sample are probably not representative of household financial decision-makers in general. Effectively, value scores obtained for any other sample are almost certainly not the same for different types of samples or a representative sample. Yet, several analyses have repeated experiments with non-student (sometimes even with non-human) participants and were able to validate the findings of the experiments with students. For instance, in dictator and ultimatum games, Carpenter, Burks, and Verhoogen (2003) find no significant differences between student and non-student participants. Falk, Meier, and Zehnder (2012) confirm this result using a trust game. In addition, even if there are differences in preferences and behavior between students and non-students, we only need the assumption that the relationship between preference parameters and financial risk taking is similar across all individuals. We therefore believe that our data set is appropriate for analyzing household financial decision-making. 4. Results To empirically capture the relationship between an individual’s willingness to invest in risky assets and the individual’s risk preferences as well as cultural values, we estimate individual-level ordinary least-squares regression models. The regression results for Hypothesis 1 can be found in Table 3. The dependent variable is risky, the behavior of an individual in terms of riskiness of portfolio choice. The independent variables are the risk aversion α, demographic variables such as sex and age, and economic variables such as monthly income and wealth. Additionally, we control for other risk and time preferences. When we include controls, we lose observations due to missing variables for some respondents. Including all controls simultaneously would reduce our sample significantly. For this reason, we chose to include the control variables separately and in groups. The table reports the ordinary least-squares estimates. We find positive relationships between risk aversion α and risky in the regression models as well, but in none of all regressions, this relationship is significant. This finding contradicts standard portfolio theory and adds to the increasing evidence that risk aversion has only very low explanatory power for individual decision-making (Fellner and Maciejovsky 2007; Guiso, Sapienza, and Zingales 2008). The very low R2 -values and the insignificant F-statistics add to the limited influence of risk and time preferences on financial risk taking. The regression results for Hypothesis 2 can be found in Table 4. The dependent variable remains risky, capturing individual attitudes toward investment risk. The independent variables are the

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Table 3. Regression results for Hypothesis 1. Independent variable Constant Sex Age Ln(month income) Ln(wealth) Risk aversion, α

Dependent variable: risky 0.4805 (0.8303) 0.1571 (0.1395) 0.0142 (0.0293) 0.1696∗ (0.0915) 0.0093 (0.0430) 0.1310 (0.2387)

Loss aversion, λ

1.1024 (0.8949) 0.1989 (0.1561) 0.0167 (0.0332) 0.0637 (0.1024) −0.0280 (0.0482) 0.4272 (0.2814) 0.0383 (0.0328)

Probability bias, γ

0.1582 0.7828 (0.8358) (0.8993) 0.0741 0.1226 (0.1435) (0.1591) 0.0115 0.0183 (0.0290) (0.0328) 0.2056 0.0870 (0.0921)∗∗ (0.1019) 0.0042 −0.0382 (0.0426) (0.0479) 0.1685 0.4468 (0.2370) (0.2783) 0.0305 (0.0326) 0.4758∗∗ 0.3773∗∗ (0.1774) (0.2398)

Present bias, β Long-term discount factor, δ R2 Adjusted R2 Number of observations F-statistic p-Value Standard error

0.0377 0.0099 173 1.3569 0.2430 0.8177

0.0420 0.0033 127 0.9277 0.4775 0.7637

0.0624 0.0297 172 1.9075 0.0822∗ 0.8095

0.0710 0.0194 126 1.3758 0.2211 0.7550

0.9371 (1.0676) 0.1295 (0.1437) 0.0123 (0.0298) 0.1572∗ (0.0941) 0.0167 (0.0450) 0.1715 (0.2462)

0.2456 (0.205) −0.6618 (0.7162) 0.0538 0.0139 166 1.3481 0.2308 0.8181

Notes: The table shows coefficient estimates for ordinary least-squares regressions at the individual level with risk aversion as the main explanatory variable for financial risk taking according to Hypothesis 1. ∗∗∗ indicates the coefficient is different from 0 at the 1% level. ∗∗ indicates the coefficient is different from 0 at the 5% level. ∗ indicates the coefficient is different from 0 at the 10% level. Standard errors are given within parentheses.

cultural dimension of individualism as well as the remaining three Hofstede cultural dimensions uncertainty avoidance, power distance, and masculinity to avoid an omitted variables bias. We again include demographic variables such as sex and age and economic variables such as monthly income and wealth. We also include risk preferences and time preferences. The table reports ordinary least-squares estimates. We find a strong relationship between individualism and risky. Controlling for different variables, individualism has a significantly positive effect on an individual’s willingness to invest in risky assets. The coefficient of individualism in our basic regression is 0.0028, which implies that, all else equal, a one-standard-deviation increase in individualism would induce a 0.0028 · 57.2 = 0.1602 increase in risky. In percentage terms, relative to the mean of risky, this corresponds to a 0.1602 : 2.09 = 8% increase, which is economically significant. Our analysis supports the behavioral view that psychological biases induced by individualism are effective predictors of financial risk taking. Individualism is linked to overconfidence and optimism and increases the willingness to invest in risky financial assets.

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Table 4. Regression results for Hypothesis 2. Independent variable Constant Sex Age Ln(month income) Ln(wealth) Uncertainty avoidance Power distance Individualism Masculinity

Dependent variable: risky 0.8547 (0.7861) 0.1050 (0.1213) 0.0329 (0.0274) 0.0511 (0.0852) 0.0114 (0.0392) 0.0000 (0.0009) 0.0003 (0.0013) 0.0028∗∗∗ (0.0010) −0.0007 (0.0006)

Risk aversion, α Loss aversion, λ

1.7406∗ (1.0099) 0.0706 (0.1650) 0.0071 (0.0361) 0.0340 (0.1091) −0.0328 (0.0505) −0.0007 (0.0012) 0.0026 (0.0017) 0.0033∗∗ (0.0014) −0.0002 (0.0008) 0.3233 (0.2898) 0.0241 (0.0334)

Probability bias, γ

1.6723∗ (0.9885) −0.0206 (0.1684) 0.0079 (0.0356) 0.0605 (0.1088) −0.0418 (0.0502) −0.0008 (0.0012) 0.0030∗ (0.0017) 0.0030∗∗ (0.0014) −0.0004 (0.0008) 0.0039 (0.0315) 0.5069∗∗ (0.2486)

1.4485 (1.0052) −0.0182 (0.1681) 0.0060 (0.0356) 0.0693 (0.1089) −0.0447 (0.0501) −0.0008 (0.0012) 0.0030∗ (0.0017) 0.0030∗∗ (0.0014) −0.0003 (0.0008) 0.3340 (0.2857) 0.0161 (0.0331) 0.5121∗∗ (0.2482)

0.1343 0.0549 109 1.6912∗ 0.0917 0.7385

0.1451 0.0581 108 1.6670∗ 0.0907 0.7372

Present bias, β Long-term discount factor, δ R2 Adjusted R2 Number of observations F-statistic p-Value Standard error

0.0603 0.0247 211 1.6936∗ 0.1014 0.7955

0.1114 0.0299 109 1.3670 0.2052 0.7482

1.3788 (0.9619) 0.0630 (0.1223) 0.0292 (0.0274) 0.0163 (0.0862) 0.0128 (0.0398) −0.0002 (0.0009) 0.0006 (0.0013) 0.0026∗∗ (0.0010) −0.0006 (0.0006)

0.3972∗∗ (0.1807) −0.5336 (0.5964) 0.0861 0.0413 204 1.9227∗∗ 0.0438 0.7843

Notes: The table shows coefficient estimates for ordinary least-squares regressions at the individual level with individualism as the main explanatory variable for financial risk taking according to Hypothesis 2. ∗∗∗ indicates the coefficient is different from 0 at the 1% level. ∗∗ indicates the coefficient is different from 0 at the 5% level. ∗ indicates the coefficient is different from 0 at the 10% level. Standard errors are given within parentheses.

Moreover, results concerning our control variables indirectly support our findings insofar, as the probability bias γ positively affects an individual’s risk attitude. Higher values of γ imply higher probability weights and thus make risky investments in comparison to riskless ones more attractive. In fact, this also hints at some kind of a positive influence of ‘overconfidence’.According to Breuer and Perst (2007), upward shifts of the probability weighting curve may be interpreted as increasing subjectively perceived competence levels – a concept which is apparently closely related to (over-) confidence. Additionally, the significance of γ as well as that of the present bias parameter β suggests that individual decision parameters are indeed of relevance, although risk aversion α is not essential at all.

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5. Discussion Until now, we have only examined the effect of differences in cultural values across individuals. We documented that individualism is positively related to attitudes toward risk. Do these differences in individual preferences have an impact on economic outcomes across countries? What is the average implication of a low level of individualism in a country? According to our reasoning above, investors will be more reluctant to invest in risky assets when the level of individualism is low. Hence, we expect that countries with low individualism have low equity holdings. We now turn to country-level data on the use of equities to test this prediction. We obtain data on portfolio shares invested in equity from EIU WorldData. For each country, we calculate the ratio of equity assets to total assets held by the household sector in 2008. Equity assets consist of claims to residual value of incorporated enterprises, after claims of all creditors, and include mutual fund holdings. Hofstede (2001) provides data on the cultural dimensions. To capture wealth and age effects, we include median household income (median nominal disposable income earned by households per annum) and the dependency ratio (the ratio of the population under age 15 and over age 64 to the population ages 15–64) in our regression analysis. We get both variables from EIU WorldData as well. We also control for the average level of trust in a country, as Guiso, Sapienza, and Zingales (2008) show that a general lack of trust can have a negative effect on stock market participation. The data are from the World Values Survey. Besides, we include stock market capitalization to GDP as a control variable, to ensure that our results are not simply driven by the general development of financial markets. This variable comes from Beck and Demirgüç-Kunt (2010). We test our hypothesis by regressing the portfolio shares of equities in each country on the cultural dimensions and control variables. We standardize the independent variables so that the coefficient estimates can be directly compared within and across regressions. The first column in Table 5 reports the results. As predicted, individualism has a positive effect on stock market participation and is statistically significant. Since the independent variables have been standardized for our regressions, the estimates are easy to interpret in economic terms. The dependent variable for investments in equities has a mean of 0.31, and individualism has a coefficient estimate of 0.0963. This estimate implies that, all else equal, a one-standard-deviation increase in individualism would induce a 0.0963 : 0.31 = 31% increase in the measure for equity shares, relative to the mean value. The coefficient of individualism has the highest absolute value, suggesting that cultural variables are as important as economic variables in understanding cross-country differences in portfolio structures. All the variables together can explain 20% of the cross-country variability in the rate of investments in equity. Our cultural analysis also sheds light on the stock market participation rates quoted at the beginning. Sweden, the UK, and the US, where participation rates are notably high, have very high levels of individualism. Concerns arise over the possibility of endogeneity. Does culture really precede economic outcomes or vice versa? Reverse causality would imply that culture would adapt rapidly to economic changes, and thus would be of no original relevance. We opt for an instrumental variable approach that establishes an exogenous source of variation in culture to address this issue. Our instrument for the cultural dimensions of individualism is derived by analyzing the language spoken in a country. A large body of work maintains that culture and language are inseparable and mutually constitute one another. Though a detailed discussion of this literature is beyond the scope of this article, there is substantial evidence that language affects people’s social beliefs and value judgments (Whorf 1956; Sapir 1970). Culture and language may be connected through the

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Dependent variable: equities 0.0907∗∗ (0.0406) 0.0875∗ (0.0484) 0.0963∗ (0.0514) −0.0625∗ (0.0306) 0.0719∗∗ (0.0298)

Uncertainty avoidance Power distance Individualism Masculinity Stock market capitalization Median income −0.0306 (0.0516) Trust −0.0351 (0.0363) Dependency ratio −0.0450 (0.0459) Constant 0.2360∗∗∗ (0.0377) R2 0.3960 Adjusted R2 0.2020 Number of observations 34 F-statistic 2.0450∗ p-Value 0.0820 Standard error 0.1630

0.0992∗∗∗ (0.0375) 0.1060∗∗ (0.0499) 0.1410∗ (0.0801) −0.0668∗∗ (0.0274) 0.0770∗∗∗ (0.0270) −0.0535 (0.0564) −0.0388 (0.0321) −0.0544 (0.0423) 0.2200∗∗∗ (0.0406) 0.3770 0.1780 34 3.0164∗ 0.0731 0.1410

Notes: The table shows coefficient estimates for ordinary least-squares regressions at the country level with individualism as the main explanatory variable for financial risk taking. The second column reports an instrumental variables regression where individualism is instrumented for. ∗∗∗ indicates the coefficient is different from 0 at the 1% level. ∗∗ indicates the coefficient is different from 0 at the 5% level. ∗ indicates the coefficient is different from 0 at the 10% level. Standard errors are given within parentheses.

conception of the person, which is coded in the use of person-indexing pronouns, such as ‘I’ and ‘you’ in English. Major differences arise from the question of ‘whether to use a pronoun’ and ‘which pronoun to use’. The cultural dimension of individualism concerns the relationship between the individual and the collective. Kashima and Kashima (1998) relate this dimension to the linguistic practice of pronoun drop, in particular the omission of the first-person singular pronoun (‘I’ in English). In some languages (like English, for example), it is mandatory to include a subject pronoun in most sentences, while it is not required in other languages (in Spanish, for example) where these pronouns can be dropped. An explicit use of ‘I’ emphasizes the speaker’s person, whereas a language that allows pronouns to be dropped reduces its prominence. Kashima and Kashima (1998) examine major languages and code a language as ‘2’ if it almost always requires a first-person singular pronoun in an independent clause and as ‘1’ otherwise, and label the variable as pronoun drop. Therefore, we expect a positive relationship between pronoun drop and individualism (ρ = 0.83, p = 0.00). In the two-stage least-squares instrumental approach, our first step is to treat individualism as a dependent variable and use pronoun drop as the instrumental explanatory variable. In the second step, we then insert the predicted values of individualism back in our regression with the portfolio shares of equities as the dependent variable, together with other explanatory variables that appear in the regression. The results are shown in the second column of Table 5. The coefficient of individualism using this instrumental variable approach is considerably bigger than the coefficient using the ordinary least-squares approach, suggesting that endogeneity is not a major concern. Economic outcomes are causally linked to cultural values and thus maintain themselves over decades. The statistical significance of individualism remains about the same.

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6. Conclusion In this article, we link individual and cultural preferences to the willingness to invest in risky assets. We collect data using a survey that allows us to test (1) standard portfolio theory that risk aversion is negatively related to financial risk taking, (2) whether individualism, which is linked to overconfidence and overoptimism, affects financial risk taking, and (3) how cultural preferences translate into economic outcomes at the country level. Our empirical evidence contradicts standard portfolio theory, as it does not indicate a significant relationship between risk aversion and risk taking in financial matters. However, our analysis supports the behavioral view that psychological factors rooted in culture can affect portfolio choice. Individualism has a significantly positive effect on financial risk attitudes. The last extension shows that cultural values are also important predictors for investments in equities across countries. Our findings add to the increasing evidence that culture matters for economic outcomes. The economics literature has hesitated to address cultural influences on economic decision-making for a long time and culture has often been treated as a ‘black box’ (Williamson 2000). Recent research has however overcome methodological challenges of operationalizing culture as well as the lack of sufficient theoretical foundation of this issue. Cultural values are linked to well-known economically relevant phenomena as overconfidence, for example, and can thus increase our understanding of economic decision-making (Chui, Titman, and Wei 2010). The evidence in the literature indicates that culture can have an important effect on economic outcomes and can explain economic phenomena across countries, which is consistent with the idea that individuals in different countries are subject to different biases. Culture constitutes a powerful determinant in economic studies and can nurture future economic discourse (Guiso, Sapienza, and Zingales 2006). As most developed countries are facing an increased population aging, households need to accumulate assets on their own in order to finance retirement (Bilias, Georgarakos, and Haliassos 2010). Despite a noticeable premium on investments in equity, worldwide participation in equity markets is still limited. Financial economists need to advance solutions to reduce the incidence of these investment mistakes (Campbell 2006). Any policy interventions aimed at fostering investment can be more effectively designed if there is a proper understanding of the underlying factors (Badunenko, Barasinska, and Schäfer 2009). References Badunenko, O., N. Barasinska, and D. Schäfer. 2009. Risk attitudes and investment decisions across European countries – Are women more conservative investors than men? DIW Discussion Paper, no. 982, German Institute for Economic Research, Berlin. Barnea, A., H. Cronqvist, and S. Siegel. 2010. Nature or nurture: What determines investor behavior? Journal of Financial Economics 98, no. 3: 583–604. Barsky, R.B., M.S. Kimball, F.T. Juster, and M.D. Sharpio. 1997. Preference parameters and behavioral heterogeneity: An experimental approach in the health and retirement survey. The Quarterly Journal of Economics 112, no. 2: 537–79. Beck, T., and A. Demirgüç-Kunt. 2010. Financial institutions and markets across countries and over time: The updated financial development and structure database. World Bank Economic Review 24, no. 1: 77–92. Bilias, Y., D. Georgarakos, and M. Haliassos. 2010. Portfolio inertia and stock market fluctuations. Journal of Money, Credit and Banking 42, no. 4: 715–42. Breuer, W., and A. Perst. 2007. Retail banking and behavioral financial engineering: The case of structured products. Journal of Banking and Finance 31, no. 3: 827–44. Campbell, J.Y. 2006. Household finance. The Journal of Finance 61, no. 4: 1553–604. Carpenter, J., S. Burks, and E. Verhoogen. 2003. Comparing students to workers: The effects of social framing on behavior in distribution games. In Field experiments in economics, ed. J. Carpenter, J. List, and G. Harrison, 261–90. Greenwich: JAI Press.

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Appendix To elicit risk preferences, we employ a simple task for every parameter with three sub questions. The main assignment for risk aversion toward gains has the following form:

For each lottery comparison, please state the amount of Z for which you are indifferent between both lotteries. Lottery A: 50% chance to gain 20 ¤, 50% chance to gain 200 ¤ Lottery B: 50% chance to gain Z ¤, 50% chance to gain nothing Z should be −−−−−−−−− ¤, such that lottery A is as attractive as lottery B. Using Prospect Theory by Kahneman and Tversky and three different lotteries of the form (xn , 0.5; yn , 0.5) and (zn , 0.5; 0, 0.5) with xn , yn > 0, we calculate the risk aversion toward gains: π(0.5) · v(xn ) + π(0.5) · v(vn ) = π(0.5) · v(zn ) + π(0.5) · v(0).

(A1)

v(xn ) + v(yn ) = v(zn ).

(A2)

With v(0) = 0, it follows:

Now the function v(xn ) has to be adjusted. For every value of the parameter α (exogenously given), the sum of the differences between the calculated value and the real value given through the questionnaire is calculated. The value of α, for which this sum is minimal, is the optimal value of the parameter. The higher the value of α, the smaller is the risk aversion toward gains, since the shape of the function is getting more concave with smaller α. For α = 1, the investor is neutral toward risk. Since we assume α + = α − , the risk aversion parameter α is set equal to the risk seeking toward gains parameter and we can use three subsequent questions of the form (xn , 0.5; −yn , 0.5) and (−zn , 0.5; 0, 0.5) with xn = yn to calculate λ,

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the loss aversion. v(xn ) + v(−yn ) = v(−zn ) +

−

⇔ xn α − λ · yn α = −λ · zn α

−

(A3)

α+

⇔λ=

x − . − yn α − zn α

For the calculation of the probability γ bias, we use a well-known formula that has been introduced by Tversky and Kahneman (1981): pn γ

πγ (pn ) =

1

.

(A4)

(pn γ + (1 − pn )γ ) γ

The treatment of probabilities differs between Expected Utility Theory and Prospect Theory. In Expected Utility Theory, the utility of an uncertain outcome is weighted by its probability. In Prospect Theory, the probability is replaced by a decision weight π(p) that is not a probability. We use questions of the form (xn , pn ; 0, 1 − pn ) and ask the respondents for the certainty equivalent zn . With v(0) = 0, it follows: π(pn ) · v(xn ) + π(1 − pn ) · v(0) = v(zn ) ⇔ π(pn ) =

v(zn ) v(xn )

(A5)

Since v(xn ) and v(zn ) are known, π(pn ) can be determined by variation of pn , using the same procedure that has been used for the calculation of risk aversion α. Concerning time preferences, we use the theory of the quasi-hyperbolic discount-model. Following this model that has been confirmed in a large number of experiments, individuals tend to prefer smaller, but earlier rewards instead of larger, but later rewards. The function describing the subjective discount factor over time does not follow the shape of exponential discounting (as implied on capital markets by arbitrage freeness conditions) but a hyperbolic shape. Mathematically, quasi-hyperbolic discounting can be described as: U(xo , x1 , . . . , xT ) = u(xo ) +

T 

βtδ

(A6)

t=1

where U is an individual’s overall utility consisting of discounted utility values u at times t = 0 to t = T that result from rewards xt . β and δ are constants between 0 and 1 utilized for subjective discounting purposes. The parameter β is called the present bias, because this factor describes the time preference of the individual between this period and the next period. A larger β implies a less present bias. The other parameter δ is called long-term discount factor and describes the time preference between any two future periods. For the calculation of these parameters, the following two questions were used:

Please consider the following alternatives: Payment A: A payment of 100 ¤ now Payment B: A payment of F ¤ in one year (ten years) F1year (F10years ) should be −−−−−−−−− ¤, such that payment A is as attractive as payment B. Both parameters can be inferred from the individual’s responses F1year and F10years : δ=



β=

100 . δ · F1year

F1year F10years

1

9

,

(A7) (A8)

Journal of Risk Research, 2014 Vol. 17, No. 3, 367–381, http://dx.doi.org/10.1080/13669877.2013.815648

Risk perception and management in smallholder dairy farming in Tigray, Northern Ethiopia Kinfe Gebreegziabhera,b* and Tewodros Tadessea,c a

Department of Natural Resource Economics and Management, Mekelle University, Mekelle, Ethiopia; bDepartment of Food Business and Development, University College Cork, Ireland; cAgricultural Economics and Rural Policy Group, Wageningen University, Wageningen, The Netherlands (Received 8 February 2012; final version received 22 May 2013) Empirical studies on smallholder dairy farmers’ risk perceptions and management strategies have still received little attention in agricultural research of developing countries. This study focuses on farmers’ risk perception and management strategies of smallholder dairy farms in urban and peri-urban areas of Tigray in northern Ethiopia. Based on data collected from a sample of 304 smallholder dairy farm households, we used descriptive statistics for analyzing farmers’ risk attitude and factor analysis for analyzing and classifying risk sources and management strategies. The majority of dairy farmers considered themselves risk takers towards farm decision that may have a positive impact on technology adoption. Factor analysis identified technological, price/market, production, financial, human, and institutional factor as major sources of risks. In addition, factor analysis indicates that disease reduction, diversification, financial management, and market network are perceived as the most effective risk management strategies. Our findings indicate that perceptions of risk and management strategies are farmer-specific; therefore, policy-makers need to consider tailor-made strategies that would address farmers’ individual motives to manage risks and shocks. Keywords: dairy farm; factor analysis; risk perception; risk management; Ethiopia

1. Introduction Smallholder dairy farms produce over 80% of the milk in Eastern and Southern Africa (ESA) (COMESA and EAC 2004), making them an important component of the dairy sector and its future development. Dairying contributes to food production, generates cash income, produces manure to support crop production, and is a means to accumulate capital assets for emergency cash needs (Bebe et al. 2003). In this regard, the dairy industry has been recognized as one of the most important industries in ESA region in the quest to attain food security and improve well-being (Mdoe and Wiggins 1996). Despite existing opportunities for smallholders to develop dairy production in ESA, current output is well below the potential level of production (Somda, Kamuanga, and Tollens 2005). The main reasons for poor productivity of the *Corresponding author. Email: [email protected] Ó 2013 Taylor & Francis

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smallholder dairy farms are risks associated with inadequate feed and water, lack of skills, problems with marketing, and poor animal health services (Kivaria, Noordhuizen, and Kapaga 2006). Cause of animal mortality and morbidity in smallholder dairy farms in ESA region were identified as tick-borne diseases (TBDs), tick infestation, mastitis, trypanosomiasis, and diarrhea (Phiri, Benschop, and French 2010). Smallholder dairy farm production is mainly constrained by TBDs which is spread in 11 countries in the Eastern, Central and Southern regions of Africa (Kivaria 2006; Okuthe and Buyu 2006). About 80% of the 18 million head of cattle were at risk in Tanzania alone, and direct economic losses of mortality due to TBDs were estimated at US$248 million per year including an estimated mortality of 0.92 million animals (Kivaria 2006). The incidence and prevalence of lameness and foot lesions was also found to be high among smallholder dairy farming system in Kenya (Gitau, McDermott, and Mbiuki 1996). Gran et al. (2002) also revealed risks of smallholder dairy processing in Zimbabwe related to hygiene practice during milking and the microbiological quality of milk at the farm and on delivery. Furthermore, credit access among smallholder dairy farms remains low in many of the African countries (Freeman and Jabbar 1998a). Studies in Uganda, Ethiopia, and Nigeria showed that collateral and minimum investment requirements as well as information problems restrict access to credit for smallholder dairy farmers (Freeman and Ehui 1998b). Staal, Delgado, and Nicholson (1997) also argued that urban and peri-urban areas of smallholder dairy farmers of East Africa have been faced with shrinking arable land which resulted in high transaction costs for dairy production and marketing. Somda, Kamuanga, and Tollens (2005) also noted that past and current field research and development policies in Africa have for long favored crops over livestock development and technical over socioeconomic solutions. In Ethiopia, market-oriented smallholder dairy farming is an emerging business and is becoming an important supplier of milk and milk products to urban centers. In Ethiopia, nearly 83% of the total milk produced is consumed at the household level and only 7% is supplied to the formal and informal markets. The demand for milk and milk products is expected to rise from 17 L per capita in 2010 to approximately 27 L per capita in 2020 (Land O’Lakes 2010). Although a lot of effort was made towards dairy development and various research projects have been undertaken in some parts of the country (Ethiopia), the outcome and impact have not been satisfactory (Yigezu 2003). In Ethiopia, smallholder dairy farm production is severely hampered by various risks related to drought, diseases, limited feeding, poor market, limited credit, limitations of land for sustainable dairy development, problems related to waste disposal, shortage of supply of genetically superior dairy animals, poor extension services, labor problems, limited infrastructure and veterinary services (Yigrem, Beyene, and Gebremedhin 2008). Dairy farmers, like other agri-business is risky, also place a greater weight on potential negative outcomes of risk and they are generally willing to sacrifice potential income to avoid either risk or uncertainty (Marra, Panell, and Ghadim 2003). Farmers’ risk perception and management responses to those risks are very important for understanding their risk behavior (Meuwissen, Huirne, and Hardaker 2001; Flaten et al. 2005; Akcaoz, Kizilay, and Ozcatalbas 2009). So far, several studies have examined farmers’ perceptions on risk and management strategies of dairy farming in developed countries (Wilson, Dahlgran, and Conklin 1993; Martin 1996; O’Connor, Keane, and Barnes 2008; Ogurtsov, Van Asseldonk, and Huirne

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2008). However, the literature in risk perception and management strategies in smallholder dairy farming in developing countries is not well developed. The fact that economic, social, and environmental realities in developing countries are often different from those prevailing in developed countries makes it difficult to generalize results from developed countries. In this regard, we are not aware of studies in Africa that attempt to explore dairy farmers’ risk perception and management strategies in urban and peri-urban areas. This study aims to fill this gap by providing empirical study through identification of major sources of risks and risk attitude, and evaluate the effectiveness of the existing management strategies in smallholder dairy farmers in socioeconomic perspective. This paper is organized as follows: the second section provides a description of the study areas and data. In addition, this section briefly discusses the statistical tools used for analysis. In section three, the results are presented and discussed. The last section concludes. 2. Materials and methods 2.1. Description of the study areas The study was conducted in three urban and peri-urban areas of Tigray, including Mekelle, Adigrat, and Alamata. Tigray National Regional State is one of the regional states within the structure of the Federal Democratic Republic of Ethiopia. It is located in the northern part of the country bordering with Eritrea in the north, Sudan in the west, Afar region in the east, and Amhara region in the southwest. The region had an estimated population of over 4.3 million at the end of 2007, of which about 19.5% lived in urban areas (CSA 2008). More than 58% of the total population lives in absolute poverty (earning less than a dollar a day), which makes the region’s situation more serious compared to the national average (44.4%) (WFP 2009). Mekelle The city of Mekelle located in the northern edge of Ethiopia is situated 773 km from the national capital, Addis Ababa. Geographically, it lies between latitude 13o29′ North and longitude 39o28′ East with an elevation of 2084 m above sea level. The population of the city in census 2007 was 215,546 (CSA 2008). Adigrat The city of Adigrat is located in northern Ethiopia around 893 km far from Addis Ababa. It lies between latitude 14o16′ North and longitude 39o27′ East with an elevation of 2457 m above sea level. The population of the town in census 2007 was 57,572 (CSA 2008). Alamata The city of Alamata is located in northern Ethiopia around 600 km far from Addis Ababa. It lies between latitude 12o25′ North and longitude 39o33′ East with an elevation of 1520 m above sea level. The population of the town in census 2007 was 33,198 (CSA 2008).

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Table 1. Socioeconomic and demographic characteristics of dairy farmers. Household and farm characteristics Sample size Average age of head (year) Gender of household head (male) Head respondent (household head) Average farm experience (year) Average family size (number) Average farm area (m2) Average cattle (number) Average monthly expenditure (Birra)

Mekelle

Adigrat

Alamata

120 (39.5%) 86 (28.3%) 98 (32.2%) 54.4 53.9 49.4 61 (30.8%)

37 (43%)

83 (69.2%) 56 (67.4%)

Average 52.7

SD 0.84 12.97

37(37.7%) 135 (44.4%)

0.50

64(65.3%) 203 (66.8%)

0.49

12.6

11.6

12.9

12.4

10.70

6

5.7

6.7

6.2

2.23

52.5 2.9 1917

67.3 4.7 1882

57.5 4.3 2434

99.77 3.90 2211

54 4.9 3255.6

Source: own survey, 2011. a 1 USD was equivalent to 17.75 Ethiopian Birr (as of February 2011).

2.2. Data We used primary data collected from smallholder dairy farmers in Tigray, northern Ethiopia. The primary data incorporated both cross-sectional household data and key informant discussion. The data were collected from the three urban and peri-urban areas introduced in Section 2.1. These urban and peri-urban areas were purposively selected as they represent urban pockets of intensive smallholder dairy farm activities. From these study areas, a total sample of 304 randomly selected dairy farmers was considered in the survey. Specifically, the total sample included a proportional sample of 120, 86, and 98 dairy farmers from Mekelle, Adigrat, and Alamata, respectively. In order to collect the required data using face-to-face interviews, six highly experienced enumerators, one coordinator, and one researcher were involved in the survey. The enumerators were given sufficient training before the questionnaire was pretested on smallholder dairy farmers and key informants. Useful feedbacks were extracted from the pretest and incorporated in the final structured questionnaire. Finally, the main survey was conducted in February 2011 through face-to-face interview between enumerators and smallholder dairy farmers. In the main survey, there were incomplete responses from a few respondents. Nevertheless, evaluation of the completed questionnaires on a daily basis during the field survey helped us locate the missing data. Since we had mechanisms (household IDs and names) to track back the respondents, the enumerators were able to revisit the respondents and obtain the relevant data. The questionnaire covered a wide range of information including, among others, household and farm characteristics, risk sources, risk attitude, and risk management strategies. Sources of risk, risk attitude, and risk management strategies were of closed-type items represented by five point Likert-type scales. Information from key informant discussion was collected from farmers and development agents. Risk sources, risk attitude, and management strategies items included in the questionnaire are based on previous studies, key informant discussion, and local context. Table 1 provides a general summary statistics of the sample dairy households.

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2.3. Statistical methods Farmers’ perception on sources of risk, risk attitude, and management strategies in the smallholder dairy farming was analyzed using factor analysis. The large numbers of variables were reduced in to smaller. This was done by means of aggregation for risk attitude, and through factor analyses for sources of risk and management strategies. Factor analysis is a popular multivariate technique used to assess the variability of variables of a data-set (in our case, risk sources and management strategies variables) through linear combination of smaller number of latent variables, called factors. The extent of variation between variables in each factor is expressed by Eigenvalues. If there is a strong relationship between variables, the first few factors explain a high proportion of the total variance and the last factors contain very little additional information. In our analysis, factors whose Eigenvalues are greater than one were retained. Total variance accounted for risk sources and management strategies was observed to be 67.40 and 71.42%, respectively (Tables 3 and 4). Orthogonal rotation was used for linear functions of independent (uncorrelated) factors to reconstruct the scores on the original variables of factor analysis. Varimax rotation was also used to maximize the variance of the squared loadings for each factor, and thus polarizes loadings (either high or low) on factors for easy interpretation. To check the internal reliabilities, we calculated Cronbach’s alpha. The Cronbach alpha values were found to be 0.70 and 0.80 for risk sources and management strategies, respectively, which is acceptable (Greiner, Patterson, and Miller 2009). Kaiser–Meyer–Olkin (KMO) measures of sampling adequacy for sources of risk gave a value of 0.70. This KMO value indicates that overall the variables have 70% in common to warrant a factor analysis. Similarly, KMO value for risk management strategies was found to be 0.77. Thus, these KMO values show that overall the items of risk source and management strategies are adequate for factor analysis due to large portion of communality. In addition, KMO measure of sampling adequacy for the individual items of risk sources and risk management strategies was also checked. Items with values less than 0.50 were omitted from the factor analysis (see also Hair et al. 2006). In interpreting the retained factors, we only used variables with factor loading greater than 0.3, which is considered satisfactory (Flaten et al. 2005). The higher load is more relevant in defining the factor’s dimensionality. In other words, it implies the degree of communality (relation) of each item to the rest of items under risk source and management strategies. To this end, at least two variables (items) with significant loadings are considered in each retained factor (see also Ahsan 2011). Table 2. Relative risk aversion by dairy farmers. Relative risk aversionb (%) Risk categorya

1

2

3

4

5

Production Marketing Financial Technological

2.63 4.61 5.92 5.59

4.28 4.61 7.24 5.92

9.21 13.16 21.38 9.21

24.34 35.86 32.57 20.07

59.54 41.77 32.89 59.21

a

In order to elicit risk attitude, dairy households were presented with the following question: ‘Are you willing to take more risk than others with respect to (each risk category)’. b Relative risk: 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = strongly agree.

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3. Results and discussions 3.1. Smallholder dairy farmers’ relative risk attitude Smallholder dairy farmers were requested to rate their willingness to take risks relative to other farmers in their district on a Likert-type scale ranging from one (strongly disagree) to five (strongly agree). The results are presented in Table 2. In this case, respondents from category 1 and 2 considered themselves as risk averse; category 4 and 5 as risk taker; and category 3 as risk neutral. About 83.88, 77.64, 65.46, and 79.28% of the dairy farmers considered themselves relatively (as compared to other dairy farmers) more willing to take production, marketing, financial, and technological risks, respectively. This shows that a large proportion of the dairy farmers considered themselves risk takers. Production risk takers are associated with farmers’ decision to prevent feed shortage, diseases, accident, and death of dairy cows. This may be due to the fact that dairy farmers have firm decision to mitigate cattle mortality and morbidity since it is difficult for such farmers to replace loss of cattle. Marketing/price risk-taking decisions are linked to the purchase and selling decisions within the environment of change in demand and supply, market access, and price variability in input and output of the dairy farms. Financial risk-taking farmers use credit to avoid cash shortage for further farm investment. Similarly, a good proportion of the dairy farmers also considered themselves as technological risk taker in which they were encouraged to adopt new technologies like better breeding cows, use of artificial insemination (AI), and supplementary feeding. Our finding revealed that the majority of dairy farmers regarded themselves as risk takers in relation to their decision of production, marketing, financial and technological risks. Credit access may be limited due to collateral problem in the urban areas. However, this access may drive farmers to adopt new technology and influence other farm-related decisions. In addition, farmers’ awareness on the benefits of technology adoption and other farm decisions is improving from time to time due to the expansion of schools and extension services.

3.2. Smallholder dairy farmers’ perception of various risk sources In total, 16 risk sources were considered in the smallholder dairy farming based on theory, empirical study, and personal observation. Farmers were asked to score each source of risk on a Likert scale to express how significant they perceived each source of risk in terms of smallholder dairy farm production performance. Table 3 shows the communalities, average scores, standard deviation, and factor analysis of smallholder dairy farmers’ perception of each source of risk. The statistics of risk sources are presented in decreasing order of relevance to the smallholder dairy farmers. Low milk yield due to feed shortage was identified as the top-rated source of risk. This implies that smallholder dairy farmers are aware of their inability to provide sufficient quantity and quality feed to their cattle, which is a prerequisite for attaining milk yield potential. Feeds and feeding interventions need to be considered in the context of the socioeconomic and environmental conditions of the smallholder dairy farmers in urban and peri-urban areas of Ethiopia. It is also probably a good signal for government to strengthen the ongoing environmental rehabilitation efforts in the country so as to widen the feed resource base and increase green fodder availability for livestock in the long run. The standard

0.64 0.60 0.57 0.70 0.76 0.77 0.72 0.74 0.58 0.66 0.79 0.74 0.74 0.51 0.63 0.60 – –

Communality 3.21 3.09 3.06 2.39 2.23 2.04 2.03 1.93 1.79 1.76 1.64 1.63 1.61 1.58 1.45 1.19 – –

Mean 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 – –

Rank 1.48 1.51 1.53 1.55 1.51 1.40 1.39 1.35 1.33 1.18 1.12 1.16 1.07 1.07 0.87 .60 – –

SD 0.04 0.32 0.21 0.03 0.17 0.83 0.17 0.04 0.50 0.01 0.89 0.12 0.04 0.13 0.34 0.04 12.97 12.97

1 0.41 0.14 0.23 0.88 0.82 0.28 0.29 0.01 0.07 0.23 0.03 0.02 0.11 0.07 0.15 0.01 12.17 25.13

2 0.03 0.06 0.49 0.10 0.21 0.05 0.16 0.01 0.25 0.71 0.00 0.01 0.84 0.48 0.03 0.21 11.76 36.89

3

0.07 0.62 0.05 0.01 0.05 0.04 0.14 0.85 0.02 0.05 0.02 0.85 0.02 0.04 0.02 0.03 11.64 48.53

4

The first 6 major factorsb

Items which are relevant in each factor and having loading greater than absolute value of 0.30 are in bold. a Sources of risk (1 = Not at all relevant 2 = Of minor relevance 3 = Moderately relevant 4 = Of major relevance 5 = Extremely relevant). b Factors 1 to 6 are technological, price/market, production, financial, human, and institutional risks.

Low milk yield due to feed shortage Low farm income Lack of government support Milk price variability Milk marketing problems Ineffective AI Low milk yield due to poor breed Changes in interest rate Availability of hired responsible labor Low milk yield due to animal disease Inadequate/inexistence of AI Credit availability Animal disease (epidemic) Disease risk in milk (mastitis/teat problem) Technology risk (vaccination, breeding) Family members’ health situation Percent of total variance explained Cumulative percent of the variance explained

Sources of riska

Sample

Table 3. Average score and varimax rotated factor loading for sources of risk.

0.58 0.28 0.19 0.06 0.02 0.02 0.04 0.02 0.48 0.27 0.03 0.01 0.07 0.49 0.21 0.74 10.11 58.64

5

0.32 0.12 0.43 0.00 0.11 0.01 0.75 0.11 0.17 0.15 0.02 0.05 0.05 0.07 0.65 0.06 8.76 67.40

6

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deviation of risk source related to low milk yield due to feed shortage is greater than 1, probably suggesting the lack of consensus among all smallholder dairy farmers. Further, low farm income and lack of government support were identified as the second and third major relevant sources of risk in the urban and peri-urban areas of smallholder dairy farmers. This emphasizes the concern of smallholder dairy farmers on the limited farm income and poor government support, especially the acute lack of land for dairy farm expansion. A factor analysis on risk sources has been conducted using principal component factor followed by varimax rotation. Six factors with Eigenvalues greater than 1 were identified (Table 3). Factors 1–6 were identified as technological, price/market, production, financial, human, and institutional risks. Among the technological risk factors were high loadings of ineffective AI and inadequate/inexistent AI. The ineffectiveness of AI in the study areas could be related to delayed detection of estrus in cows/heifers which is sometimes beyond the knowledge of farmers. Failure to detect estrus instantly is probably the main cause of suboptimal fertility in dairy cows. In addition, ineffective AI may be related to poor quality of semen, availability of processing and storage of semen, inadequate manpower and commitments, and inefficiencies of AI technicians. Gebremedhin (2008) also noted that the most important constraints associated with AI in Ethiopia include poor structural linkage between AI centers and service-giving units. In this case, the focus group discussion indicated the absence of collaboration and regular communication between the center and stakeholders. Moreover, lack of breeding policy and herd-recording system, inadequate resource in terms of inputs and facilities, and absence of incentives to AI technicians are the major constraints while providing AI service. Price/market risk had high loadings on milk price variability and marketing problems in factor 2. Milk price variability and marketing problem is encountered by the smallholder dairy farm households that could be correlated with availability of feed, disease prevalence like mastitis, low milk demand during fasting time, lack of market access, and low milk price. In line with this, a recent study by Akcaoz, Kizilay, and Ozcatalbas (2009) about risk management strategies in dairy farming in Turkey indicated that milk price variability was the most important source of risk in small, medium, and large farm producers. An inefficient milk collection and distribution system to distribute milk from the producer to the consumer is a critical problem in dairy development. The existence of relatively high transaction costs coupled with the perishable nature of milk plays a central role in limiting marketing of smallholder dairy farmers (Tassew and Seifu 2009). Under such conditions, smallholder dairy milk cooperatives that are operating in some urban areas of Ethiopia need to expand and strengthen. Smallholder dairy farmers would be advantageous to supply their milk to the milk cooperatives that may encourage dairy farmers to increase milk supply and sufficiently reduce transaction costs. A production risk on factor 3 is associated with low milk yield due to nonepidemic and epidemic dairy cattle diseases including mastitis/teat problem. The literature indicates similar results of the high loading of diseases in production risks (Meuwissen, Huirne, and Hardaker 2001; Flaten et al. 2005). The high loadings of cattle diseases on production risk are likely to reflect smallholder dairy farmers’ concern on cattle mortality and morbidity. The key informant discussion also confirmed the prevalence of diseases such as blackleg, foot and mouth disease (FMD), anthrax, bovine tuberculosis, lump skin disease, tick-borne disease, and lice

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infestation in the study areas, which hinders the realization of increased milk production. Summary values in the literature for losses of milk production due to mastitis were estimated at 375 kg for a clinical case (5% at the lactation level) (Seegers, Fourichon, and Beaudeau 2003). A study conducted at Andassa Government Dairy Farm (Ethiopia) indicated that the incidence of FMD in dairy cattle was found to be 14.5% and the average milk production for cows that contracted FMD was low. The average milk yield ahead of 10 days of FMD infection was found to be 1.183 kg while 10 days after infection was 0.603 kg, the difference of which was found to be statistically significant (Mazengia et al. 2010). In relation to this, the discussion of key informants indicated that problems related to access to veterinary service, medication supply, and cost of medication aggravated milk productivity in particular and dairy farm production in general. Financial risk of smallholder dairy farmers in factor 4 is affected by low farm income, changes in interest rate, and credit availability. Smallholder dairy farmers’ financial risk sources were mainly associated with increasing interest rate and limited credit availability as a result of government regulation of the financial institutions. Financial risk is one of the fundamental problems hampering agricultural production and productivity of the farmers. In the rural areas of Ethiopia, the group lending methodology solves the problem of collateral requirement by lending institutions even if there are practical limitations. However, in the urban areas, collateral is still a precondition for getting credit from formal financial institutions. Policy aimed at accelerating smallholder dairy farm development could be improved if the financial problem is taken into consideration to relax credit access from the formal financial sources. Namely, livestock insurance as a pilot has already started in some urban and peri-urban areas of Ethiopia by private companies such as Nyala and Oromia Insurance S.C. It is, therefore, possible to devise a policy environment whereby smallholder dairy farmers may have access to formal credit without forming groups, by means of using cattle insurance certificates as collateral for credit. Smallholder dairy farmers are also affected by risks associated with human risk. Human risk of factor 5 revealed high loadings of variables such as availability of hired responsible labor and family members’ health situation. The results related to human risk in this case is consistent with other studies (Meuwissen, Huirne, and Hardaker 2001; Flaten et al. 2005). Smallholder dairy farmers are likely to be concerned with their farm regarding labor that engage in activities such as feeding cattle, cleaning, milking, herding, selling/processing milk, and managing all other activities. Hence, human risks associated with inadequate hired/family labor, divorce, illness, death, disability, and conflict on member households are the real constraints for the development of the smallholder dairy farms. This result could signify the importance of pro-poor health care policy by expanding health services, strengthening the community health insurance, and devising formal health insurance. For the majority of urban people, the health care costs are often very high as compared to annual personal income. Thus, governmental support for the traditional community health insurance (Iddir) and introducing formal health insurance scheme at micro level might minimize the severity of human risks associated with sickness or accidental death of the household members. Lack of government support and low milk due to poor breed loaded strongly on factor 6 of the institutional risks. Key informants indicated that the government provides little attention towards urban agriculture. In particular, government support in terms of dairy farm land, agricultural extension, and dissemination of modern

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breeding is very limited in the urban areas. As the dairy industry in Ethiopia is based largely on indigenous cattle breeds (MoARD 2007), improved breeding need to focus mainly at cross-breeding activities through research centers and government stock multiplication centers. Some items which carried factor loading greater than 0.3 in two or more factors were included in one of the relevant factors (categories). We argue that the variables that load more than one factor are not relevant in more than one factor; however, it is relevant (pure measure) in one of the factors. In our analysis, we do have two reasons not to drop variables that load more than one factor. First, the result of item-test correlation and item-rest correlation for each of the variable (item) shows a good correlation coefficient. The item-test correlation shows how highly correlated each item is with the overall scale while item-rest correlation shows how the item is correlated with a scale computed from only the rest items. Along with this, including each of these variables increases the Cronbach’s alpha. The second reason is justifiable for theoretical reasons. For instance, availability of hired responsible labor is theoretically accepted to be categorized in factor 5 (human risk) instead of factor 1(technological risk). Vaccination and breeding failure is theoretically supported to be classified under factor 1(technology risk) rather than factor 6 (institutional risk). Similarly, lack of government support is also part of factor 6 (institutional risk) instead of factor 3 (production risk) and the relevance for the rest of variables loaded in two or more factors are placed in accordance to the statistical and theoretical support. 3.3. Smallholder dairy farmers’ perception of risk management strategies For the risk management strategies, we also considered 12 items in the smallholder dairy farming based on theory and empirical study. Based on the ranking, applying strict hygiene, main operator’s working off-farm and use of veterinary service were perceived as the three most effective risk management strategies of the smallholder dairy farmers (see Table 4). In factor analysis, four factors with Eigenvalues greater than 1 were identified. Factor 1 included reducing cattle diseases by applying strict hygiene, use of veterinary service, getting animal health information through phone, social network, and extension services. Disease control and hygiene remained the most effective risk management strategies in smallholder dairy farmers to reduce cattle diseases. This is also supported by the top ranking of strict cattle hygiene as an effective risk management strategy. This implies the need for governmental support in providing adequate veterinary services, competent veterinary staff, and affordable treatment and drug costs to overcome animal health problems. Factor 2 represented diversification by working off-farm, off-farm investment, and crop–animal integration. Income diversification was perceived as an effective strategy in the smallholder dairy farmers to manage risk. Thus, development policy should support and motivate self-employment and wage employment and strengthen micro and small enterprises in urban and peri-urban areas of Ethiopia. Income diversification as risk management strategies will continue to increase, especially in light of future risk and uncertainties about climate change and skyrocketing food prices in the agricultural sector. Financial management in the form of solvency (debt management) and keeping debt low was represented along factor 3. In Turkey, Akcaoz, Kizilay, and

0.72 0.74 0.66 0.78 0.70 0.67 0.61 0.63 0.50 0.93 0.92 0.82 – –

Communality 3.89 3.47 3.42 3.41 3.36 3.21 2.94 2.63 2.57 2.42 2.35 1.42 – –

Mean 1 2 3 4 5 6 7 8 9 10 11 12 – –

Rank 1.29 1.71 1.18 1.65 1.38 1.23 1.31 1.89 1.51 1.63 1.68 1.03 – –

SD

1 0.82 0.23 0.78 0.27 0.79 0.75 0.59 0.07 0.13 0.09 0.17 0.05 25.20 25.20

Items which are relevant in each factor and having loading greater than absolute value of 0.30 are in bold. a Risk management strategies (1 = least effective 2 = less effective 3 = moderately effective 4 = more effective 5 = most effective). b Factors 1–4 are reducing disease, diversification, financial management, and market network.

Applying strict hygiene Main operator working off-farm Use of veterinary service Off-farm investment Reducing livestock disease Information (phone, social network and extension service) Gathering market information Crop–animal integration Farm members working off-farm Solvency (debt management) Keeping debt low Cooperative marketing Percent of total variance explained Cumulative percent of the variance explained

Risk management strategiesa

Sample

Table 4. Average score and varimax rotated factor loading for risk management strategies.

0.16 0.82 0.14 0.83 0.10 0.23 0.19 0.70 0.65 0.01 0.10 0.04 20.30 45.50

2

0.14 0.01 0.05 0.10 0.15 0.17 0.20 0.15 0.00 0.96 0.94 0.03 16.18 61.69

3

The first 4 major factorsb 0.01 0.04 0.14 0.02 0.16 0.13 0.42 0.09 0.23 0.01 0.02 0.93 9.73 71.69

4

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Ozcatalbas (2009) found that keeping debt low was the most significant risk management strategy for all dairy farmers. Similarly, smallholder dairy farmers in our study perceived proper debt management as an effective risk management strategy. This may imply the need to settle initial loan of farmers that will help them get further credit from the lending institutions (such as microfinance institutions). However, smallholder dairy farmers’ access to formal financial service is very limited, which is a major constraint to urban dairy farm development. In addition, households’ repayment rate of microfinance credit in urban areas of Ethiopia is low (Addisu 2006). Hence, devising sound financial policy which improves dairy farmers’ credit access to cope with emerging risk would be important. Financial institutions’ initiative to foster financial literacy and opportunity on credit access and debt management can contribute further to more effective risk management in smallholder dairy farmers that is equally important for the financial institutions. High loadings on gathering market information and cooperative marketing lied along factor 4 (market network). The absence of well-organized network of milk and dairy products has adversely affected smallholder dairy farmers’ benefit and created disincentive to seriously invest in the industry. Likewise, the situation has deprived the consumers’ accessibility to milk and milk products. This implies the need for further attention towards the expansion of milk cooperatives and milk marketing facilities so as to improve the dairy industry in the long run. In addition, government role in expanding extension service and infrastructure help to strengthen the risk management strategy in smallholder dairy farmers. Empirical study in different countries also indicated similar findings on how farmers manage their livestock risks (Hall et al. 2003; Akcaoz and Ozkan 2005; Flaten et al. 2005). In this regard, there are very few items that had factor loading larger than 0.3 in more than one factor that has been categorized in relevant factors. Namely, an item such as gathering market information had factor loading more than 1; however, this variable is categorized under factors 4. The reason for categorizing such variable under the relevant factor emanates from statistical and theoretical reasons discussed earlier in this section. 4. Conclusions This study identifies perception on sources of risk, risk attitude, and risk management strategies in smallholder dairy farming in Tigray, northern Ethiopia. The major sources of risk in smallholder dairy farmers were identified to be of technological, price/market, production, financial, human, and institutional nature. Identifying such sources of risks can contribute to a better understanding of the nature and dynamics of risk and uncertainty in smallholder dairy farming systems. Besides, it can contribute to the designation of dairy farm risk-based interventions to put in place sound coping and management strategies. From our analyses, the majority of the respondents were found to be risk takers towards their decisions on risks related to production, marketing, finance, and technology. This behavior implies an interesting result that improving the poorly developed credit and market access in Ethiopia may motivate farmers further to undertake optimal farm input decision to achieve higher dairy output. This would likely have the effect of increasing the efficiency associated with the dairy enterprise, again rendering the overall risk management strategy more effective.

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The major risk management strategies that dairy farmers practice include reducing cattle disease, income diversification, financial management, and expanding market networks. Scaling up the most effective risk management strategies can, therefore, improve livelihood security in smallholder dairy farm households. In general, identifying dairy farmers’ specific risk sources and risk attitude and suggesting effective dairy risk management strategies at household level could contribute to poverty alleviation in the country; since poverty reduction depends not only on growth but also on capacity to absorb and manage shocks. To conclude, the findings may be valuable for detailed analysis of risk perceptions and management strategies in smallholder dairy farm systems so as to design policies and strategies to support smallholder dairy development programs in urban and peri-urban areas of Ethiopia. Acknowledgments Financial support from the NORAD III (2010) project of Mekelle University is gratefully acknowledged. However, the views expressed in the paper are those of the authors. We would particularly like to thank Dr Yayneshet Tesfaye for his valuable comments and suggestions on the earlier draft version. The authors also wish to acknowledge the great accuracy of the enumerators (Aregawi Abreha, Samson Hailu, Tsehaynesh Weldegiorgis, Tadesse Gidey, Tekeste Negash, and Bereha Alemu) and many urban respondents who were actively involved in the research process. We would like to thank Dr Seamus O’Reilly, Dr Edward Lahiff and Dr Bodo Steiner for their useful ideas and contributions, and Dr Girmay Gebresamuael for his encouragement and unreserved support for this paper. We also received helpful comments from two anonymous referees. Any remaining errors are our own.

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by H. A. Freeman, M. A. Jabbar, and S. K. Ehui, 16–35. Addis Ababa: International Livestock Research Institute. Socioeconomics and Policy Research Working Paper No. 22. Freeman, H. A., S. K. Ehui, and E. N. Betubiza. 1998b. “Supply of Institutional Credit for Smallholder Livestock Producers in Uganda, Ethiopia, and Nigeria.” In Role of Credit in the Uptake and Productivity of Improved Dairy Technologies in Sub-Sahara Africa, edited by H. A. Freeman, M. A. Jabbar, and S. K. Ehui, 6–15. Socioeconomics and Policy Research Working Paper No. 22. Ethiopia: International Livestock Research Institute. Gebremedhin, D. 2008. “Assessment of Problems/Constraints Associated with Artificial Insemination Service in Ethiopia.” MSc thesis, Faculty of Veterinary Medicine, Addis Ababa University. Gitau, T., J. J. McDermott, and S. M. Mbiuki. 1996. “Prevalence, Incidence, and Risk Factors for Lameness in Dairy Cattle in Small-Scale Farms in Kikuyu Division, Kenya.” Preventive Veterinary Medicine 28 (2): 101–115. Gran, H. M., A. N. Mutukumira, A. Wetlesen, and J. A. Narvhus. 2002. “Smallholder Dairy Processing in Zimbabwe: Hygienic Practices during Milking and the Microbiological Quality of the Milk at the Farm and on the Delivery.” Food Control 13: 41–47. Greiner, R., L. Patterson, and O. Miller. 2009. “Motivations, Risk Perceptions and Adoption of Conservation Practices by Farmers.” Agricultural Systems 99 (2–3): 86–104. Hair, J. F., W. C. Black, B. J. Babin, R. E. Anderson, and R. L. Tatham. 2006. Multivariate Data Analysis. 6th ed. Upper Saddle River, NJ: Pearson Education. Hall, D. C., T. O. Knight, K. H. Coble, A. E. Baquet, and G. F. Patrick. 2003. “Analysis of Beef producers’ Risk Management Perceptions and Desire for Further Risk Management Education.” Review of Agricultural Economics 25: 430–448. Kivaria, F. M. 2006. “Estimated Direct Economic Costs Associated with Tick-Borne Diseases on Cattle in Tanzania.” Tropical Animal Health and Production 38 (4): 291–299. Kivaria, F. M., J. P. T. M. Noordhuizen, and A. M. Kapaga. 2006. “Prospects and Constraints of Smallholder Dairy Husbandry in Dar Es Salaam Region, Tanzania.” Outlook on Agriculture 35 (3): 209–215. Land O’Lakes. 2010. The Next Stage in Dairy Development for Ethiopia. Dairy Value Chains, End Markets and Food Security: Cooperative Agreement 663-A-00-05-0043100. Addis Ababa: Land O’Lakes. Marra, M., D. J. Panell, and A. A. Ghadim. 2003. “The Economics of Risk, Uncertainty and Learning in the Adoption of New Agricultural Technologies: Where Are We on the Learning Curve?” Agricultural Systems 75 (2–3): 215–234. Martin, S. 1996. “Risk Management Strategies in New Zealand Agriculture and Horticulture.” Review of Marketing and Agricultural Economics 64: 31–44. Mazengia, H., M. Taye, H. Negussie, S. Alemu, and A. Tassew. 2010. “Incidence of Foot and Mouth Disease and Its Effect on Milk Yield in Dairy Cattle at Andassa Dairy Farm, Northwest Ethiopia.” Agriculture and Biotechnology Journal of North America 1 (5): 969–973. Mdoe, N., and S. Wiggins. 1996. “Dairy Products Demand and Marketing in Kilimanjaro Region, Tanzania.” Food Policy 21 (3): 319–336. Meuwissen, M. P. M., R. B. M. Huirne, and J. B. Hardaker. 2001. “Risk and Risk Management: An Empirical Analysis of Dutch Livestock Farmers.” Livestock Production Science 69 (1): 43–53. MoARD (Ministry of Agriculture and Rural Development). 2007. Livestock Development Master Plan Study-Phase I Report. Ogurtsov, V. A., M. A. P. M. Van Asseldonk, and R. B. M. Huirne. 2008. “Purchase of Catastrophic Insurance by Dutch Dairy and Arable Farmers.” Review of Agricultural Economics 31 (1): 143–162. Okuthe, O. S., and G. E. Buyu. 2006. “Prevalence and Incidence of Tick-Borne Diseases in Smallholder Farming Systems in the Western-Kenya Highlands.” Veterinary Parasitology 141 (3–4): 307–312.

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O’Connor, D., M. Keane, and E. Barnes. 2008. “Managing Price Risk in a Changing Policy Environment: The Case of the EU Dairy Industry.” In Income Stabilization in a Changing Agricultural World: Policy and Tools, edited by E. Berg, R. B. M. Huirne, E. Majewski, and M. P. M. Meuwissen, 74–83. Warsaw: Wies Jutra. Phiri, B. J., J. Benschop, and N. P. French. 2010. “Systematic Review of Causes and Factors Associated with Morbidity and Mortality on Smallholder Dairy Farms in Eastern and Southern Africa.” Preventive Veterinary Medicine 94 (1–2): 1–8. Seegers, H., C. Fourichon, and F. Beaudeau. 2003. “Production Effects Related to Mastitis and Mastitis Economics in Dairy Cattle Herds.” Veterinary Research 34 (5): 475–491. Somda, J., M. Kamuanga, and E. Tollens. 2005. “Characteristics and Economic Viability of Milk Production in the Smallholder Farming Systems in the Gambia.” Agricultural Systems 85 (1): 42–58. Staal, S., C. Delgado, and C. Nicholson. 1997. “Smallholder Dairying under Transactions Costs in East Africa.” World Development 25 (5): 779–794. Tassew, A., and E. Seifu. 2009. “Smallholder Dairy Production System and Emergence of Dairy Cooperatives in Bahir Dar Zuria and Mecha Woredas, Northwestern Ethiopia.” World Journal of Dairy and Food Sciences 4 (2): 185–192. Wilson, P. N., R. G. Dahlgran, and N. C. Conklin. 1993. “Perceptions as Reality on LargeScale Dairy Farms.” Review of Marketing and Agricultural Economics 15: 89–101. WFP. 2009. Food Security and Vulnerability in Selected Towns of Tigray Region, Ethiopia. Addis Ababa: WFP-Ethiopia Vulnerability Assessment and Mapping (VAM). Yigezu, Z. 2003. “Imperative and Challenges of Dairy Production, Processing and Marketing in Ethiopia.” In Challenges and Opportunities of Livestock Marketing in Ethiopia, edited by Jobre Y and G. Gebru. Proceedings of the 10th Annual Conference of the Ethiopian Society of Animal Production (ESAP) held in Addis Ababa, Ethiopia, 22–24 August 2002. Addis Ababa, Ethiopia: ESAP, 61–67. Yigrem, S., F. Beyene, A. Tegegne, and B. Gebremedhin. 2008. Dairy Production, Processing and Marketing Systems of Shashemene-Dilla Area, South Ethiopia. IPMS (Improving Productivity and Market Success) of Ethiopian Farmers Project Working Paper 9. Nairobi, Kenya: ILRI (International Livestock Research Institute), 62.

Accounting and Business Research Vol. 42, No. 3, August 2012, 295 –324

Risk reporting quality: implications of academic research for financial reporting policy STEPHEN G. RYAN∗ Stern School of Business, New York University, NY, USA In this paper, I survey empirical research on the relevance of firms’ financial report information for the evaluation of their risk. I recommend that financial reporting policymakers require or encourage firms to enhance their risk reporting quality in four ways. First, firms should report comprehensive income statements that: (1) use fair value or a similarly informationrich accounting measurement attribute and (2) separate the components of comprehensive income that are primarily driven by variation in cash flows from those that are primarily driven by variation in costs of capital. Such comprehensive income statements would provide users of financial reports with the flexibility to calculate alternative summary accounting numbers and to perform different types of risk assessment analyses. Second, firms should conduct and disclose the results of back-tests of prior significant accrual estimates, indicating any identified trends in and drivers of revisions to those estimates, and describing the effects of those revisions on current or future summary accounting numbers. Third, firms should aggregate and present risk disclosures in tabular or other well-structured formats that promote the usability of the information. Identifying existing best disclosure practices and encouraging new best practices are the most natural way to do this. Fourth, for model-dependent risk disclosures, firms should disclose the primary historical and forwardlooking attributes of the models and their implementation in practice, sensitivity of the model outputs, and benchmarking of the models to standard portfolios of exposures. Keywords: disclosure; financial reporting; risk reporting.

1.

Introduction and background on risk and risk disclosures

I appreciate the opportunity provided to me by the organisers of the The Institute of Chartered Accountants in England and Wales (ICAEW) Information for Better Markets Conference 2011 to write this paper on the implications of academic research on risk assessment for how financial reporting policymakers can improve risk reporting quality. Almost 15 years ago, the organisers of the American Accounting Association/Financial Accounting Standards Board (AAA/FASB) Financial Reporting Issues Conference provided me with a similar opportunity (Ryan 1997). ∗

Email: [email protected]

ISSN 0001-4788 print/ISSN 2159-4260 online # 2012 Taylor & Francis http://dx.doi.org/10.1080/00014788.2012.681855 http://www.tandfonline.com

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I summarise in Section 2 the primary conclusions of that paper, most of which remain relevant today. However, theoretical and empirical researches on the determinants of market, credit, liquidity, and information risks have become both broader and deeper over the intervening period, rendering the paper decidedly limited in the scope of its research coverage and policy implications. Three primary developments have enabled and motivated this research. First, financial reports now contain more risk-relevant information, due to the expansion of fair value accounting and disclosures that either supplement for fair value accounting when it is required or substitute for that accounting when it is not required.1 Second, market sources now provide more data about the pricing of risk, due to the rapid expansion of markets to trade risk, particularly credit risk. Third, the still looming financial crisis has jumpstarted and made prominent a pre-existing stream of research on liquidity and information risks. In addition, my own thinking about risk reporting quality has evolved over this period due to my teaching, research, and other activities, particularly those related to financial reporting for financial instruments by financial institutions. In completing the task the ICAEW has assigned to me, I face the difficulty of conveying the nature and implications for risk reporting quality of a large and varied body of research within a paper of manageable length and complexity. In an attempt to do this, I begin with definitions of risk, risk reporting quality, and the risk-relevance of financial report information, distinctions pertaining to risk disclosures and risk assessment, and some drawing of boundaries for the scope of this paper, which is more selective and idiosyncratic in its research coverage than my prior paper. As I lay out these preliminaries, I provide general background about risk disclosures, a sense for my overall views about risk reporting quality, and my recommendations for how financial reporting policymakers can improve it. I define risk as random variation in firms’ future economic performance, given currently available information. I emphasise ‘variation’ in this definition, because I view risk as a two-sided phenomenon. In general, the research I survey also does that. I acknowledge that individuals, policymakers, and academics, particularly those trained in psychology, such as Koonce et al. (2005), are often primarily concerned with downside risk. In my opinion, separate risk disclosures should be required for upside and downside risk when outcome distributions are sufficiently asymmetric. However, downside risk cannot exist without upside risk unless the benchmark outcome against which risk is assessed is the maximum possible outcome. My definition of risk encompasses Knight’s (1921) distinct notions of risk and uncertainty. Knightian risk pertains to ‘known unknowns’ for which decision-makers know all possible outcomes and their probabilities of occurrence, which allows for precise quantification of random variation. Knightian uncertainty pertains to ‘unknown unknowns’ for which decision-makers have incomplete knowledge of either the set of possible outcomes or their probabilities of occurrence, which generally does not allow for such precise quantification. Most of the empirical literature that I discuss does not distinguish these notions. Obviously, it is far more feasible for firms and researchers to construct models of Knightian risk than of Knightian uncertainty. For this reason, firms’ model-based risk disclosures, such as value at risk, correspond more closely to Knightian risk. I emphasise ‘firms’ in this definition because it is essential, in assessing their risk, to distinguish firms as portfolios of exposures from their individual exposures. A firm’s exposures can be risky, but the firm need not be so to the extent that economic net gains on those exposures offset (i.e. covary negatively).2 This is most likely to be the case for financial institutions that engage in asset – liability management and other forms of economic hedging. I define enhanced risk reporting quality as the provision of financial report information that better identifies the economic drivers (e.g. exposures with market, credit, liquidity, or information risks) and/or conveys the statistical properties (e.g. variances and relevant covariances) of the variation in firms’ future economic performance. Identification of these economic drivers is

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critical for users of financial reports to understand why this variation arises. Conveyance of these statistical properties is important for users to estimate the level of this variation. Enhanced risk reporting quality allows users to better predict the variability of firms’ future performance, which is what most of the empirical research I summarise tries to do. It also allows users to evaluate the consequences of future events for firms’ performance as the events occur, which is what financial analysis of individual firms often involves. The empirical research that I summarise usually documents risk-relevance, not risk reporting quality per se. Researchers say a disclosure is risk-relevant if it has explanatory power for measures of firms’ systematic/priced risk (e.g. beta, cost of capital, and valuation multiples), total risk (e.g. share return variance), or downside risk (e.g. probability of default and loss given default). This research typically assesses risk-relevance using linear regression models with many explanatory variables, so that a variable is risk-relevant only if it has incremental explanatory power beyond the other included variables.3 Consistent with the inferences made in this research, I assume that higher risk-relevance implies enhanced risk reporting quality. While necessarily informing about the future, risk-relevant financial report information could be historical to the extent that the past helps predict the future. For example, banks could provide information about how their underwriting criteria, loan and borrower attributes, loan status, and macroeconomic conditions have predicted the level and dispersion of their loan losses in the past. To the extent that the past does not predict the future, however, then these disclosures need to be more explicitly forward-looking, such as stress tests of loan losses to potential future downturns of real estate prices or other events. This distinction between historical and forward-looking risk disclosures is one of degree rather than nature. Moreover, the two types of disclosures generally are complementary, and specific disclosures may have both historical and forward-looking aspects. For example, firms could supplement historically derived accrual estimates of future losses with disclosures of the results of back-tests of those estimates, e.g. analyses of how predictive firms’ accrual estimates in periods prior to the current balance sheet date turned out to be given subsequently available information, in order to provide a sense for how well their accrual estimates likely will predict beyond the current balance sheet date. Similarly, forward-looking disclosures can be informed by historical experience. For example, stress tests usually involve potential future events that at least are somewhat analogous to events that have occurred in the past. I distinguish risk assessment based on summary accounting numbers from risk assessment based on other financial report disclosures. Summary accounting numbers may be recognised bottom-line financial statement amounts or analogous amounts calculated from required disclosures, such as disclosures of the fair values of financial instruments under FAS 107. Risk assessment based on summary accounting numbers may use these amounts either in isolation (e.g. earnings variance or book return on equity) or in combination with market numbers (e.g. market-to-book and price-to-earnings ratios). Other financial report disclosures include specific accrual estimates and market risk disclosures, among many others. I make this distinction because it is a convenient way to discuss the research literature. It also corresponds imperfectly to the distinction between recognised and disclosed information. Schipper (2007) discusses the differential reliability of and salience to users of financial reports of these two types of financial report information.4 Summary accounting numbers and other financial report disclosures have distinct attributes that make them amenable to different sorts of risk assessment analyses. Although more limited in the range of information that they can convey, summary accounting numbers typically enable simpler and less context-dependent analyses for at least two reasons. First, summary accounting flow measures such as earnings and operating cash flows inform about firms’ historical performance without much influence from changes in their costs of capital,5 because of the predominant use of the historical cost basis to measure earnings and the cash basis to measure

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operating cash flow. Second, summary accounting numbers such as earnings and book value of equity tend to be far more comparable across firms, time, and economic contexts than are other disclosures, because of the standardising effects of accounting rules and the normalising effects of competition on profitability. For these reasons, summary accounting numbers play fundamental roles in risk assessment, an often unappreciated point that I emphasise in Ryan (1997) and again in this paper, as do other authors elsewhere.6 I briefly describe two recent empirical literatures that individually demonstrate this point, but that collectively have conflicting implications about whether these accounting numbers should incorporate unexpected changes in costs of capital. First, finance researchers increasingly employ summary accounting numbers rather than share returns to assess firms’ systematic/priced risk. For example, Cohen et al. (2009) estimate a firm’s ‘cash flow beta’ (CFB) as the correlation of its book return on equity with market book return on equity.7 Cohen et al. (2009) provide evidence that CFBs are better measures of firms’ systematic risk than are standard betas estimated using high-frequency firm and market share returns. Apparently for this reason, they find that CFBs explain a sizeable portion of the long-standing book-tomarket ratio anomaly (Fama and French 1992). Cohen et al. (2009) claim that this predictive ability results from CFBs being less influenced by transitory changes in the cost of equity capital and security market imperfections than are standard betas. They further claim that CFBs have greater interest to long-term investors – presumably the class of investors of most concern to financial reporting policymakers – because [a]s the holding period increases, news about cash flows begins to dominate the second moments (covariances and variances) of returns. As a consequence, the risk in a company’s cash flows is really what matters to an investor with a buy-and-hold perspective. (Cohen et al. 2009, p. 2740)

Second, Hodder et al. (2006) show that banks’ fair value gains and losses on financial instruments are risk-relevant beyond reported net income, most strongly so for interest rate risk. Fair value gains and losses reflect both unexpected changes in cash flows and unexpected changes in the costs of capital for the corresponding instruments during the period. Hence, in apparent tension with the findings of Cohen et al. (2009), the findings of Hodder et al. (2006) suggest that accounting numbers that reflect changes in costs of capital are risk-relevant. My belief is that this tension is only apparent, and that unexpected changes in cash flows and unexpected changes in costs of capital are both risk-relevant, just not in the same ways or for the same types of risk assessment analyses. Reflecting this belief, in my opinion it is not useful for financial reporting policymakers to try to identify the accounting measurement attributes that yield ‘the best’ measurements of income and other summary accounting numbers, a one-sizesuits-all approach. Instead, I recommend that policymakers require firms to report comprehensive income statements that: (1) measure comprehensive income based on fair value or a similarly information-rich accounting measurement attribute and (2) present the components of comprehensive income that are primarily driven by variation in cash flows from those that are primarily driven by variation in costs of capital. Such comprehensive income statements would provide users of financial reports with the flexibility to calculate alternative income numbers and thereby to perform different types of risk assessment analyses that Cohen et al. (2009) and Hodder et al. (2006) show to be useful.8 The recent push back against fair value accounting and deactivation (and perhaps demise) of the joint Financial Accounting Standards Board/ International Accounting Standards Board (FASB/IASB) project on financial statement presentation both work in opposition to this recommendation. I do not mean to suggest that other financial report disclosures are unimportant for risk assessment. On the contrary, these disclosures are essential forward-looking complements to primarily

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historical accounting numbers. However, these disclosures are limited in the following ways, among others. The disclosures are required by financial reporting policymakers and presented by firms in increasingly lengthy and poorly integrated financial reports. They are non-comparable across exposures, firms, and time. The assumptions underlying model-dependent disclosures are opaque. These limitations make it difficult for users of financial reports to identify the risk-relevance of the disclosures, at least prior to the occurrence of specific events that focus users’ risk assessment analyses on well-defined subsets of the disclosures. This difficulty helps explain why the empirical research surveyed in this paper collectively provides fairly weak evidence that these disclosures are risk-relevant. For this reason, I believe it is less important for policymakers to require new risk disclosures in financial reports and more important for them to impose structure on, and increase the comparability of, existing disclosures. I recommend that policymakers do so in three primary ways. First, they should attempt to maximise the ties of other financial report disclosures with summary accounting numbers. My primary specific recommendation in this vein is to require firms to conduct back-tests of prior significant accrual estimates and to disclose the results of those tests, indicating any observed trends in and identified drivers of revisions to those estimates, and describing the effects of those revisions on current and, if possible, future summary accounting numbers. Second, policymakers should encourage and, to the extent feasible, require firms to aggregate and present risk disclosures in tabular or other well-structured formats that promote the usability of the information. Identifying and propagating the use of existing best disclosure practices and encouraging new best practices is the most natural way to do this. Third, for modeldependent risk disclosures, policymakers should encourage and, if feasible, require disclosures of the primary historical and forward-looking attributes of the models and their implementation in practice, sensitivity of the model outputs to common variants of those attributes, and benchmarking of the models to standard portfolios of exposures. I limit the scope of this paper in the following five respects.9 First, after a brief summary of the primary conclusions in Ryan (1997), I do not repeat the summary of risk research in accounting and finance up to the mid-1990s in that paper. I emphasise, however, that the results of that research should be understood by anyone interested in improving risk reporting quality. Second, I limit my discussion to recent research on the risk-relevance of summary accounting numbers to one paper examining implied costs of capital, two papers examining CFBs, and three papers examining fair value accounting. I believe the reader is better served by a solid understanding of these papers than by a casual acquaintance with a laundry list of the results of many studies. Third, my discussion of the risk-relevance of other financial report disclosures is mostly limited to financial instruments and/or financial institutions, my area of expertise. Naturally, non-financial firms have meaningful risks that can and should be disclosed in financial reports. In addition, while I discuss many of these studies, I do not try to be comprehensive in my research coverage, but rather focus on papers with direct implications for risk reporting quality. Fourth, I limit my discussion to information that financial reporting policymakers naturally would require, with some specificity, firms to disclose in footnotes to their financial reports. I do not discuss disclosures that, while nominally required, allow firms to have substantial control over their content, such as lists of risk factors in the front end of financial reports.10 I do not discuss disclosures that firms naturally would provide through timelier, less-structured, and typically more voluntary information channels outside of financial reports.11 Indeed, I try not to wade into the swampish issue of when disclosures should be mandated versus voluntarily provided, given that firms have incentives to disclose information voluntarily when it reduces their costs of capital. I acknowledge, however, that my recommendation for financial reporting policymakers to impose structure and comparability on financial report disclosures betrays my belief that some disclosure requirements are necessary for this to occur.

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Fifth, I do not discuss cost –benefit trade-offs that are outside my expertise to evaluate. However, I do believe that the recommendations I make in this paper are not excessively costly. The remainder of this paper is organised as follows. Section 2 summarises the primary conclusions in Ryan (1997). Sections 3 and 4 address the risk-relevance of summary accounting numbers. Section 3 summarises empirical research examining the risk-relevance of estimates of implied costs of equity capital (ICOC) and CFBs derived from summary accounting numbers. Section 4 summarises empirical research examining the risk-relevance of fair value gains and losses. Sections 5 and 6 address the risk-relevance of other financial report disclosures. Section 5 defines the five major types of risk I consider, namely: market, credit, information, liquidity, and estimation, and overviews existing disclosure requirements for these risks. Section 6 summarises empirical research examining the risk-relevance of these disclosures. Section 7 contains my recommendations of how financial reporting policymakers can try to improve risk reporting quality. Section 8 concludes. 2. Brief summary of Ryan (1997) In Ryan (1997), I summarise theoretical and related empirical research in accounting and financing on equity risk up to about the mid-1990s. The theoretical research (Hamada 1972, Lev 1974, Mandelker and Rhee 1984) generally assumes perfect markets. It focuses on the individual and joint effects of three broad risk components, namely, operating risk (i.e. contribution margin variance), operating leverage (i.e. fixed operating costs, which increases contribution margin variance due to the tradeoff between fixed and variable costs), and financial leverage (i.e. fixed financing costs), on two measures of the firms’ equity risk, namely, beta (systematic risk) and share return variance (total risk). Under simplifying assumptions, systematic and total equity risk can be mathematically represented as systematic and total operating risk, respectively, times one plus operating leverage times one plus financial leverage. Hence, the equity risk measures increase if any one of their three risk components increases while the other two components are held constant. Motivated by this theoretical research, the empirical research provides the following primary results: .

.

.

Firms’ reported earnings variance is the primary accounting predictor of both measures of firms’ equity risk (Beaver et al. 1970, Rosenberg and McKibben 1973). This finding is somewhat surprising for systematic equity risk, because the covariance of firm earnings with market earnings (then called ‘accounting beta’, now called ‘CFB’) is in theory more analogous to systematic risk than is earnings variance.12 Market participants incorporate into their estimates of firms’ financial leverage the following estimates with respect to individual firms: (1) off-balance sheet financing (Bowman 1980a, Dhaliwal 1986, Imhoff et al. 1993), (2) risk-concentrated positions such as derivatives (McAnally 1996), and (3) market value of equity (Bowman 1980b). Firms’ equity risk increases with each of their operating risk, operating leverage, and financial leverage, controlling for the other components (Lev 1974, Bowman 1980a, 1980b, Mandelker and Rhee 1984, Dhaliwal 1986). However, firms choose their financial leverage to yield an appropriate level of overall risk given their operating risk and operating leverage, so this control is necessary to observe these positive relationships. In particular, financial leverage has little relationship with equity risk without such control.

In Ryan (1997), I propose two primary ways that financial reporting policymakers could use to improve the risk-relevance of financial report information. First, policymakers could require firms to provide better information about their ex post realisations of risk. I cite the example of

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disclosures required for property casualty insurers in the USA to report revisions of their loss reserves for each of the prior nine years of insurance coverage written for each fiscal year up to the current year (so-called ‘loss reserve triangles’). Insurance analysts examine these disclosures carefully for their risk implications and also for evidence of discretionary management of loss reserves. Policymakers could easily require similar disclosures for other types of firms’ significant accrual estimates, such as banks’ allowances for loan losses. For almost all accrual estimates, users of financial reports currently cannot distinguish revisions of the accrual estimates for preexisting exposures from the initial accrual estimates for newly assumed exposures. The proposed disclosures would convey the economic and estimation risks of accrual estimates to users of financial reports in a simple and useful fashion. My recommendation in the current paper that financial reporting policymakers should require firms to conduct and disclose the results of back-tests of their accrual estimates effectively extends this prior recommendation. Second, policymakers could revise aspects of accounting that blur the three broad components of equity risk described above. The primary example I cited is absorption costing, which obscures operating risk by combining fixed and variable costs in the unit product costs used to estimate cost of goods sold. The FASB mitigated this issue somewhat with the issuance of FAS 151 (ASC 33010-30),13 which constrains firms’ absorption of fixed production costs into unit product costs to reflect the range of normal capacity.14 Obviously, many other income statement line items, such as selling, general, and administrative expenses, combine fixed and variable costs. My recommendation in the current paper for financial reporting policymakers to distinguish components of income that are primarily driven by variation in cash flows from those that are primarily driven by variation in costs of capital is in a similar vein to this prior recommendation. 3.

Recent research on the risk-relevance of estimates of ICOC and CFBs

The research on the risk-relevance of estimates of ICOC and CFBs discussed in this section extends earlier research on accounting predictors of firms’ equity risk described in Section 2 to incorporate structure from formal accounting-based valuation models.15 While various models have been used for this purpose, I limit my discussion to the simplest and most widely used model, referred to as the residual (or abnormal) income model (‘RIM’). Although its initial development occurred no later than 1888,16 RIM had been little appreciated and rarely used until rigorously reframed in setting of uncertainty by James Ohlson in an influential series of sole and coauthored theoretical papers (Ohlson 1995, Feltham and Ohlson 1995, 1999). As discussed below, RIM has provided the basis for two distinct risk-assessment literatures: (1) papers estimating and using ICOC and (2) papers estimating and using CFBs.17 Both literatures focus on explaining firms’ future long-horizon share returns, which in efficient markets should on average reflect the firms’ costs of equity capital. It is necessary for me to first explain RIM in order to convey an adequate understanding of these papers. RIM recasts the standard discounted dividends valuation model used in finance in terms of summary accounting numbers. It does so using the ‘clean surplus relation’, which specifies that dividends net of equity issuance and redemption in a period equal net income minus the change in book value of equity in the period.18 For this reason, some academics have criticised RIM as not a theoretical advance, because it derives algebraically from the assumptions of dividend discounting and the clean surplus relation. However, RIM is practically useful because it substitutes accounting numbers for dividends – the latter being zero for the foreseeable future for many firms – and thereby yields considerably more accurate valuations over the short-tomedium horizons typically used in valuation analyses (Penman and Sougiannis 1998). This usefulness is evident from the highly economically meaningful results in the risk assessment literatures described below.

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The ICOC literature employs a simple version of RIM with a constant periodic cost of equity capital, while the CFB literature uses a more sophisticated version of RIM with a stochastic cost of equity capital. I describe both versions below, using the following common notation: MVEt denotes market value of equity at the end of period t, BVEt denotes book value of equity at the end of period t, and NIt denotes net income during period t. In the simple version of RIM used in the ICOC literature, r denotes the assumed intertemporally constant periodic cost of equity capital. Abnormal income in period t, denoted NIat , is defined as net income in period t minus the product of the cost of equity capital and the book value of equity at the end of period t21, i.e. NIat = NIt − rBVEt−1 . Using this assumption and notation, this version of RIM is:19 1  ˜ a } E{NI t+s MVEt = BVEt + . (1 + r)s s=1

(1)(RIM − ICOC)

Intuitively, RIM-ICOC decomposes MVEt into two parts: BVEt, which reflects the portion of MVEt that is currently recognised on the balance sheet; and discounted abnormal income, which reflects the portion of MVEt that is not yet recognised on the balance sheet (but on average will be recognised as the firm earns income in the future). To employ RIM-ICOC to estimate firms’ ICOC, researchers must:20 . .

.

obtain or generate forecasts of firms’ future abnormal income over some horizon, estimate the terminal value at the end of the horizon (which involves specifying the growth path for abnormal income beyond the horizon), and impute the ICOC as the internal rate of return that equates the currently unrecognised portion of MVEt, i.e. MVEt – BVEt, to the present value of expected future abnormal income.

To forecast future abnormal income, researchers typically use analysts’ earnings forecasts. They sometimes employ statistical models of the time series of income, either to avoid the known biases in analyst forecasts or to increase the sample beyond the firms covered by those forecasts. Researchers employ various approaches to specify the terminal value, as described in Lee et al. (2011). Lee et al. (2011) ran a horse race in which they use alternative ICOC estimates based on earnings forecasts from time-series models as well as standard finance approaches to estimating the cost of equity capital (i.e. the capital asset pricing model and two multi-factor models attributed to Fama and French) to predict firms’ future realised share returns, which they assume on average reflect firms’ equity costs of capital, albeit noisily. The results from Lee et al. (2011) effectively summarise the results of the now large ICOC literature. These results demonstrate convincingly that ICOC estimates using any method of implementing RIM handily outperform the standard finance approaches.21 They also demonstrate that the performance of ICOC estimates varies somewhat depending on the growth assumptions used to estimate the terminal value. The results in Lee et al. (2011) suggest that financial report information that better enables users of financial reports to forecast future abnormal net income, including the terminal value, should enable more accurate estimates of firms’ ICOC. While it might seem that requiring accounting approaches that smooth abnormal net income up to the horizon would do this, in fact, such approaches would tend to push economic variance into the relatively hard-to-specify

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and growth-rate-sensitive terminal value. Instead, ICOC estimation likely would become more accurate if the terminal value became smaller or more specifiable in terms of simple growth paths. How financial reporting policymakers could enhance ICOC estimation through improved accounting or disclosure raises difficult issues worthy of future research. For example, perfect fair value accounting for all of the firm’s exposures, if it were possible, would make each element of its expected stream of future abnormal income, including the terminal value, equal to zero. While this might be desirable from a valuation perspective, it would make ICOC estimation using RIM impossible. I recommend that financial reporting policymakers consider whether it is possible to improve the forecasting of firms’ future abnormal income, and thus ICOC estimation, through alternative accounting measurement bases, financial statement line item classifications, and other disclosures in financial reports. I recommend they start by requiring informationally rich comprehensive income statements of the sort described in the introduction. Feltham and Ohlson (1999) and Nekrasov and Shroff (2009) derive a more sophisticated formulation of RIM that allows for stochastic costs of equity capital that are reflected in an adjustment to the numerator of a present value calculation that uses risk-free interest rates. Rubenstein (1976) shows that risk adjustment in the numerator is the conceptually more general approach. In f denotes the (multi-period, for s . 1) risk-free rate from the end of this formulation of RIM, rt,t+s period t to the end of period t + s. Abnormal income in period t is defined as net income in period t minus the product of the risk-free rate for period t and the book value of equity at the end of t BVEt−1 . mt,t+s denotes the stochastic market discount factor period t21, i.e. NIat = NIt − rt−1,t from the end of period t to the end of period t + s.22 The covariances of mt,t+s with a firm’s NIat+s , 1≤s,1, determine the firm’s systematic risk, with (more) negative covariances indicating greater systematic risk.23 Covariances using the information available at the end of period t are denoted as covt. Using this notation, Nekrasov and Shroff (2009) express RIM as

MVEt = BVEt +

 a  1 ˜  E NI t+s s=1

1+

f rt,t+s

+

1 

a

˜ covt (mt,t+s , NI t+s ).

(2)(RIM − CFB)

s=1

RIM-CFB expresses MVEt as equal to BVEt plus the present value of the future stream of abnormal income discounted at the relevant risk-free rates, plus a systematic risk adjustment term that captures the covariances of mt,t+s with NIat+s , 1≤s,1. This risk adjustment term is the focus of the CFB literature. The empirical work in this literature typically uses abnormal book return on equity above the risk-free rate, rather than abnormal income, in applying RIM to estimate CFBs. Defining book return on equity in period t as ROEt ¼NIt/BVEt21, abnormal income can be re-expressed as f a NIat = BVEt−1 (ROEt − rt−1, t ). Using this expression to substitute for NIt in the risk-adjustment term in RIM-CFB yields the equivalent expression 1  s=1

a

˜ covt (mt, t+s , NI t+s ) =

1 

˜ t+s − rf ˜ t+s−1 (ROE covt (mt, t+s , BVE t+s−1, t+s )).

(3)

s=1

Equation (3) is difficult to apply in practice because it expresses systematic risk in terms of an infinite series of covariances. Similar to the derivation of the capital asset pricing model, Nekrasov and Shroff (2009) assume that these covariances are all equal (as well as various less important things), which simplifies the expression to the right of the equal sign of Equation (3) to a single covariance of the market discount factor m with abnormal ROE. Consistent with much

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of modern finance, Nekrasov and Shroff (2009) also assume that the risk adjustment term can be expressed as a linear function of the covariances of abnormal ROE with market abnormal ROE and other risk factors. Based on these assumptions, Nekrasov and Shroff (2009) estimate a firm’s CFB as the correlation of its abnormal ROE with the abnormal ROE for the market. A higher CFB implies higher systematic risk. Cohen et al. (2009) estimate CFBs using a similar approach.24 With regard to the risk-relevance of estimated CFBs, Cohen et al. (2009) and Nekrasov and Shroff (2009) both find that CFBs have significant explanatory power over: (1) previously anomalous long-horizon share returns on portfolios formed based on the level of the market-to-book ratio, because CFBs are higher for lower market-to-book ratios and (2) the risk adjustment (i.e. ICOC) reflected in the level of equity price. Additionally, Cohen et al. (2009) find that CFBs predict changes in standard betas over a horizon of at least eight years. Standard betas rise (fall) over this horizon for low (high) market-to-book ratio firms. This finding is consistent with standard betas reflecting changes of cost of capital or market imperfections in the short term but ultimately being driven by accounting-related fundamentals. These findings are consistent with CFBs capturing priced risk better than standard betas and sensitivities to other previously identified risk factors, the market-to-book ratio in particular. Overall, the results of this research indicate that CFBs are highly risk-relevant, considerably more so than one would expect from the prior research examining similar measures summarised in Ryan (1997). I recommend that financial reporting policymakers consider whether it is possible to improve the risk-relevance of CFBs. As with the ICOC literature, however, this question raises difficult issues worthy of future research. For example, perfect fair value accounting might induce CFBs to reflect changes in costs of capital rather than cash flows or related performance measures, thereby rendering CFBs more like standard betas. I again recommend policymakers start by requiring informationally rich comprehensive income statements of the sort described in the introduction. In summary, research on estimates of ICOC and CFBs summarised in this section shows that existing summary accounting numbers are highly risk-relevant. Hence, financial reporting policymakers should focus on enhancing the current financial reporting system, not reinventing the wheel. 4.

Recent research on the risk-relevance of fair value accounting

In this section, I describe the results of three relatively recent papers (Hodder et al. 2006, Barth et al. 2008, Blankespoor et al. 2011) that examine the risk-relevance of calculated income or financial leverage measures based on disclosed or estimated fair values of financial instruments.25 I describe Hodder et al. (2006) in particular detail because this paper considers banks’ entire portfolios of financial instruments and a wide set of risk measures. Blankespoor et al. (2011) also consider banks’ entire portfolios of financial instruments but they focus only on credit risk. Barth et al. (2008) examine the highly controversial issue of the incorporation of non-financial firms’ own credit risk in the measurement of fair value for their debt. Fair values may be recognised or disclosed. Disclosure need not be an adequate substitute for recognition. This is because firms do not appear to estimate, and auditors, analysts, and others do not appear to evaluate, fair values with as much effort if these values are disclosed rather than recognised. Moreover, firms rarely provide meaningful discussion or analysis of fair value disclosures in financial reports. Presumably, if researchers find disclosed fair values to be riskrelevant, then recognised fair values would be even more so. The restrictions in the samples of both Hodder et al. (2006) and Blankespoor et al. (2011) to coverage of banks are important, because banks’ balance sheets primarily comprise financial instruments for which firms must disclose the fair and carrying values under FAS 107 (ASC

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825-10-50),26 annually prior to 2009 and quarterly thereafter. Fair value accounting generally is most desirable when applied to firms’ entire economic balance sheets, so that any offsetting economic gains and losses on the firms’ exposures do not yield variability in the income measure, while any non-offsetting gains and losses do yield such variability. This should occur to a reasonable approximation for banks. In contrast, when fair value accounting is applied only to a portion of a firm’s economic balance sheet, the accounting generally will not capture any economic offsetting across the subset of the exposures that is fair valued and the subset of the exposures that is not fair valued, or within the latter subset. Accordingly, fair value accounting applied to a portion of a firm’s balance sheet could make the firm appear either more or less risky than it actually is.27 This is a more significant problem for non-financial firms. However, Barth et al. (2008) provide evidence that this problem is not overwhelming for the average non-financial firm with regard to fair value accounting for liabilities, as discussed below. As described more fully below, Hodder et al. (2006) examine the total and incremental riskrelevance of banks’ fair value income beyond their reported net income and other comprehensive income. They calculate annual unrecognised fair value gains and losses for each bank for each year from 1996 to 2004 using FAS 107 disclosed fair values of financial instruments.28 Specifically, Hodder et al. calculate unrecognised fair value gains and losses on economic assets (liabilities) as (minus) the change over the reporting period in the excess of the fair value over the book value. Annual fair value net income equals reported net income plus reported other comprehensive income plus calculated unrecognised fair value gains and losses during the year. They calculate bank-specific variances of the following income measures or components across their nine-year sample period: reported net income, reported comprehensive income, calculated fair value income, reported other comprehensive income, and calculated unrecognised fair value gains and losses. Hodder et al. (2006) report the following interesting descriptive analyses of the variances and correlations of their income measures and components. .

.

.

On average, fair value income is over five (three) times more variable than reported net income (comprehensive income), and these differences in variances are highly significant. Hence, fair value accounting for banks’ entire portfolios of financial instruments would make these firms appear considerably riskier than does their current accounting. On average, the correlations of reported net income, other comprehensive income, and unrecognised fair value gains and losses are all close to zero, suggesting minimal offsetting across these income components. This explains the results of the analysis described in the prior item. In contrast, on average unrecognised gains and losses on liabilities are strongly negatively correlated (approximately 20.4) with both recognised (in other comprehensive income) and unrecognised fair value gains and losses on assets. This is consistent with banks actively engaging in asset – liability management, but not coming close to fully hedging.

Hodder et al. (2006) examine various market-based measures of bank risk: return variance, beta, long-term interest-rate beta, and the valuation multiple on abnormal income in a simplified version of RIM-ICOC. They correlate these risk measures with the income variance measures in two separate types of analyses. First, they evaluate the total explanatory power over their risk measures of each of reported net income variance, reported comprehensive income variance, and calculated fair value income variance, without controlling for the other two income variance measures. A measure of income variance with more explanatory power over risk is more risk-relevant. Second, Hodder et al. (2006) evaluate the incremental explanatory power over

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their risk measures of reported net income variance, reported other comprehensive income variance, and variance of calculated fair value gains and losses, controlling for the other two income component variance measures. An income component variance measure with significant explanatory power over risk controlling for the other measures is incrementally risk-relevant. Hodder et al. (2006) report the following primary empirical results. .

.

.

.

.

Regardless of the risk measure, calculated fair value income variance is always riskrelevant and the variance of calculated fair value gains and losses is always incrementally risk-relevant. When the risk measure is return variance, reported net income variance is the most riskrelevant of the income measures, followed closely by calculated fair value income variance. The variance of calculated fair value gains and losses is highly significantly incrementally risk-relevant. When the risk measure is beta, none of the variance measures are particularly significant, likely due to well-known problems in measuring beta. However, the calculated fair value variance measures are the only ones that consistently are significant and have the right sign. When the risk measure is long-term interest-rate beta, the calculated fair value income variance measures are highly significant and essentially dominate the other measures. When the risk measure is the valuation multiple on abnormal income, the calculated fair value income variance measures have the right signs and are most significant, and the reported net income variance measure has the right sign and is significant.

In summary, the findings of Hodder et al. indicate that fair value accounting yields highly riskrelevant income measures, especially regarding interest rate risk.29 Using methods similar to those of Hodder et al. (2006), Blankespoor et al. (2011) calculate a measure of banks’ financial leverage that incorporates the disclosed or estimated fair values of financial instruments. They examine two measures of banks’ credit risk: publicly traded bond yield spreads (over comparable maturity US Treasuries) and failure as defined by the presence on the FDIC’s failed bank list. It is relatively difficult to predict failure due to its infrequency; in particular, almost all bank failures occurred during the recent financial crisis. Blankespoor et al. (2011) regress each of these credit risk measures on each of banks’ fair value leverage and reported leverage ratios. They estimate these regressions both with and without an extensive set of controls for banks’ profitability, size, interest rate sensitivity, asset quality, and, in the yield spread regressions, bond characteristics. Blankespoor et al. (2011) find that the fair value leverage measure exhibits far more explanatory power over banks’ credit risk than does reported leverage. For example, in the bond yield regressions without controls, the R2 is 37% using the fair value leverage measure, versus 25% using the reported measure. In the regressions with controls, the R2 is 45% for the model using fair value leverage, versus 35% for the model with reported leverage. These results indicate that fair value leverage not only dominates reported leverage in explaining banks’ yield spreads, it also dominates their extensive set of control variables. These findings illustrate the power of well-chosen summary accounting numbers for risk assessment. Reflecting the difficulty of predicting failure discussed above, Blankespoor et al. (2011) provide somewhat weaker evidence that fair value leverage predicts bank failure better than does reported leverage. The most interesting aspect of their results is that the predictive power of fair value leverage increases as the bank failure horizon is increased from 1 to 3 years, whereas the predictive power of reported leverage falls as this horizon lengthens. This reflects

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the forward-looking nature of fair value and the historical nature of the measurement attributes used to determine the recognised amounts of the most part of banks’ exposures. As discussed above, the attractiveness of fair value accounting diminishes when applied to subsets of firms’ exposures. Barth et al. (2008) examine this issue in the specific setting of fair value accounting for financial liabilities, in particular, the effect of firms’ own credit risk on the fair value of their debt. Fair value accounting for debt is a particularly contentious topic within the overall contentious arena of fair value accounting, because firms whose own credit risk deteriorates (improves) generally will record gains (losses) on their debt. This would yield counterintuitive effects on their net income unless they record offsetting losses (gains) on their assets of at least equal magnitude. In their empirical analysis, Barth et al. (2008) choose not to use reported or disclosed fair values of debt. This is attributable to two facts. First, the fair value option for debt did not exist during their 1986 – 2003 sample period.30 Second, FAS 107 did not require firms to incorporate their own credit risk in the disclosed fair values of their debt prior to the effective date of FAS 157 (ASC 820 – 10), and firms do not appear to have done so except for publicly traded bonds for which the disclosed fair value was based on the observed market price.31 Instead Barth et al. (2008) either infer changes in the value of debt from changes in actual or modelled credit ratings or estimate the change in the value of debt using Merton’s (1974) option-pricing-based model of the value of risky debt. Barth et al. (2008) primarily provide evidence that the value of non-financial firms’ equity is sensitive to their own credit risk and that this effect is stronger for more levered firms. Intuitively, equity is protected against downside risk to the extent that firms’ debt absorbs that risk, which is greater for firms with more debt. These findings suggest that fair value accounting for debt conveys information about firms’ own credit risk, an important risk. Barth et al. (2008) also provide descriptive analysis that if firms recognised decreases in the fair value of their debt estimated using the Merton model, then on average those firms’ net income would not change sign. This descriptive analysis is consistent with fair value gains on debt being on average more than offset by impairment write-downs on assets or other sources of negative income, and with fair value losses on debt being on average more than offset by other sources of positive income. Hence, fair value accounting for debt would not on average dominate net income for non-financial firms, obscuring their performance and risk. These results likely would change if firms were grouped based on the types of assets they hold, which affects the required accounting for assets, as well as on their financial leverage, however. The empirical research summarised in this section shows that fair value gains and losses are highly risk-relevant, particularly for interest rate risk, but also for systematic/priced risk, total risk, and firms’ own credit risk. Because fair value gains and losses are significantly affected by changes in costs of capital, the results of this research may appear to be in tension with those of the empirical research on CFBs summarised in Section 3. I discuss how financial reporting policymakers should view this apparent conflict and provide recommendations for how they can enhance the risk-relevance of financial report information in Section 7. 5.

Existing requirements for financial report risk disclosure

In this section, I overview the primary existing risk disclosure requirements in the financial reporting rules of US GAAP and the US Securities and Exchange Commission (SEC).32 I focus on US disclosure rules because almost all of the research I summarise examines US firms only. I indicate International Financial Reporting Standards (IFRS) analogues to US rules, most of which are contained in IFRS 7,33 when they exist and I am aware of them; however, I emphasise that I am not an IFRS expert. With a few exceptions, I limit this overview to disclosure requirements for financial

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instruments as they are defined in US GAAP (i.e. financial assets and liabilities, not the firm’s own equity) or for financial institutions. I organise this section into subsections for four primary types of economic risk, namely, market, credit, liquidity, and information,34 as well as the overarching notion of estimation risk. While this is a convenient structure, I note that many disclosures (e.g. notional amounts of derivatives) provide information pertinent to multiple types of risk. As indicated in the introduction, I define risk as random variation in firms’ future performance. Types of risk pertain to specific drivers or aspects of this variation. Market risk

5.1

Market risk is a random variation in firms’ future performance resulting from unexpected future changes in market prices. Examples of market prices include interest rates, foreign exchange rates, equity prices, and commodity prices. SEC FRR 48 contains the primary market risk disclosure requirements for financial instruments and derivatives.35 IFRS 7, paragraphs 40– 42, requires similar disclosures. FRR 48 exhibits a number of features that reduce the comparability and interpretability of market risk disclosures and thereby compromise risk reporting quality. I describe these features in some detail, because they generalise to most other types of risk disclosures and any attempt by financial reporting policymakers to improve risk reporting quality must make choices about these or similar features. Reflecting policymakers’ emphasis on downside risk discussed in the introduction, FRR 48 defines risk as the possibility of loss, not gain. Possibility of loss is a one-sided notion of risk that translates into overall risk (variance) only when returns are distributed symmetrically. Derivatives and other financial instruments that are or include options are particularly likely to exhibit asymmetric return distributions. FRR 48 only requires annual disclosures as of the balance sheet date. Given the infrequency of these disclosures, they are unlikely to be useful for firms that speculate, since speculative positions tend to change frequently. End-of-year window dressing by firms is also a concern. IFRS 7, paragraph 42, specifically requires firms to disclose when their end-of-year exposures are not representative of their normal exposures. FRR 48 disclosures need not be comparable across firms, exposures for a given firm, or time, for the following four reasons. First, firms may define loss in terms of reduction of value, earnings, or cash flow, and the three definitions of loss are not identical and can be inconsistent. Moreover, value and earnings depend on the accounting measurement bases used. Second, firms may disclose their exposures to each type of market risk using any of the following approaches: .

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Tabular format. This approach reports fair values and information sufficient to estimate the expected cash flows over each of the next five years and beyond five years for derivatives and other financial instruments grouped based on common characteristics. Sensitivity approach. This approach reports the loss of value, earnings, or cash flow attributable to a specific adverse market price movement chosen by management, subject to the constraint that the movement be at least 10% of the beginning value of the market price. Value-at-risk (VaR) approach. This approach reports the loss of value, earnings, or cash flow that occurs over a certain period with a certain probability, usually 5%.

These approaches have distinct strengths and weaknesses. The tabular format provides less processed and more disaggregated data than the other approaches, and so it allows users of financial reports more freedom to develop their own risk measures. However, this approach does not clearly convey the nature of dynamic exposures such as options or the covariances among

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exposures, so users typically have to make assumptions about these exposures and covariances to use these disclosures. The sensitivity and VaR approaches provide aggregate risk measures that embed management’s knowledge of individual exposures and covariances among exposures. However, these measures are difficult for users of financial reports to interpret without a full understanding of the assumptions involved in the aggregation. Moreover, the VaR approach does not indicate the direction of market price movements that causes loss, so it does not help users estimate the effects of subsequent changes in market prices. Although FRR 48 requires that firms discuss the assumptions and limitations of their chosen approaches, in practice these disclosures tend to be boilerplate. The fact that firms need not and often do not choose the same approach for each type of market risk makes it difficult for users of financial reports to develop measures of firms’ aggregate market risk, since different types of market risk may be correlated. For example, interest rates are correlated with exchange rates. Third, the period over which loss is measured in the sensitivity and VaR approaches varies across firms and across different risks for a given firm. Sensitivity and VaR estimates need not increase linearly or in any other simple fashion with the length of the measurement time period, due to portfolio changes over time and for other reasons, although under simplifying assumptions (e.g. intertemporal independence of returns) it usually is possible to make these estimates more comparable through appropriate transformations. Fourth, the size of the market price movements in the sensitivity approach and the confidence level used in the VaR approach vary across firms. Different-size market price movements in the sensitivity approach do not raise comparability problems if exposures are linear, but they do if exposures are nonlinear. Different confidence intervals in the VaR approach are easily adjusted for if the shapes of the distributions of the returns on the portfolios under consideration are known, but not otherwise. Market risk disclosures provided using the sensitivity or VaR approaches generally do not distinguish market risks before versus after any risk management activity (e.g. hedging using derivatives). Hence, these disclosures do not indicate how firms’ manage their market risks. However, FAS 161 (ASC 815-10-50)36 requires firms to disclose qualitative information about their risk management strategy as well as the volume of their use of derivatives (e.g. notional amounts) by type of risk, distinguishing those used for risk management purposes versus other (e.g. trading) purposes. The standard also encourages but does not require firms to make quantitative disclosures of their risk management. These disclosures give some sense for the volume of risk management activity using derivatives, but not for other forms of risk management, in particular, the far more important asset – liability management. Although FRR 48 is the primary source of market risk disclosures, other specific disclosure requirements related to market risk exist. Two examples of these are: .

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FAS 133 (ASC 815-10-50)37 requires disclosures of hedge ineffectiveness, as does IFRS 7, paragraphs 22– 24. These disclosures give some sense for the quality of risk management using derivatives, but not for other forms of risk management. FAS 140 and 166 (ASC 860-20-50)38 require disclosures of the sensitivity of the valuations of retained interests from securitisations accounted for as sales to changes in significant assumptions, including market prices such as interest rates, as does IFRS 7, paragraphs 42A-H. While these sensitivities to different parameters may interact in fashions not portrayed in the disclosures,39 and these limitations may hinder the aggregation necessary for firm-level analysis, these disclosures provide some sense for the degree of risk concentration in these often first-loss or otherwise highly sensitive positions.

310 5.2

S.G. Ryan Credit risk

Firms, particularly lenders, generally view their credit risk as variation in their future performance resulting from uncertainty about the occurrence and timing of default by their borrowers and other counterparties, including third-party guarantors, as well as the percentage loss in the event of default. I limit my discussion to this type of credit risk. Alternatively, firms may themselves default if they become insolvent or illiquid, and their own credit risk affects the values of their liabilities and equity. I do not discuss this type of credit risk because it results from myriad firm attributes. Many disclosures in firms’ financial reports potentially could inform about this type of credit risk. As discussed in Section 4, Barth et al. (2008) examine this type of credit risk. Existing credit risk disclosure requirements are considerably less comprehensive than are market risk disclosure requirements. They can be divided into three primary groups. First, two standards require disclosures of known significant concentrations of credit risk in individual or groups of exposures. The inclusion of ‘known’ is important in this requirement, because the identification of concentrations of credit risk is difficult, and reasonable people can differ about what constitutes a concentration. For example, the sharp decline in real estate prices during the financial crisis affected a wide range of banks’ exposures and devastated the overall economy, particularly in certain regions of the country. Did this constitute a concentration for a bank with exposure to real estate prices? If so, how should the bank determine which of its exposures to include in the concentration? FAS 107 (ASC 825-10-50) requires the following disclosures for identified concentrations of credit risks in all financial instruments: qualitative disclosures about the nature of the concentration, quantitative information about the maximum possible amount of loss, and quantitative information about collateral and master netting agreements. Although very terse, in principle IFRS 7, paragraph 34(c), requires disclosures of concentrations of any risk in financial instruments, including credit risk. SOP 94-6 (ASC 275-10-50)40 requires disclosures of concentrations of any risk for which it is reasonably possible that the firm will experience a near-term (within one year), severe (higher than material) impact. Reflecting FAS 107’s more extensive disclosure requirements, SOP 94-6’s requirement explicitly applies only to risk concentrations involving non-financial instruments.41 Second, FAS 5 (ASC 450-20)42 requires firms to accrue for loss contingencies, including credit loss contingencies, that probably have been incurred at the balance sheet date based on information available at that date, and that the firms can reasonably estimate at that date. The standard also requires disclosure for credit losses that are only reasonably possible or that cannot be reasonably estimated at the balance sheet date.43 IAS 37,44 paragraphs 85 and 86, contains similar disclosure requirements. Third, SEC Industry Guide 345 requires disclosures relevant to the assessment of credit risk for banks and thrifts. Specifically, it requires those financial institutions to disclose quantitative information about non-accrual, past due, troubled debt restructured, and potential problem loans; loan balances and allowances for loan losses by type of loan; loan charge-offs and recoveries by type of loan; and loan concentrations. FAS 118 (ASC 310-10-50)46 requires lenders to provide disclosures about the amount of impaired loans (as well as the allowance and accrued interest for those loans). In practice, impaired loans correspond closely to non-performing loans. IFRS 7, paragraph 37, requires similar disclosures. In addition, FSP FAS 133-1 and FIN 45-4 (ASC 815-10-50)47 and various other standards require disclosures of collateral, netting agreements, and credit contingent features of derivatives and similar exposures that may enhance or mitigate credit risk. IFRS 7, paragraphs 14, 36, and 38, contains similar requirements.

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Liquidity and information risks

I describe liquidity and information risks together because they are both trading-related risks that overlap considerably but are not the same and are often confused. The overlap and confusion result in part because both types of risk tend to be higher for credit riskier exposures that are traded less frequently, particularly during credit crises, and for which asymmetric information is more likely to exist. Illiquidity exists when trading impacts the price of the traded item or, in the limit, trading is impossible. Liquidity risk is random variation in the level of liquidity in the market for an item or of the firm’s need for liquidity, both of which may vary over time. Information risk usually is said to exist when a party is uncertain about whether it is trading with a better-informed counterparty. Distinct kinds of information risk arise in cases when both trading parties lack relevant information due to Knightian uncertainty or for other reasons. Relatively few disclosure requirements currently exist for liquidity and information risks. FAS 157 requires firms to disclose the amounts of balance sheet items measured at fair value using Level 1, Level 2, and Level 3 inputs.48 (These disclosures also pertain to estimation risk, which is discussed in Section 5.4.) Level 1 inputs are market prices for the same item in active markets. Items measured using Level 1 inputs usually have relatively low liquidity and information risks. Level 2 inputs are other observable market information. Items measured using Level 2 inputs usually have higher liquidity risk, but possibly moderate information risk depending on the level of transparency of the market inputs. Level 3 inputs are unobservable firmsupplied inputs. Items measured using significant Level 3 inputs usually have relatively high-liquidity and information risks, due to the absence of significant observable market inputs and unobservability of firm-supplied inputs, respectively. FAS 157 requires additional disclosures (e.g. rollforwards of the account balances from the beginning to the end of the fiscal period) for items measured at fair value using significant Level 3 inputs. Item 303 of SEC Regulation S-K49 requires firms to disclose known trends or demands, commitments, events, or uncertainties that will result in or that are reasonably likely to result in the registrant’s liquidity increasing or decreasing in any material way. 5.4

Estimation risk

Estimation error is the difference between reported measurements of items and the ideal measurements given the relevant accounting standards for the items and currently available information. Estimation risk is a random variation in estimation error. Such variation results from economic risk, the unobservability of transactions involving the same or similar items, imperfect modelling of the value of the item or relevant data to apply the model, and various other factors. Such variation does not include the average error, i.e. estimation bias. In principle, while estimation risk tends to be higher for economically riskier positions, it is different from the economic risks described above. Economic risks pertain to uncertainty about outcomes that have not yet been realised and that in principle are described by distributions. Accounting measurements do not (and logically cannot) involve prediction of the one possible outcome within each distribution that will ultimately be realised. Instead, accounting measurements typically involve estimation of the central tendencies (e.g. expected values) or other points in the distributions of possible outcomes. Accounting measurements generally are revised over time as relevant new information becomes available. Such revisions typically move the accrual measurement towards the outcomes that will be ultimately realised, but they do not imply that estimation errors previously existed. SOP 94-6 contains the primary disclosure requirements for estimation risk in US GAAP. This standard requires qualitative disclosures of the use of estimates in financial reporting, the nature of

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any uncertainty about those estimates, and reasonably possible near-term material changes in estimates. In addition, as discussed in Section 5.1, FAS 166 and various other standards now require firms to disclose the sensitivity of estimated fair values of items to important parameters. Higher estimation sensitivities likely are correlated with higher estimation risk. 5.5

Summary

Even the incomplete summary of the primary existing risk-relevant financial report disclosure requirements in US GAAP and SEC rules in this section indicates that firms are required to provide many such disclosures in their financial reports. This is particularly true for market risk. While the volume of risk disclosures is high, these disclosures are not well integrated either within or across types of risk. Disclosure options yield inconsistency across firms and exposures. For these reasons, my belief is that users of financial reports generally do not appreciate the information in these disclosures, at least not prior to the occurrence of specific events that focus their attention on well-defined subsets of the disclosures. I emphasise, however, that immediate appreciation is not necessary for risk disclosures to be useful. One of the primary roles of financial reporting is to archive information that is not immediately informative but will be useful in evaluating information that becomes available or events that occur subsequently. Moreover, my own teaching experience suggests that these disclosures are susceptible to careful, if time consuming, financial analysis of individual financial reports. 6. Recent research on the risk-relevance of other financial report disclosures In this section, I summarise recent empirical research on the risk-relevance of other financial report disclosures. The subsections are again organised by the type of risk, with a subsection on research on securitisations added at the end because the economics of these transactions and this research span the types of risk. To keep this section to a manageable length and with reasonable focus, with a few exceptions I do not discuss papers in three large and well-established literatures that bear on firms’ risk assessment using financial report information. First, I exclude ‘value-relevance’ papers, in which the dependent variable is market value or returns rather than a risk measure, even if the primary explanatory variables are risk disclosures, such as Venkatachalam (1996) (examining derivatives disclosures) and Lim and Tan (2007) (examining VaR disclosures). Naturally, market values and returns are sensitive to disclosures that inform about priced risk. Second, I exclude papers in which the primary explanatory variables are measures of past performance rather than risk per se, even if the dependent variable is a risk measure. For example, this excludes papers that correlate measures of banks’ risk, such as bond yields or CDS spreads with their provisions or allowances for loan losses, non-performing loans, or loan charge-offs, such as Jagtiani et al. (2002). Naturally, superior past performance tends to be associated with lesser risk. Third, I exclude papers in the bankruptcy, credit ratings, loss given default, and similar default variable prediction literatures. Good recent summaries exist for most of these literatures, such as Beaver et al. (2010) for bankruptcy prediction. 6.1 Market risk A reasonably large number of papers empirically examine the risk-relevance of FRR 48 market risk disclosures for samples of financial institutions and/or non-financial firms. Most of these studies primarily examine disclosures in the sensitivity format, partly because it is the most common format, and partly because the simplicity of these disclosures renders them most

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amenable to large-sample empirical analysis. Several papers examine VaR disclosures, which with few exceptions are made only by trading-oriented firms. Overall, the evidence for the risk-relevance of market risk disclosures is fairly weak. It is strongest in cases of simple exposures for homogeneous sets of firms, in particular, specific commodity price risk for commodity firms exposed in the same direction to the same commodity. For example, Rajgopal (1999) (using tabular and sensitivity disclosures that pre-dated FRR 48) and Thornton and Welker (2004) (using reported sensitivity and VaR disclosures under FRR 48) both find that disclosures of energy commodity price sensitivity by oil and gas producers are commodity price risk-relevant. By comparison with commodity price risk, interest rate risk and exchange rate risk are complex. Interest rate risk involves yield curves that can change both location and shape and that interact with firms’ exposures in each maturity or repricing interval. Most firms with exchange rate risk are exposed to multiple correlated currencies. As a consequence, firms’ modelling of these exposures is likely to be judgmental, limited, and opaque to users of financial reports. Consistent with this point, empirical evidence for the risk-relevance of disclosures of these market risks is much weaker than for commodity price risk. For example, Hodder (2002) examines interest-rate-sensitivity disclosures by banks. Sribunnak and Wong (2004) examine exchange-rate-sensitivity disclosures by non-financial firms. Both papers find only weak and inconsistent risk-relevance for the disclosures they examine.50 A partial exception to these relatively weak findings is research examining trading-oriented banks’ disclosures of VaR for their trading portfolios, which usually are primarily exposed to interest rate risk. Both Jorian (2002) and Liu et al. (2004) find that these banks’ VaR disclosures explain cross-sectional differences in banks’ unsystematic risk as measured by the variances of their trading income and stock returns as well as cross-sectional differences in banks’ systematic risk as measured by their betas. In addition, Liu et al. find that these results are more significant for the largest, more technically sophisticated banks that are better able to estimate VaR, and they have strengthened over time as VaR becomes better measured by firms and understood by users of financial reports. More recently, however, Perignon and Smith (2010) find banks’ VaR does not predict the variance of trading income, a change from prior findings that they ascribe to banks’ increasing tendency to estimate VaR using historical simulation. Historical simulation rapidly loses predictive power when economic conditions change from the period over which historical data are sampled. A consistent difficulty faced by these papers is developing cross-sectional samples of sufficient homogeneity and size to yield powerful statistical tests. This difficulty results in part from the varied approaches and other choices that firms are allowed to and do make in their market risk disclosures under FRR 48 described in Section 5.1. Accordingly, a frequent conclusion of this research is that the risk-relevance of these disclosures is diminished by their lack of comparability and also by the low quality of firms’ disclosures of their exposures before the use of derivatives or other risk management. This literature generally predates FAS 161, however, which has modestly improved firms’ disclosures of their exposures before derivatives. This large-sample, cross-sectional research does not have direct implications regarding whether market risk disclosures are useful for assessing the risk of individual firms. I believe it is likely that these disclosures are more useful for this purpose than the research shows, but less useful than they would be if firms reported them more coherently and consistently. In this regard, psychology-motivated research indicates that users of financial reports have difficulty using firm-level disclosures and are highly influenced by the reporting format. For example, Hodder et al. (2001) and Koonce et al. (2005) argue that the limitations of market risk disclosures are exacerbated by various well-known cognitive biases regarding risk evaluation and that these biases interact with disclosure alternatives. Koonce et al. (2005) and Nelson and Rupar (2011) provide experimental evidence that fairly subtle changes in format, e.g. Nelson and Rupar

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(2011) examine disclosures expressed in dollars versus percentages, significantly influence individuals’ interpretation of market risk disclosures. Empirical research also examines market risk disclosures provided under rules other than FRR 48, often disclosures about firms’ use of derivatives. For example, Wong (2000) provides inconsistent evidence (depending on the year examined) that the notional amounts of exchange-rate derivatives disclosed under FAS 11951 are relevant for the assessment of exchange rate sensitivity. Ahmed et al. (2011) provide evidence that banks’ disclosures of their derivatives use became more risk-relevant after the issuance of FAS 133. This standard expanded derivative disclosures, particularly regarding hedge ineffectiveness. They find that interest rate derivatives used for hedging are more negatively associated with bond spreads after FAS 133, consistent with the standard increasing bond investors’ confidence in the effectiveness of banks’ hedging activity. 6.2 Credit risk Aside from research employing disclosures related to banks’ loans that I do not discuss and of securitisation retained interests that I discuss in Section 6.4, I am unaware of any research that uses credit risk disclosures to explain any measure of firm risk.52 For example, to my knowledge, no empirical study has examined the risk-relevance of concentration of credit risk disclosures. Bhat (2009) comes closest to examining this question. She finds that fair value estimates are more value-relevant for banks with more extensive credit risk disclosures. I briefly describe two studies that suggest banks’ disclosures of their credit risk modelling exhibit general usefulness. While these papers do not examine risk-relevance per se, their findings suggest that disclosures of credit risk modelling are likely to be risk-relevant. First, Bhat et al. (2012) provide evidence that banks with better historical credit risk modelling disclosures (estimation of credit loss parameters based on current loan status and underwriting criteria) on average record timelier provisions for loan losses across the business cycle and did so late in the financial crisis once credit loss parameters had stabilised. They also show that banks with better forward-looking credit risk modelling disclosures (stress testing) recorded timelier provisions for loan losses when credit loss parameters changed sharply at the beginning of the recent financial crisis. Second, Bhat et al. (2011, 2012) provide evidence that banks with better credit risk modelling disclosures make less procyclical53 and higher-quality loan originations, respectively. Specifically, Bhat et al. (2012) provide evidence that banks with better credit risk modelling disclosures rely less on summary underwriting criteria in accruing for loan losses and, as a consequence of their better ability to assess credit losses, make less procyclical loan originations. They find that the mitigating effects of historical credit risk modelling disclosures on loan origination procyclicality are observed for total loans and for homogeneous consumer loans in particular. They find that the mitigating effects of forward-looking credit risk modelling disclosures are observed only for heterogeneous commercial and industrial loans. Bhat et al. (2011) provide evidence that banks that more fully disclosed their credit risk modelling were better able to sell mortgages during the financial crisis and had fewer non-performing mortgages. The results of Bhat et al. (2011, 2012) indicate that credit risk modelling disclosures are associated with higher loan quality. These findings work against the claim that modelling disclosures are net costly to banks; any cost of producing these disclosures is offset by the benefits to the firm and society of higher quality loan originations. 6.3

Liquidity and information risks

Most accounting research on liquidity and information risks examines the association of proxies for those risks with measures of financial reporting quality (mostly commonly, measures of

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earnings smoothness or persistence), rather than measures of firm risk (see Lang and Maffett 2011, Ng 2011 for recent examples). Yu (2005) constitutes somewhat of an exception, however, showing that financial reporting transparency as measured by Association for Investment Management and Research disclosure rankings (which are affected by footnote disclosure quality) tend to have lower credit spreads, particularly for short-term bonds. Yu interprets these results as higher disclosure quality reducing information risk. Linsmeier et al. (2002) show that after firms began providing market risk disclosures under FRR 48, the sensitivity of their trading volume to changes in the relevant market prices declined. This finding is consistent with market risk disclosures reducing informational differences across traders and thus information risk. Two recent studies examine firms’ disclosures of the amounts of assets and liabilities that are measured at fair value using FAS 157’s three levels of inputs. First, Lev and Zhou (2009) argue that these levels indicate liquidity risk, with Level 1 being the least risky and Level 3 the riskiest. Consistent with this argument, they find that, for economic events during the recent financial crisis that impaired market liquidity, the share price reaction is most negative for financial institutions with Level 3 items, followed by Level 2 items and then Level 1 items. The reverse holds for events that improved market liquidity. Second, Riedl and Serafeim (2011) argue that the three FAS 157 levels indicate information risk, with Level 1 again being the least risky and Level 3 the riskiest. They provide evidence that firms with greater exposure to Level 3 financial assets exhibit higher betas than those designated as Level 1 or Level 2, with these differences being more pronounced for firms with poorer information environments. 6.4 Estimation risk I am unaware of any empirical research directly pertaining to the risk-relevance of estimation risk disclosures under SOP 94-6 or any other disclosure rule. 6.5

Securitisations

Empirical research examining the risk-relevance of securitisation footnote disclosures spans the market, credit, and information risks discussed in Sections 6.1– 6.3. This research employs disclosures of the characteristics, volume, and retained interests in securitisations accounted for as sales under FAS 140 and 166, as well as similar disclosures in banks’ regulatory filings. For example, Chen et al. (2008) identify the amounts of retained interests by type of assets securitised (distinguishing mortgages from consumer loans from commercial loans) and the riskiness of the interests (distinguishing less risky contractual subordinated interests from more risky contractual credit-enhancing interest-only strips as well as contractual interests from non-contractual implicit recourse). Chen et al. provide evidence that these retained interests are positively associated with banks’ systematic risk (beta) and total risk (return variance), with the magnitude of the associations rising with the credit risks of the securitised loans and the retained interests. The results of Chen et al. (2008) could reflect securitising banks’ retention of credit or any other risks of the securitised loans (e.g. prepayment risk for fixed-rate mortgages). Barth et al. (2012) focus on credit risk, and provide evidence that disclosures of retained interests are positively associated with banks’ credit risk as measured by their credit ratings and bond spreads. Barth et al. also find that banks’ sold interests are risk-relevant with respect to bond spreads (though not credit ratings), consistent with the bond market treating securitisations accounted for as sales as economic secured borrowings. Focusing instead on information risk, Cheng et al. (2011) provide evidence that banks that securitise loans for which it is more difficult to evaluate the degree of risk transfer, as proxied

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by the volume of securitisations, credit riskiness of the assets securitised, and the amount of retained interests, face greater information risk as measured by larger bid – ask spreads and analyst forecast dispersion. Oz (2012) provides evidence that these effects diminished after the January 2010 effective date of FAS 166 and FAS 167, due to the enhanced information provided by those standards. Overall, the research on securitisations obtains the strongest results of all the areas of risk research summarised in this section.54 While this likely reflects the economic importance of securitisations to the banks that engage in these transactions, I believe it also reflects the consistent tabular formatting and other features of these disclosures that enhance their useability. 7. Recommendations to improve risk reporting quality I first state my recommendations for how to enhance risk reporting quality through summary accounting numbers and related financial statement presentation. These recommendations are motivated by the research summarised in Sections 3 and 4. I then state my recommendations for how to enhance risk reporting quality through other financial report disclosures. These recommendations are motivated by the research summarised in Section 6. The research summarised in Sections 3 and 4 indicates that summary accounting numbers are highly risk-relevant. This research focuses on measures of income and book return on equity, possibly adjusted to include calculated unrecognised fair value gains and losses or to remove the normal rate of return on book value. Corresponding with this research, I limit my recommendations to summary measures of income. The CFB research summarised in Section 3 suggests that income is risk-relevant because it is sensitive to unexpected changes in cash flows but relatively insensitive to unexpected changes in costs of capital. The fair value research summarised in Section 4 suggests that unexpected changes in cost of capital, which have significant effects on fair value gains and losses, are risk-relevant. The most sensible interpretation of these literatures collectively is that the components of income that are driven by unexpected changes in cash flows versus unexpected changes in costs of capital are both risk-relevant, just differently and in ways that depend upon the risk assessment analysis involved. To preserve and enhance the risk-relevance of income, I forward the following two interrelated recommendations. First, firms should measure comprehensive income using an informationally rich measurement attribute such as fair value that incorporates both unexpected changes in cash flow and unexpected changes in costs of capital.55 Second, firms should present the components of comprehensive income in a financial statement (or, less desirably, through prominent footnote disclosure) to inform users of financial reports about the effects of changes in unexpected cash flows from the changes in costs of capital. This presentation would serve as a good platform for qualitative discussion by management of the events that occurred during the period, in particular, for distinguishing the events that management believes are specific to the business from those that are primarily macroeconomic and, relatedly, those that are permanent from those that transitory or cyclical. The research summarised in Section 6 indicates that other risk disclosures in financial reports are risk-relevant but less so than one would expect given the high-volume of and strong economic bases for those disclosures. The most sensible interpretation of these results is that these disclosures are less useful to investors than they should be due to their diffuse and non-comparable presentation in financial reports. To enhance the risk-relevance of these disclosures, I forward the following three recommendations. First, financial reporting policymakers should attempt to maximise the ties of other financial report disclosures with summary accounting numbers. Whatever approach they take to do this, financial reports must clearly indicate and explain past revisions of significant accrual

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estimates, so that users of financial reports are informed about the variance of those estimates and how that variance results from the economic and estimation risks that firms face. I made this recommendation in Ryan (1997) to little or no effect, but I remain convinced it is the single most important thing that financial reporting policymakers could do. Extending this prior recommendation, I recommend that financial reporting policymakers require firms to conduct back-tests of their significant accrual estimates and to disclose the results of those tests, indicating any observed trends in and identified drivers of revisions to those estimates. For example, a bank’s back-test at the end of 2006 of its reported allowance for loan losses at the end of 2005 would examine how well its 2005 estimates of loan transition probabilities (e.g. from current to delinquent to default statuses) and losses given default performed over the following year. Such back-tests would reveal the trends in accrual estimates as soon as they become observable to management. In my opinion, back-tests constitute the best possible platform for management to discuss their current understanding of the risks that face their businesses. Koonce et al. (2010) provide experimental evidence that these disclosures should be made consistently across firms, because users of financial reports need comparable firm benchmarks to effectively distinguish environmental factors from firm-specific factors. By extension, the availability of comparable firm benchmarks should also help users to distinguish economic risks from estimation risks. In the same vein, a large literature in judgment and decisionmaking (see, for example, Yates 1990, Chapter 4) provides evidence that disclosures used to facilitate prediction should be made frequently (in financial reporting this means quarterly), because decision-makers need regular feedback to hone their prediction models. No doubt some will deem this recommendation too costly or proprietary to reporting firms. I disagree. If firms are not already conducting back-tests of significant accrual estimates for their own internal decision-making, then they should be, and for them the net cost of this recommendation is probably negative. For example, in my experience banks that conducted back-tests of their allowances for loan losses prior to the financial crisis were in informationally stronger positions to react to the crisis when it occurred. If a firm’s competitive advantage is dependent on it not disclosing high aggregated financial information about the unexpected consequences of past decisions, then in my view it does not have such an advantage. Second, policymakers should encourage and, to the extent feasible, require firms to aggregate and present risk disclosures in tabular or other well-structured formats that promote the usability of the information. Identifying and propagating the use of existing best practices and encouraging new best practices is the most natural way to do this. I would direct readers to two specific examples of actual best-practice tabular format disclosures. First, JPMorgan Chase in its 2010 Form 10-K filing,56 provides tabular format disclosures of its exposures to credit losses in the various portions of its highly diverse portfolio (e.g. loans, loan commitments, off-balance sheet securitisations, and credit derivatives). Second, Countrywide Financial,57 prior to its acquisition by Bank of America, provided tabular format disclosures of the sensitivities of its various on- and off-balance sheet exposures to two upward and two downward parallel shifts in yield curves, aligning the economic hedged items and hedges. I believe that securities’ regulators, such as the SEC, are in the best position to promote such best practices. The SEC is charged with monitoring firms’ financial reporting on a regular basis, plus it has the huge hammer of potential rejection of financial report filings with which to influence firms’ reporting behaviour. Third, for model-dependent risk disclosures (e.g. market risk sensitivities or VaR), policymakers should encourage and, if feasible, require disclosures of the primary historical and forward-looking attributes of the models, the essential decisions made in implementing the models in practice, the sensitivities of model outputs to alternative decisions, and benchmarking of the models to standard portfolios of exposures. Absent such explanatory and comparability-

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enhancing disclosures, my belief is these disclosures are almost useless to users of financial reports. 8. Concluding remarks As explained in detail in ICAEW Financial Reporting Faculty (2011), the current widespread desire to improve firms’ risk reporting quality stems in no small part from the financial crisis. Financial institutions’ woeful performance during the crisis and investors’ inability to perceive the potential for such performance before the crisis hit has led many accounting observers to conclude that these institutions’ risk reporting must have been inadequate. While I do not dispute this conclusion, it obscures two fundamental facts regarding the nature of business and the feasible goals of risk reporting. First, firms face business risks that are significantly unknowable ex ante and that evolve dynamically and behaviourally ex post. Second, while firms can and must attempt to manage these risks, risk management is inherently limited both ex ante and ex post. Liquidity risk exemplifies these facts, particularly for banks, whose business models inherently involve assuming this risk. For banks to eliminate liquidity risk entirely, if it were possible, would be for them to cease to be banks. Given these facts, the goals of risk reporting cannot be to inform users of financial reports about events that no one can anticipate to any significant degree. These goals must be feasible, indeed, humbler. One feasible goal is for firms to explain what they currently believe about variation in their future performance based on current information, and how those beliefs have changed during the reporting period as a result of new information. A second feasible goal is for firms to indicate the assumptions upon which their current beliefs are based, how those assumptions yield those beliefs, and why those assumptions may turn out to be incorrect. A third feasible goal is for firms to explain how and why their prior assumptions turned out to be incorrect ex post. I intend the recommendations forwarded in this paper to help financial reporting policymakers develop risk disclosure requirements that advance the practice of financial reporting towards these feasible yet still difficult-to-attain goals. I also hope to encourage reporting firms to provide open and informative discussion of their risks in their financial reports. Acknowledgements I prepared this paper for the 19 –20 December 2011 ICAEW Information for Better Markets Conference. I appreciate useful comments from Gauri Bhat, Leslie Hodder, Lisa Koonce, Jim Ohlson, and Dushyant Vyas. The comment from an anonymous reviewer is also appreciated.

Notes 1.

2.

3.

In a number of recent US standards – for example, FAS 166, Statement of Financial Accounting Standards No. 166, Accounting for Transfers and Servicing of Financial Assets and Extinguishments of Liabilities, June 2009, paragraph 16-A (ASC 860-20-50) – the FASB states disclosure requirements in terms of general objectives before requiring specific disclosures. I believe this is a good idea, although it remains to be seen how great an impact it will have on firms’ disclosures. To illustrate, assume a firm holds only two exposures, whose economic performances are denoted A and B, so that firm’s performance is A+B. The variance of the firm’s performance equals the variance of A plus the variance of B plus two times the covariance of A and B. Perfect offsetting occurs (i.e. the variance of the firms’ performance is zero) if the covariance of A and B equals minus the average of the variances of A and B. For reasons of space, I do not attempt in this paper to evaluate individual research studies based on the specification of the regression models or the statistical power of the research designs. I personally prefer studies that more fully incorporate accounting structure and economic or decision-making context into their models and designs.

Accounting and Business Research 4. 5. 6. 7.

8.

9. 10. 11. 12.

13. 14. 15.

16. 17.

18.

19. 20. 21. 22.

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Libby et al. (2006) conduct an experimental study demonstrating auditors’ lesser concerns about disclosed than recognised information. For simplicity, throughout the paper I use the term ‘cost of capital’ to refer to any discount rate (e.g. for any financial instrument held by the firm), not just the discount rates for the firm’s debt or equity capital. See, for example, Penman (2010) and Pope (2010). From accountants’ perspective, the term ‘CFB’ is unfortunate, as its estimation involves accrual accounting numbers. A more accurate term would be ‘accounting beta’; in fact, this term is used in the empirical literature summarised in Ryan (1997) and Section 2. Despite this, I use the former term in conformance with the usage in the recent literature. I do not dispute that more and less risk-relevant accounting measurement attributes exist in specific economic or decision-making contexts. For example, long-standing literatures in accounting argue and provide evidence that: (1) the conservative deferral of revenue and gains is a prudent response to uncertainty about the realisation of income (Penman 2011) and (2) that conservative accounting is associated with lower bankruptcy risk (Biddle et al. 2011). My point is that a measurement attribute that yields highly risk-relevant financial report information in one context may not in another. For example, Penman explains that conservatism can yield earnings growth that is ‘not to be paid for,’ i.e. that should not increase valuation multiples for earnings. I direct the reader to ICAEW Financial Reporting Faculty (2011) for a broad-ranging discussion of the issues related to risk reporting quality. See Campbell et al. (2011) and Kravet and Muslu (2011) for evidence that firms that provide longer lists or expanded discussion of risk factors in financial reports exhibit higher risk, as proxied by return variance, beta, and bid–ask spread. See Beyer et al. (2010) for a recent survey of research on voluntary disclosures. It is unclear to me whether this finding would remain if researchers used more recent data and approaches to estimate accounting beta. As discussed in Section 3, Cohen et al. (2009) find that CFB has strong predictive power for standard beta, although they do not run a horse race assessing the relative predictive powers of CFB and earnings variance for standard beta. Statement of Financial Accounting Standards No. 151, Inventory Costs – an amendment of ARB No. 43, Chapter 4, November 2004. As a certifiable accounting dinosaur, I refer to US GAAP standards that predate the Accounting Standards Codification (ASC) by their original type of standard and number designation, as with FAS 151 here. I indicate the relevant portion of the ASC the first time I refer to a standard. On the other hand, the research on implied costs of equity capital and CFBs deemphasises the distinctions among operating risk, operating leverage, and financial leverage made in the earlier research, although some accounting-based valuations models distinguish net operating assets (which are associated with operating risk and operating leverage) from net financial liabilities (which is associated with financial leverage). To the best of my knowledge, no extant accounting-based valuation models distinguish operating risk from operating leverage. See Preinreich (1941), footnote 14. In addition, RIM has been used to develop trading strategies based on the difference between modelled (based on accounting numbers) and observed market values of equity. The risk-assessment and trading strategy literatures appear to largely explain the same phenomena; in particular, the results of Cohen et al. (2009) and Nekrasov and Shroff (2009) strongly suggest that the estimated abnormal returns generated by the RIM-derived trading strategies can be significantly explained by RIM-derived CFBs. RIM may also be derived from the discounted free cash flow valuation model using both the clean surplus relation and the cash conservation relation (i.e. free cash flow equals operating cash flow minus each of capital expenditures, cash dividends, and principal and interest payments to debtholders). See Ohlson (1995) for the derivation of this version of RIM. Easton (2007) provides a comprehensive discussion of the conceptual and practical issues involved in ICOC estimation. In particular, he discusses the fundamental issue of the dependence of estimates of ICOC on assumed growth rates. Admittedly, this is a rather low bar for success given the dismal performance of standard finance approaches to estimating the cost of equity capital to date. The market discount factor mt,t+s reflects investors’ state-dependent willingness to defer consumption from period t to period t+s. The expectation in period t of mt,t+s – which is probability-weighted

320

23. 24.

25. 26. 27. 28.

29.

30. 31. 32. 33. 34. 35.

36. 37. 38.

39.

40. 41.

S.G. Ryan across all possible future states that might occur in period t+s – equals 1/(1+r tf,t+s). Intuitively, a security that pays a constant amount in each possible future state must yield the risk-free rate. More negative covariances with mt,t+s indicate greater systematic risk because mt,t+s is inversely related to expected consumption in period t+s. More positive covariances with consumption indicate greater systematic risk. Cohen et al.’s (2009) approach to estimating CFB is somewhat less directly motivated by RIM than is Nekrasov and Shroff (2009)’s approach. Cohen et al. estimate CFB by correlating time-aggregated ROE with time-aggregated market ROE. They allow the time-aggregation horizon to vary from 1 to 15 years, under the idea that a longer horizon allows errors in the measurement of ROE to be more fully mitigated. Roughly speaking, this yields CFBs that reflect a weighted-average of the covariances in Equation (2) across the time-aggregation horizon. I do not describe a related and far larger body of ‘value-relevance’ research that examines the explanatory power of accounting numbers for the market value of equity or share returns. Statement of Financial Accounting Standards No. 107, Disclosures about Fair Value of Financial Instruments, December 1991. Francis (1990), Barth et al. (1995), and Bernard et al. (1995) provide evidence that fair value accounting for a portion of firms’ economic sheets yields more variable owners’ equity and net income than does amortised cost accounting, and that this variance is only partly reflective of economic variance. International Financial Reporting Standard No. 7, Financial Instruments: Disclosures, August 2005 (but frequently amended), paragraphs 25 –27, requires similar disclosures. Hodder et al. (2006) directly estimate gains and losses on non-term deposits, an important financial instrument for banks, because FAS 107 does not require banks to disclose the fair values of these deposits. I believe that Hodder et al.’s (2006) findings regarding the risk-relevance of fair value gains and losses likely generalise to firms’ revisions of accrual estimates generally. For example, Petroni et al. (2000) provide evidence that discretionary revisions of loss reserves by property-casualty insurers are positively associated with the insurers’ beta and return variance. Even if the fair value option had existed then, as it does now under Statement of Financial Accounting Standards No. 159, February 2007, relatively few firms have elected that option to date. Statement of Financial Accounting Standards No. 157, Fair Value Measurements, September 2006. Most SEC disclosure rules are not included in the ASC. International Financial Reporting Standard No. 7, Financial Instruments: Disclosures, August 2005 (frequently amended). I do not discuss operational risk in this paper, despite considerable disclosure of this risk in banks’ financial reports. This risk is outside my expertise, has a quite different nature from the risks I discuss, and is not subject to any research related to the themes in this paper of which I am aware. Financial Reporting Release (FRR) 48, Disclosure of Accounting Policies for Derivative Financial Instruments and Derivative Commodity Instruments and Disclosure of Quantitative and Qualitative Information about Market Risk Inherent in Derivative Financial Instruments, Other Financial Instruments, and Derivative Commodity Instruments (1997), 17 CFR 229.305. Statement of Financial Accounting Standards No. 161, Disclosures About Derivative Instruments and Hedging Activities – An Amendment of FASB Statement 133, March 2008. Statement of Financial Accounting Standards No. 133, Accounting for Derivative Instruments and Hedging Activities, June 1998. Statement of Financial Accounting Standards No. 140, Accounting for Transfers and Servicing of Financial Assets and Extinguishments of Liabilities, September 2000; Statement of Financial Accounting Standards No. 166, Accounting for Transfers and Servicing of Financial Assets and Extinguishments of Liabilities, June 2009. For example, mortgage banks’ fixed-rate mortgage servicing rights are sensitive to interest rates through their direct effect on discounting and indirect effect on prepayment. Ideally, mortgage banks’ disclosed sensitivities to interest rates would reflect both effects. Instead, their disclosed sensitivity to interest rates reflects only the discounting effect and they make separate disclosures of prepayment sensitivity. American Institute of Certified Public Accountants (AICPA) Statement of Position 94-6, Disclosure of Certain Significant Risks and Uncertainties, December 1994. In addition, FASB Staff Position SOP 94-6-1, Terms of Loan Products That May Give Rise to a Concentration of Credit Risk, November 2005 (ASC 825-10-55) indicates that non-traditional mortgage products may constitute a credit risk concentration. (This FSP is misdesignated; it primarily provides guidance for FAS 107, not SOP 94-6, because mortgages are financial instruments.)

Accounting and Business Research 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

53. 54. 55.

56. 57.

321

Statement of Financial Accounting Standards No. 5, Accounting for Contingencies, March 1975. In my experience, financial institutions evidence little or no compliance with this disclosure requirement. International Accounting Standard No. 37, Provisions, Contingent Liabilities, and Contingent Assets, September 1998. SEC, Securities Act Industry Guide 3, Statistical Disclosure by Bank Holding Companies, http://sec. gov/about/forms/industryguides.pdf. Statement of Financial Accounting Standards No. 118, Accounting by Creditors for Impairment of a Loan-Income Recognition and Disclosures – An Amendment of FASB Statement No. 114, October 1994. FASB Staff Position FAS 133-1 and FIN 45-4, Disclosures About Credit Derivatives and Certain Guarantees: An Amendment of FASB Statement No. 133 and FASB Interpretation No. 45; and Clarification of the Effective Date of FASB Statement No. 161, September 2008. International Financial Reporting Standard No. 7, paragraphs 27A-B, requires similar disclosures. SEC Regulation S-K, Management’s Discussion and Analysis of Financial Condition and Results of Operations 17 CFR 229.303. A related literature examines the value-relevance of banks’ maturity or repricing gap disclosures, e.g. Flannery and James (1984), Schrand (1997), and Ahmed et al. (2004). Statement of Financial Accounting Standards No. 119, Disclosure About Derivative Financial Instruments and Fair Value of Financial Instruments, October 1994. This standard was superceded by FAS 133. This dearth of research may reflect the difficulty of proxying for banks’ credit risk using their financial report disclosures, as discussed in Knaup and Wagnerz (2009). Banks have diverse loan portfolios and other credit risky exposures, including complex ones such as retained interests from securitisations and credit derivatives. Beatty and Liao (2011) provide evidence that banks that make timelier provisions extended more loans during the financial crisis. The results for commodity price risk discussed in Section 6.a are also strong but apply to fewer firms. As discussed in Ronen and Ryan (2010), when markets are illiquid, fair value is just one sensible measurement basis among several others with similar informational richness but different approaches to dealing with illiquidity. I am not wedded to fair value as it is currently defined, but I believe whatever measurement attribute is chosen should be applied consistently across exposures. See page 119 of Annual report 2010, JP Morgan Chase & Co, available at http://files.shareholder.com/ downloads/ONE/1750571523×0x458380/ab2612d5-3629-46c6-ad94-5fd3ac68d23b/2010_JPMC_ AnnualReport_.pdf (accessed 28 March 2012). See page 139 of Annual report 2007, Countrywide Financial Corporation, available at http://www.sec.gov/ Archives/edgar/data/25191/000104746908002104/a2182824z10-k.htm (accessed 28 March 2012).

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Journal of Risk Research Vol. 15, No. 9, October 2012, 1101–1116

Risk selection in the London political risk insurance market: the role of tacit knowledge, trust and heuristics Lijana Baublyte*, Martin Mullins and John Garvey The Department of Accounting and Finance, Kemmy Business School, University of Limerick, Limerick, Ireland (Received 23 March 2011; final version received 15 April 2012) This study demonstrates that the basis of decision-making and risk selection in the London Political Risk Insurance (PRI) market is a combination of Art and Science with such factors as trust and reputation playing an important role. The study breaks new ground by uncovering and examining different methods and strategies of political risk underwriting employed in the insurance market, which does not rely on statistical tools as seen in more traditional insurance types. Adopting a grounded theory approach, the data was generated through 14 semistructured and unstructured interviews conducted with PRI experts from five PRI companies and two leading political risk broking houses. The data also included documentation reviews and observations. Keywords: political risk; political risk insurance; risk selection; tacit knowledge; heuristics; grounded theory

1. Introduction Political risk insurance (PRI) is a relatively new business line with origins that can be traced back to the Marshall Plan in 1948. Multinational enterprises and banks use PRI to manage and mitigate risks arising from adverse host governments’ actions and inactions. The role of both public and private PRI programmes in promoting foreign direct investment (FDI) in developing countries has long been recognised by governments and investors. Nevertheless, there is a substantial gap in the risk and insurance literature regarding political risk underwriting. This article analyses the risk selection process in the London PRI market which, to date, has received little attention from researchers and regulators alike. The paper outlines the underwriting process and discusses issues faced by political risk underwriters. Overall, the paper demonstrates that tacit knowledge, heuristics and trust play an important role in risk selection. 2. Underwriting process in PRI Underwriting is the process of deciding what risks to accept and, if so, at what premium rates, terms and conditions (Bennett 2004). Political risks considered by insurers are diverse in nature and vary widely by geographical region and industry. *Corresponding author. Email: [email protected] ISSN 1366-9877 print/ISSN 1466-4461 online Ó 2012 Taylor & Francis http://dx.doi.org/10.1080/13669877.2012.705312 http://www.tandfonline.com

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Underwriters have to evaluate the degree of risk in each case and assign a particular risk category. Through classification, underwriters ensure that clients with similar probabilities of loss are placed in a distinct class and are charged a premium corresponding to the degree of risk they embody i.e. the higher the risk, the greater the premium rate. Risk classification is a standard procedure in conventional insurance business where underwriters have well established underwriting criteria and actuarial models to help them price individual risks. Political risk underwriting is more complex as it deals with rare and unique risks. Political risk is influenced by a number of political, economic, financial, social and cultural drivers where the strength of different drivers varies from country to country and from one potential insured to the other. There is no universal method of quantifying or evaluating political risks. It is a class of business which in many cases requires creative underwriting decisions and solutions. Underwriters frequently encounter unique and uncommon risks in the PRI market and must undertake decision in the absence of strong historical and statistical data. Even for the risks where limited historical data are available, the political environment surrounding the risk can change abruptly as a result of political turmoil, unforeseen host government’s actions or inactions and less obvious triggers like the extreme weather conditions which can force a change in political risk level of a particular country. The PRI market is a distinctive phenomenon in the insurance sector, political risks rarely satisfy the principles of insurability, that is; risks only can be insured when they are: (1) sufficiently separable (e.g. risks have to be random and independent); (2) dispersed and calculable (e.g. it is possible to statistically gauge the probability of event happening); (3) there is a large number of identical or similar risks to be covered at the same time; and (4) the value of a risk is not in excess of the insurer’s capacity (Coomber 2006). If all four conditions are satisfied, a risk is considered to be insurable and where one or more insurability conditions are violated the insurer may opt not to provide coverage against the risk as it can threaten its solvency or simply it would not yield desired return. In the field of risk management and insurance, political risk presents a set of distinctive challenges relating to the dynamic uncertainty associated with the political structures, the existence of interdependent risks due to inherent human nature, and the potential of catastrophic losses. In subsequent sections, we discuss how underwriters employ heuristics and tacit knowledge to assess the subjective probability of a particular political risk and/or visualise possible outcomes and scenarios in order to make an underwriting decision under uncertainty or incomplete information. This study contributes new insights into political risk perception and underwriting practices that take place within the London PRI market. The next section describes the methodology used to investigate the nature of PRI underwriting. Section four reports the grounded theory analysis results of the risk selection process in the London PRI market. Finally, the discussion and conclusion sections outline the contributions to theory and implications for practice. 3. Research methodology This study was carried out during 2009–2010 in five different PRI companies in both the Lloyd’s market, the London Company market and in two leading political

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risk broking houses.1 The London PRI market is a niche market where the total population of PRI community, including both PR underwriters and PR brokers does not exceed 200 individuals. Table 1 provides a summary of study participants and their area (some interviewees have an expertise on both political risk underwriting and broking sides). The London PRI market was selected as the venue for this research for its size and reputation as a leading market for PRI. The research approach applied here is based on grounded theory methodology (Glaser and Strauss 1967; Charmaz 1983; Strauss and Corbin 1990). This method is well suited to those domains that are under-theorised, in that it allows the researcher to identify the thinking/behaviour set that drives decision making in a particular field. Given the lack of integrated theory in the risk and insurance literature regarding PRI and political risk underwriting, an inductive approach that allows theory to emerge from empirical data/research seemed the most appropriate. Grounded theory research design is particularly suited for this study for a number of reasons. First, it has a set of established guidelines both for conducting research and for interpreting the data that can offer a sense of security when delving into unknown research territory which is the case for PRI underwriting. There have been no qualitative studies carried out in the field of PRI. Second, grounded theory is especially renowned for its application to the study of human behaviour. Human nature plays an increasingly important role in PRI market in two ways: political risk is essentially a socio-political phenomenon and also the London PRI market is a market where business activities are conducted in a close proximity which adds an extra human interdependence dimension to the underwriting process. Finally, it is a methodical approach to theory building in an under-theorised area. 3.1. Data collection For the purpose of this study, a variety of ‘engaged’ data gathering methods were employed. This involved semi-structured and unstructured interviews that were supplemented by documentation reviews, observations and informal discussions. Thirteen interviews were conducted each lasting an average of an hour. The sources were not selected randomly but were chosen carefully to ensure that they were true representatives of the research domain. Several features distinguish the informants Table 1. Number of interviews conducted at the London PRI market and the position of the interviewees. Participant Participant Participant Participant Participant Participant Participant Participant Participant Participant Participant Participant Participant Participant

Area of expertise 1 2 3 4 5 6 7 8 9 10 11 12 13

Underwriter Underwriter Junior Underwriter Risk Analyst Underwriter Junior Underwriter Underwriter Underwriter Broker Underwriter/Broker Underwriter/Broker Broker/Underwriter Underwriter

Number of interviews 1 1 1 1 1 1 1 1 2 1 1 1 1

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included in the study. They were active PRI market participants and had a variety of roles such as underwriters, brokers and risk analysts. They also represented a range of managerial levels. Political risk brokers were included in the study in order to reach theoretical saturation (Glaser and Strauss 1967) and to get an extra perspective on political risk underwriting. This allowed both PR underwriters and PR brokers to identify the key aspects and issues in the political risk underwriting process. The interviews were focused on understanding the political risk underwriting process from the perspective of underwriters and brokers and sought to illicit their perception of the changes in underwriting practices that have taken place over time. All interviews were taped, transcribed and subsequently analysed in accordance with the guidelines of grounded theory research methodology. Detailed notes were also taken during the interviews. In addition, some of the study participants agreed to provide documentation on policy wordings, internal presentations on political risk business and other relevant documents that were subsequently included into data analysis. Observations were carried out in Lloyd’s insurance market where underwriters agreed to be studied while doing business as normal at a Lloyd’s box. Such grounded theory studies are often used in system development, organisational culture, marketing, consumer behaviour and social sciences (Goulding 2002) as a method to explore and understand the research phenomenon within a particular context. In this paper, the grounded theory analysis is used to describe the political risk selection process within London PRI market.

3.2. Data analysis This study employed the techniques of grounded theory for constructing research design, collecting and analysing the qualitative data. Analysis involved a number of cycles of data collection, coding, analysis, writing, design and theoretical categorisation. The concepts that emerged from the empirical findings were constantly compared and contrasted. As new data were added and analysis progressed, some concepts were reorganised under different labels. The size of a sample was determined by the ‘theoretical saturation’ of categories i.e. the data collection process was carried out until the point where new cases yield no ‘additional information’ (Glaser and Strauss 1967; Strauss and Corbin 1990). Glaser and Strauss (1967, 45) define theoretical sampling as follows: A process of data collection for generating theory whereby the analyst jointly collects, codes, and analyses his data and decides what data to collect next and where to find them, in order to develop his theory as it emerges. This process of data collection is controlled by the emerging theory in other words rather than by the need for demographic ‘representativeness,’ or simply lack of ‘additional information’ from new cases.

The resulting theories were developed inductively from data rather than tested by data. Categories and concepts emerged from data and were not imposed a priori upon it. Table 2 provides an illustration of examples of categories and concepts as implied by the empirical findings. The categories and concepts helped to explain the process of risk selection in London PRI market. It is a pioneering study and no claim is made that categories and concepts are complete.

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4. Risk selection in PRI As the qualitative data analysis progressed, a number of common influencing factors emerged from analysis of the qualitative data. Participants identified explicit risk properties such as a host country and implicit factors like trust and reputation playing a key part in selecting risks for the portfolio of political risks. Four key categories and their underlying concepts and relationships provide good insights into the risk selection process in London PRI market (see Table 2). The following subsections describe our theoretical conceptualisation of the dynamic process of the political risk selection. 4.1. Country The first key category in risk selection process was a country. It had become very clear from the very start of the data collection and analysis phase that host country played a major role. All study participants uniformly agreed that it was one of the first factors that were taken into consideration by political risk underwriters. In addition to the fact that some countries are perceived as higher political risk and others lower political risk, country factor is also crucial to a risk selection process due to the diversification effects. Generally, underwriters have different levels of capacity allocated for particular host countries where underwriters are not supposed to exceed those country limits as it can result in imbalanced portfolios and in turn lead to insolvency in the face of political catastrophe. Consider the following: Participant 9: There is a fundamental problem of imbalance of demand in the PRI market: there is too much demand for capacity in countries like Russia, Turkey, and China, where the market participants often have to turn away good, well priced business because they are at or close to their country limits, and where further exposure would unbalance their portfolio. Nor could you really set up a specialist insurer or reinsurer just to write business in those capacity constrained countries because that (re)insurer would then itself have a very unbalanced book. It is quite difficult to organise an unbalance world into a balanced insurance book.

A host country category has a number of concepts which were identified through the grounded theory analysis process (see Table 2). Political risk underwriters when evaluating political risks look at properties like political stability, the economic situation and the legal environment of a host country in order to assess the relative degree of risk. 4.2. Client The second category of political risk selection criteria was the client. As in any other business line of insurance, prospective insured is a key consideration in the underwriting process. Moral hazard and adverse selection problems associated with insureds are well known in insurance markets and are also present in the London PRI market. Political risk underwriters seldom have direct contact with the risks they are considering. They do not usually interview potential insureds personally. They gather information from other sources, most commonly from brokers and various media sources. Client specific factors that play important role in the risk selection process are: the financial position where a higher financial status of a client signals a lower risk; overseas experience, is perceived as a positive feature if the

Implicit risk properties

Explicit risk properties

Heuristics

Tacit knowledge

Client

Country

Categories

Field data from interview notes: examples

Participant 5: Whether someone is a dictatorship, or democracy, a monarchy, has its own dynamics. But from our point of view, the thing we’re looking for is stability if we know it is bad but it is stably bad we can price it. If it is very volatile that is very difficult Legal Participant 1: What is the legal setup in the country? How easy is to defend against expropriation? environment Whether or not they are signed up to EXIM, New York convention, you know, arbitration and all that kind of thing. Which can be very important and it gives you an idea of the attitude of the country Economic Participant 4: We look at inflation, domestic interest rates. And then the external account - how do situation they get hard currency, how much do they have? We look at various ratios which shows sustainability of the private sector, public sector, is there currency free floating, is it open to a speculative attack, is it pegged what supports that pegged? That comes with the economics Company Participant 6: So insured are crucial both in terms of their experience and their financial strength. As financials we talked earlier if they are short of money that limits their options. It means it is harder for them to get out of the trouble or to deal with the problem proactively Intuition Participant 12: The actual process will vary from syndicate and company. Some people have very structured processes; some tend to do it in a more … It is more of judgemental way. They have gut feeling or instinct Trust Participant 13: There is trust. I mean you are underwriting as if you were a bank, a trader but you can’t see everything that your client sees. So there has to be trust between you and your insured and there has to be trust between you and your broker Memorability Participant 10: You never forget your basics – the world always finds an excuse why he should do something. And that is where you have to be careful. Argentina is a great example. Argentina has gone nowhere in the last 9 years. It hadn’t really dealt with its foreign debt at all and it dealt with it very badly and yet you have banks flooding in there again doing money. On what basis? Reputation Participant 3: From the contract frustration risk point of view, you know, not paying on your loan or whatever … Or not meeting oil delivery … Reputation is a massive factor

Political stability

Concepts: examples

Table 2. Risk selection in political risk insurance: categories, concepts and field data.

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prospective insured has been trading internationally or successfully operating its business in a foreign market for a number years; and social parameter where risk levels are believed to be lower if client makes a positive contribution to a local community or region in terms of philanthropy and other means. When a PR underwriter was asked to identify his risk selection criteria, he explained: Participant 7: First of all we look at the client, what do they do as an industry. What is their profile globally within that industry? Do they have a social perimeter? You know, do you build schools and medical facilities for local villages? Actually, a lot of clients are much more informed and educated on the importance of that side of things now then they have ever been. We want clients that are proactive on that side of things. Actually, being realistic about the risk they face … I like dealing with clients for whom insurance is the last thing they think about rather than the first. Because, if it is last thing that they think about, they have thought about how to mitigate it, how to protect … Certainly, for some of our clients you can’t prevent this risk but you can prepare and you can mitigate. You know, for a lot of companies if you avoid the risk, you avoid the opportunity.

In addition, the type of coverage demanded by a client has a role to play in the risk selection process. The underwriting policy may indicate the kinds of coverage a company writes and does not write. Data suggest that political risk insurers generally have a preference for trade insurance products (e.g. supplier or buyer credit business policies) to investment insurance products (e.g. confiscation, expropriation or currency inconvertibility policies). This is partially due to the fact that trade insurance policy wordings are standardised and better defined and so the ‘grey protection’ issue is reduced to a minimum.

4.3. Tacit knowledge During the grounded theory analysis, ‘tacit knowledge’ emerged as the third key research theme providing a conceptualisation for the risk selection process. Polanyi (1962, 1966, 1976) was the first to introduce the tacit knowledge concept which he describes as follows: ‘I shall reconsider human knowledge by starting from the fact that we can know more than we can tell’ (1966, 4) or we have a ‘power to know more than we can tell’ (1976, 336). Tacit knowledge has several characteristics. First, it is difficult to write down or indeed to formalise (Nonaka 1991). People that own tacit knowledge cannot articulate or explain the rules that underlie their decisions and might refer to it as an intuition or gut feeling. Second, tacit knowledge is personal knowledge i.e. it consists of mental models that an individual follows in certain situations which are deeply ingrained that they tend to be taken for granted (Nonaka 1991; Sternberg 1994; Ambrosini and Bowman 2001). Third, it is context specific as Sternberg (1994, 28) describes it ‘is a knowledge typically acquired on the job or in the situation where it is used.’ Finally, Nonaka (1991) argues that tacit knowledge can be a synonym of ‘know-how’ and vice versa. In the words of Nonaka (1991, 98) ‘tacit knowledge consists partly of technical skills – the kind of informal, hard-to-pin down skills captured in the term “know-how.’” ‘Tacit’ skills are best understood as a continuum of different degrees of tacitness (Ambrosini and Bowman 2001). Where on one end you have ‘explicit skills’ that can be articulated, shared and codified and on the other you have tacit skills that are not accessible to the knowers because they are deeply embedded e.g. taken for

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granted truths. In between two extremes one can find at least two other degrees of tacitness: (a) tacit skills that can be imperfectly articulated through the use of metaphors and storytelling; and (b) tacit skills that could be articulated if the organisational members were simply asked the question: how do you do that? (Berry 1987) As data analysis progressed, some of the tacit knowledge and skills were uncovered and reorganised under different labels like trust and reputation. However, some elements of tacitness remain within the risk selection process that falls under the category of intuition. 4.4. Intuition When political risk underwriters were asked to describe a risk selection process they said that in a number of cases it was based on their instinct, sixth sense, gut-feeling or simply how comfortable they were with a risk. As one underwriter explained: Participant 7: There is no matrix. Which in some ways is good, because the next step from there is having a computer to do my job. But actually the personal sort of being objective about the risk being written, being realistic. A lot of this is a gut feel. Actually the more informed you are the harder you work to make sure you know what is going on. But at the end of a day the humans have evolved. Sort of gut feel – the sixth sense – when some things aren’t right.

Political risk brokers were also asked to describe a risk selection process and to identify what they thought were important properties of a political risk that would help them to persuade underwriters to take on a risk. Among a number of explicit risk factors like the potential insured, host country and coverage brokers also said that in some cases underwriters would base their decision partially or solely on their instincts. As one PR broker explained that: Participant 12: Generally who is the potential client, potential insured … Underwriters may also pay some passing attention to the broker because if they don’t know the insured but they know that the broker has a good reputation – so probably he made the right checks. And then there is bigger picture what is the country, what is the risk itself. Often an underwriter will not want to deal with some companies because maybe they had bad experience with them in the past. But the actual process will vary from syndicate and company. Some people have very structured processes and some tend to do it in a more … It is more of judgemental way … They have gut feeling or instinct.

Intuition as well as gut-feeling, instinct and hunches cannot be passed over to others by those involved in the specific situation, as in this case of political risk underwriting, and therefore can be considered intangible form of information (McAdam, Manson, and McCrory 2007). A number of researchers argue that tacit knowledge plays a key role in organisation growth and economic competitiveness (Howells 1996; Brown and Duguid 1998; Kreiner 2002). 4.5. Trust Trust plays a vital role in risk selection process in the PRI market. This finding is in line with a number of studies that see trust as one of the key success factors in business (Macalay 1963; Glover 1994). Economists define trust as ‘implicit contracting’ where a firm or an individual trust the other firm or individual to fulfil its

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promises and obligations (Zucker 1986). Axelrod (1984) argues that trust can be earned through repeated games like the prisoner’s dilemma which create the effects of learning, communication and the ‘shadow-of-the-future’ i.e. the possibility of future transaction which in turn encourages a cooperation between the two parties if it pays them to do so. Thus, trust can be seen as a reaction to expected future behaviour (Blomqvist 1997). Political risk underwriters, generally, feel more comfortable accepting a risk if they can trust a broker; where trust is earned through shared experiences in both the profession and the social realms. Moreover, underwriters try to develop and maintain long-term relationships with their favoured brokers with an aspiration to earn their loyalty i.e. ‘shadow-of-the-future’ effect. A PR underwriter explains when asked if he would deal with a broker with whom he has no previous work experience: Participant 5: No with our core brokers. Because, you know, we know them very well and they know us on social and professional basis. And we understand the risk and they understand the risk. You know, they have to represent their clients but you still can have a straightforward conversation … And I suppose, at the end of a day, you have brokers you trust and you have brokers you don’t.

Trust can also be seen as a function of imperfect information (e.g. see Lewis and Weigert 1985; Oakes 1990). Blomqvist (1997, 272) borrows Simmel’s words: ‘the person who knows completely need not trust; while the person who knows nothing, can no on rational grounds afford even confidence’. In other words, under perfect information there would be no need for trust as decisions would be made based on rational calculations as there would be no adverse selection problem. Thus, it can be argued that trust is used as a tool to manage adverse selection problem in the London PRI market. The importance of interpersonal relations in the PRI market was also raised by PR brokers: Participant 12: The broker itself has a role to play in the established relationships with market and persuading them to do business. Knowing their appetites. Basically, you know, if you find that the underwriter is very difficult, is not very helpful, the broker tends to say ‘well I’m not going use that underwriter.’ So the underwriters themselves if they want to be the players in the market and get to the good clients – have to have a good humour, be reasonably friendly.

It can be argued, that it is not only a risk itself is judged by a PR underwriter but also a broker himself has an important role to play in regards to shaping underwriter’s risk perception and selection process. From the underwriter’s point of view, a risk can be perceived as more acceptable if presented to an underwriter by a broker with whom he or she has a successful underwriter–broker relation that is built on trust which is a function of underwriter’s expectation of broker’s goodwill and competence. This finding is in line with Lundvall’s (1990) argument that states that in imperfect competitive markets, in which a small number of traders build long-term relationships and make relation-specific investments, trust is a significant factor. 4.6. Heuristics and memorability The fourth category of the risk selection process is heuristics. Political risk underwriters more than often have to make an underwriting decision under uncertainty or

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incomplete information. The risks being underwritten in the PRI market are usually unique in their nature where there is no historical data available on the loss experience. In the case of uncertainty surrounding a particular risk, underwriters tend to employ memorability, imaginability and similarity as cues for subjective probability which is in line with Tversky and Kahneman’s (1973) theory of availability heuristics. They argue that people often rely on heuristics to reduce complex tasks of predicting values and assessing probabilities. PR underwriters assess the subjective probability of a possible loss by the ease with which examples or occurrences of similar event can be brought to mind or envisaged, according to the data analysis. Thus, it can be argued that PR underwriters judge a particular risk to be higher if the instances of a loss from a similar risk can be recalled or imagined. This is not to be confused with the experience rating in insurance, which incorporates the history of the policyholder in the ratemaking process to arrive at a fair premium price, even though the connection between the two could be made (Pinquet 2000). Participant 1: Well you can use your knowledge of previous losses and why they happened I suppose. Or a political risk underwriter will do that. Participant 10: Well it does but, also most importantly, I think the reason you never forget your basics is – the world always find an excuse why he should do something. And that is where you have to be careful. Argentina is a great example. Argentina has gone nowhere in the last 9 years. It hadn’t really dealt with its foreign debt at all and it dealt with it very badly and yet you have banks flooding in there again doing money. On what basis? You know, the economy is awful. But they [foreign investors] somehow found an excuse why it is a good idea to start lending back into Argentina again.

Our data also suggests that political risk underwriters are heavy users of media and resource agencies. PR insurers gather information from credit rating agencies, intelligence agencies, journals, news watch services, consultancy companies and international governing bodies which is then used for ranking the political risks or assessing the probability of unique PR risks. Study participants identified the external information sources they were using to analyse the level of political risk: Participant 2: We use the credit agencies on a country level basis, solvency ratings and then we use internal Company X’s database for political risk. Then we use consultancy companies and intelligences agencies. Participant 4: Well our own view we have been around for a while and with the a lot of information now being fed through by NGOs and all sorts of kinds of agencies we think there is a way we might be able to separating those factors and forecast a little bit better [refers to PR rating]. Participant 6: We do. We do use all sorts of credit agencies, news watch services, etc.

Both intentional and non-intentional use of media can have an impact on risk perception. PR insurers are intense users of media and resource agencies which can introduce some degree of risk selection bias. Linchtenstein et al. (1978) study has showed that media coverage caused people were more inclined to overestimate the frequency of highly publicised causes of death like homicides, tornadoes and cancer and under

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publicised causes like diabetes, stroke and asthma were underestimated. Certain host countries that attract more negative publicity are perceived to belong to a category of higher political risk, everything else held constant, and this may or may not be a true representative of a real risk. Power (2004) states that the media is an important source of amplification, which is in line with Combs and Slovic’s (1979) argument that frequent media exposure gives rise to a high level of perceived risk. Furthermore, Mazur and Lee (1993) conclude that it is not the content that influences people’s beliefs, but the sheer amount of coverage. According to the data analysis, media and resource agencies do influence PR underwriters’ perception. How strong this influence is – and what its properties are – outside the scope of this study. Suffice to say here that memorability and media content are intimately related. 4.7. Reputation Reputation is intimately related to heuristics. All study participants identified client reputation as a major determinant in the risk selection process. From the data analysis, it is apparent that if the potential insured has a long history of successful cross border trading, a record of doing business in the foreign markets, and is also a well known/established entity it will be perceived by the political risk underwriters as less susceptible to a loss. PR underwriters argued: Participant 2: From contract frustration risk point of view, you know, not paying on your loan or whatever … Or not meeting oil delivery … The reputation is a massive factor. Participant 3: And I think, you know, when those risks were actually written most people just said yeah that is risk but this company has been going for 50 years it has a fantastic reputation it’s a company that, you know, trading with Western Nations for years with no difficulties culturally. So, you know, there was no question whatever or not they would default on their obligation … And I think that may have encouraged people to put that risk in a positive light and maybe turned a blind eye to certain lack of disclosure or, you know, financials that kind of thing.

A number of researchers consider a reputation as emerging over time from observed performance about exogenous characteristics of agents like borrower’s repayment record and accounting information (Kreps and Wilson 1982; Milgrom and Roberts 1982; Diamond 1984). In the PRI market the exogenous characteristics of agents can take such forms as trading record and payment history, if potential insured participates in philanthropic activities or have a well-established brand name. Consider the following: Participant 10: Depends on how well the name is known. Because with a well known name on short term basis you can make a decision very quickly. If it is not – we have to do work and we have to do work. And it will take time. Participant 7: Okay a key one is who is the client? We truly believe … The breath of business that we do I think good clients are good risks from my point of view even in the most difficult parts of the world.

Client’s reputation is a significant factor that influences and shapes underwriters’ perception. The issue of reputation manifests itself throughout the data in a

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number of different contexts. Reputation effects in insurance markets are underresearched. To date, most studies have primarily focused on the insurer’s reputation and how it influences client’s confidence and a choice of coverage provider (e.g. Gaultier-Gaillard and Louisot 2006; Zboron 2006). There is even less research done on how insured’s reputation impacts on risk perception, selection and pricing in the specialist insurance markets. Quinn (1998) argues that the presence of a potential reputation loss lessens the moral hazard problem in the medical malpractice insurance market which in turn allows insurance companies to adopt community rating without fear that the physician will behave in a more risky way. According to the grounded theory analysis, an interrelationship between the client’s reputation and a moral hazard problem coexists in the London PRI market. As one broker argued: Participant 9: Actually, one of the things that underwriters of investment insurance are worried about is moral hazard: how will the insured behave in a potential loss situation if their business is not doing well commercially. I am not suggesting that they would that easy to organise an expropriation of the struggling business; but some of the coverage we have in the policy could introduce real moral hazard. For example, Forced Abandonment insures you when you cannot operate a business due to political violence, and the trigger is ‘abandonment’ with no need for the political violence to have caused physical damaged to the businesses property. The moral hazard arises because if you have an investment which is not making money, then if there is civil unrest or a little local war, the investor could see this as an opportunity to close the business down and claim the net investment from the policy, even though, in reality, the political violence did not necessitate the abandonment.

Diamond (1991) argues that reputation alone can eventually reduce the moral hazard issue because a better reputation achieved over time would lead to less severe adverse selection. In addition, there is an incentive for a borrower to maintain his or hers good reputation as it provides a better access to lower cost funds. This is in line with data analysis results that suggest that reputation is a significant factor in risk selection process especially when the risks underwriters are dealing with have no historical record. The PRI market is highly selective and both client’s and broker’s reputations matter when it comes to risk perception and concurrently risk selection process.

5. Grounded theory analysis and discussion Political risk underwriters’ perceptions are alerted by both explicit and implicit aspects of political risk (see Figure 1). Explicit aspects refer to those characteristics that can be articulated, shared, codified and stored in some form of media like host country political and economic factors, client’s financial strength and policy features such as tenor and coverage. For example, some underwriters would not write political risk policies that are five years or longer in term. These explicit risk properties have an immediate impact on risk selection process and are usually used as the reference points in an underwriting policy as to what risks are acceptable for a book of business. The impact of individual risk properties on selection process can differ from one PRI provider to another and from one underwriter to the other. Explicit risk factors can be considered as the primary political risk selection criteria but they are not the only factors that are taken into consideration in an underwriting

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Political situation Legal environment

Extracting industry Key FX generator

Financial strength Social parameter

Coverage

Figure 1. Implicit and explicit risk selection criteria in the London PRI market.

decision. Implicit risk properties have an important role to play in risk selection process. Implicit risk properties refer to qualities that are indirect and only observable independently of conscious attempts at risk assessment. These properties trigger personal beliefs and biases that cannot be readily transmitted to others and are difficult to articulate or quantify. According to the grounded theory analysis results, a number of intangible/implicit factors such as intuition, trust, memorability and reputation have a significant impact on risk selection process and these in some instances can override the strength of explicit risk factors. If one is to understand a risk selection process in the London PRI market, both explicit and implicit properties of a risk as well as their interplay must be

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considered (see Figure 1). Explicit factors are at the centre of political risk selection criteria and in most instances form a foundation of an underwriting policy. When an underwriter is introduced to a new risk, he or she is going to evaluate it in terms of how well it fits risk selection criteria. Concurrently, an underwriter’s decision is affected by implicit factors such as trust, intuition and reputation. Grounded theory analysis shows that both explicit and implicit risk selection criteria have to be met for a particular political risk to be accepted to the portfolio of business. For example, if a risk satisfies technical underwriting requirements i.e. explicit selection criteria but an underwriter has no trust in a third party, e.g. the PR broker, then an underwriter may opt to reject otherwise acceptable risk. Overall, a risk is assumed to be acceptable if it satisfies both explicit and implicit criteria. Implicit criteria can be dominant if risk is relatively more complex in comparison to other political risks. Finally, explicit and implicit risk selection factors complement each other whilst addressing two different aspects in a risk selection process. Explicit risk properties are analysed by PR underwriters in order to arrive at a relative level of political risk or at a subjective probability of the loss event happening, whereas implicit risk properties are utilised in order to manage adverse selection and moral hazard problems within the London PRI market which arise from the uncertainty surrounding political risks. Arguably, this two-dimensional political risk selection model helps PR underwriters to convert uninsurable risks into insurable risks even though in most instances they violate insurability principles. 6. Conclusion In conceptualising this particular insurance market, the findings contained in this paper demonstrate that orthodox paradigms which privilege the idea of rational economic actor and notion of efficient markets are insufficient. The risks being insured in the PRI market are marked by their specificity and historical context and are difficult to assess using probability models. As a consequence, PRI providers must rely more on interpretative techniques in order to decide on whether or not to underwrite certain political risks. Grounded theory analysis shows that underwriters rely on intuition and tacit knowledge in order to arrive at the underwriting decision. For academics seeking to understand the operation of the London PRI market, positivist methodologies will provide, at best, only a partial picture of the underwriting process in this market. Instead, more interpretative techniques are required to gain a complete understanding of the type of thought processes at the heart of political risk underwriting. It is clear that the risk perceptions of the market participants are central to understanding this activity. Therefore, those same participants should be the centre of any analysis of the PRI market. There is a need to understand the nature of decision making amongst this group of insurers, hence the requirement for a well grounded appreciation of their thought processes and the environment in which they operate. It is clear that alongside more conventional information flows, market participants rely to large degree on how they feel about particular opportunities and that these feelings are based upon such phenomena as trust and heuristics as well as their own tacit knowledge. The fact that, in the main, market participants are expected to serve a long apprenticeship before writing PRI is testament to an awareness within the industry of the importance of experience and so-called soft skills to this activity. Whilst there are clearly other influences at work in this process, such as portfolio management and solvency considerations, the subjective judgement of a

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relatively small group of individuals is at the heart of this line of insurance business, a business that helps to promote FDI which in turn contributes to the development of emerging and developing countries. Note 1. The names of the study participants and their companies remain confidential on request.

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Quantitative Finance, 2014 Vol. 14, No. 1, 29–58, http://dx.doi.org/10.1080/14697688.2013.822989

Robust risk measurement and model risk PAUL GLASSERMAN† and XINGBO XU∗ ‡ †Columbia Business School, Columbia University, New York 10027, NY, USA ‡IEOR Department, School of Engineering and Applied Sciences, Columbia University, 500 West 120th Street, New York 10027, NY, USA (Received 18 September 2012; accepted 2 July 2013) Financial risk measurement relies on models of prices and other market variables, but models inevitably rely on imperfect assumptions and estimates, creating model risk. Moreover, optimization decisions, such as portfolio selection, amplify the effect of model error. In this work, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worstcase error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging and counterparty risk measured through credit valuation adjustment. Keywords: Risk measures; Validation of pricing models; Derivatives risk management; Risk management JEL Classification: C63, D81

1. Introduction Risk measurement relies on modelling assumptions. Errors in these assumptions introduce errors in risk measurement. This makes risk measurement vulnerable to model risk. This paper develops tools for quantifying model risk and making risk measurement robust to modeling errors. Simplifying assumptions are inherent to all modelling, so the first goal of model risk management is to assess vulnerabilities to model errors and their potential impact. We develop the following objectives: • to bound the effect of model error on specific measures of risk, given a baseline nominal model for measuring risk and • to identify the sources of model error to which a measure of risk is most vulnerable and to identify which changes in the underlying model have the greatest impact on this risk measure. For the first objective, we calculate an upper or lower bound (or both) on the range of risk values that can result over a range of model errors within a certain ‘distance’ of a nominal ∗ Corresponding author. Email: [email protected]

model. These bounds are somewhat analogous to a confidence interval; but whereas a confidence interval quantifies the effect of sampling variability, the robustness bounds we develop quantify the effect of model error. For the second objective, we identify the changes to a nominal underlying model that attain the bounds on the measure of risk—in other words, we identify the worst-case error in the nominal model. This step is crucial. Indeed, simply quantifying the potential magnitude of model risk would be of limited value if we could not point to the sources of model vulnerability that lead to the largest errors in measuring risk. Asimple example should help illustrate these ideas. Standard deviation is a conventional measure of risk for portfolio returns. Measuring standard deviation prospectively requires assumptions about the joint distribution of the returns of the assets in the portfolio. For the first objective listed above, we would want to bound the values of standard deviation that can result from a reasonable (in a sense to be quantified) degree of model error. For the second objective, we would want to identify which changes in the assumed joint distribution of returns have the largest impact on the portfolio standard deviation. In practice, model risk is sometimes addressed by comparing the results of different models—see Morini 2011 for an extensive treatment of this idea with applications to many

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different markets. More often, if it is considered at all, model risk is investigated by varying model parameters. Importantly, the tools developed here go beyond parameter sensitivity to consider the effect of changes in the probability law that defines an underlying model. This allows us to identify vulnerabilities to model error that are not reflected in parameter perturbations. For example, the main source of model risk might result from an error in a joint distribution of returns that cannot be described through a change in a covariance matrix. To work with model errors described by changes in probability laws, we need a way to quantify such changes, and for this we use relative entropy following Hansen and Sargent (2007). In Bayesian statistics, the relative entropy between posterior and prior distributions measures the information gained through additional data. In characterizing model error, we interpret relative entropy as a measure of the additional information required to make a perturbed model preferable to a baseline model. Thus, relative entropy becomes a measure of the plausibility of an alternative model. It is also a convenient choice because the worst-case alternative within a relative entropy constraint is typically given by an exponential change of measure. Indeed, relative entropy has been applied for model calibration and estimation in numerous sources, including Avellaneda (1998), Avellaneda et al. (2000, 1997), Buchen and Kelly (1996), Cont and Deguest (2013), Cont and Tankov (2004, 2006), Gulko (1999, 2002), and Segoviano and Goodhart (2009). In working with heavy-tailed distributions, for which relative entropy may be undefined, we use a related notion of α-divergence, as do Dey and Juneja (2010) in a portfolio selection problem. The tools we develop for risk measurement are robust in a sense similar to the way the term is used in the optimization and control literature. Robust optimization seeks to optimize against worst-case errors in problem data—see Ben-Tal et al. (2000), Bertsimas and Pachamanova (2008) and Goldfarb and Iyengar 2003), for example. The errors in problem data considered in this setting are generally limited to uncertainty about parameters, though distributional robustness is considered in, e.g. El Ghaoui et al. (2003) and Natarajan et al. (2008). Our approach builds on the robust control ideas developed in Hansen and Sargent (2007), Hansen et al. (2006), and Petersen et al. (2000), and applied to dynamic portfolio selection in Glasserman and Xu (forthcoming). Related techniques are used in Boyarchenko et al. (2012) and Meucci (2008). In this line of work, it is useful to imagine an adversary that changes the probability law in the model dynamics; the robust control objective is to optimize performance against the worstcase change of probability imposed by the adversary. Similarly, here we may imagine an adversary changing the probability law of the inputs to a risk calculation; we want to describe this worst-case change in law and quantify its potential impact on risk measurement. In both settings, the degree of robustness is determined through either a constraint or a penalty on relative entropy that limits the adversary’s ability to make the worst case arbitrarily bad. Our approach combines conveniently with Monte Carlo simulation for risk measurement. At the same time that we simulate a nominal model and estimate a nominal risk measure, we can estimate a bound or bounds on model risk with virtually no additional computational effort: we simply multiply the

nominal risk measure on each path by a factor (a likelihood ratio or Radon-Nikodym derivative) that captures the adversary’s change of probability measure. To understand how the adversary’s choice changes the model, we need to simulate under the worst-case model. This is again straightforward because simulating under the original model and then multiplying any output by the adversary’s likelihood ratio is equivalent to simulating the output from the worst-case model. This is similar to importance sampling, except that the usual goal of importance sampling is to reduce estimation variance without changing the mean of the estimated quantity; here, the objective is to understand how the change in probability measure changes the means and other model properties. This simulation-based approach also allows us to limit which stochastic inputs to a model are subject to model error. Our focus, as already noted, is on bounding worst-case model error. An alternative approach to model uncertainty is to mix multiple models. This idea is developed from a Bayesian perspective in, for example, Draper (1995) and Raftery et al. (1997) and applied to portfolio selection in Pesaran et al. (2009). For risk measurement, the added conservatism of considering the worst case is often appropriate and can be controlled through the parameter that controls the degree of robustness by penalzing or constraining relative entropy. The rest of the paper is organized as follows. Section 2 provides an overview of our approach and develops the main supporting theoretical tools. In Section 3, we discuss the implementation of the approach through a set of techniques we call robust Monte Carlo. The remainder of the paper is devoted to illustrative applications: Section 4 considers portfolio variance; Section 5 considers conditional value-at-risk; Section 6 examines the Gaussian copula model of portfolio credit risk; Section 7 investigates delta hedging, comparing the worst-case hedging error with various specific sources of model error; and Section 8 studies model risk in the dependence between exposures and default times in credit valuation adjustment (CVA). 2. Overview of the approach We begin by introducing the main ideas of the paper in a simple setting. Let X denote the stochastic elements of a model—this could be a scalar random variable, a random vector or a stochastic process. Let V (X ) denote some measure of risk associated with the outcome X. We will introduce conditions on V later, but for now we keep the discussion informal. If the law of X is correctly specified, then the expectation E[V (X )] is the true value of the risk measure of interest. We incorporate model uncertainty by acknowledging that the law of X may be misspecified. We consider alternative probability laws that are not too far from the nominal law in a sense quantified by relative entropy. For probability densities f and f˜ with a well-defined likelihood ratio m = f˜/ f , we define the relative entropy of f˜ with respect to f to be  ˜ f (x) f˜(x) ˜ log f (x) d x.† R( f, f ) = E[m log m] = f (x) f (x) †In some references, relative entropy is defined under the alternative model f˜ by a change of measure, i.e. R( f, f˜) = E[m log m] =  ˜ log f (x) f˜(x) d x, which is equilavent to our definition. f (x)

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Robust risk measurement and model risk In Bayesian statistics, relative entropy measures the information gain in moving from a prior distribution to a posterior distribution. In our setting, it measures the additional information that would be needed to make an alternative model f˜ preferable to a nominal model f. It is easy to see that R ≥ 0, and R( f, f˜) = 0 only if f˜ and f coincide almost everywhere (with respect to f). Relative entropy is not symmetric in f and f˜ and does not define a distance in the usual sense, but R( f, f˜) is nevertheless interpreted as a measure of how much the alternative f˜ deviates from f. (Our views of f and f˜ are generally not symmetric either: we favour the nominal model f but wish to consider the possibility that f˜ is correct.) The expression E[m log m], defining relative entropy through a likelihood ratio, is applicable on general probability spaces and is thus convenient. Indeed, we will usually refer to alternative models through the likelihood ratio that connects an alternative probability law to a nominal law, defining f˜(x) to be m(x) f (x). With the nominal model f fixed, we write R(m) instead of R( f, f˜). To quantify model risk, we consider alternative models described by a set Pη of likelihood ratios m for which E[m log m] < η. In other words, we consider alternatives within a relative entropy ‘distance’ η of the original model. We then seek to evaluate, in addition to the nominal risk measure E[V (X )], the bounds inf E[m(X )V (X )] and sup E[m(X )V (X )].

m∈Pη

(1)

m∈Pη

The expression E[m(X )V (X )] is the expectation under the alternative model defined by m. For example, in the scalar case m = f˜/ f ,   ˜ f (x) V (x) f (x) d x = V (x) f˜(x) d x. E[m(X )V (X )] = f (x) The bounds in (1) thus bound the range of possible values for the risk measure consistent with a degree of model error bounded by η. The standard approach to the maximization problem in (1) is to form the dual problem   1 inf sup E mV (X ) − (m log m − η) . θ >0 m θ (We will often suppress the argument of m to simplify notation, as we have here.) For given θ > 0, the inner supremum problem has as solution of the form exp(θ V (X )) , (2) m ∗θ = E[exp(θ V (X ))] provided the expectation in the denominator is finite. In other words, the worst-case model error is characterized by an exponential change of measure defined through the function V and a parameter θ > 0. The lower bound in (1) is solved the same way but with θ < 0. The explicit solution we get in (2) is the greatest advantage of working with relative entropy to quantify model error. In Section 3, we will apply (2) at multiple values of θ to trace out bounds at multiple levels of relative entropy.

2.1. A first example: portfolio variance To help fix ideas, we introduce a simple example. Let X denote a vector of asset returns and suppose, for simplicity, that X

is modelled by a multivariate normal distribution N (μ, ),  > 0, on Rn . We consider a portfolio with weights a = (a1 , . . . , an )⊤ summing to 1, and we use portfolio variance as our risk measure   E[V (X )] = E a ⊤ (X − μ)(X − μ)⊤ a .

We are interested in the worst-case variance   sup E[mV (X )] = sup E ma ⊤ (X − μ)(X − μ)⊤ a . m∈Pη

m∈Pη

In formulating the problem this way, we are taking μ as known but otherwise allowing an arbitrary change in distribution, subject to the relative entropy budget of η. From (2), we know that the worst-case change of measure has the form    m ∗θ ∝ exp θ a ⊤ (X − μ)(X − μ)⊤ a .

We find the worst-case density of X by multiplying the original N (μ, ) density by the likelihood ratio; the result is a density proportional to    exp θ a ⊤ (x − μ)(x − μ)⊤ a 

1 × exp − (x − μ)⊤  −1 (x − μ) . 2 In other words, the worst-case density is itself multivariate ˜ normal N (μ, ), −1  ˜ =  −1 − 2θaa ⊤ , 

with θ > 0 sufficiently small that the matrix inverse exists. For small θ , ˜ =  + 2θ aa ⊤  + o(θ 2 ),  and the worst-case portfolio variance becomes ˜ = a ⊤ a + 2θa ⊤ aa ⊤ a + o(θ 2 ) a ⊤ a  2 = a ⊤ a + 2θ a ⊤ a + o(θ 2 ).

That is, the resulting worst-case variance of the portfolio is increased by approximately 2θ times the square of the original variance. This simple example illustrates ideas that recur throughout this paper. We are interested in finding the worst-case error in the risk measure—here given by portfolio variance—but we are just as interested in understanding the change in the probability law that produces the worst-case change. In this example, the worst-case change in law turns out to stay within the family of multivariate normal distributions: we did not impose this as a constraint; it was a result of the optimization. So, in this example, the worst-case change in law reduces to a parametric change—a change in . In this respect, this example is atypical, and, indeed, we will repeatedly stress that the approach to robustness we use goes beyond merely examining the effect of parameter changes to gauge the impact of far more general types of model error. The worst-case change in distribution we found in this example depends on the portfolio vector a. Here and throughout, it is convenient to interpret model error as the work of a malicious adversary. The adversary perturbs our original model, but the error introduced by the adversary is not arbitrary—it is tailored to have the most severe impact possible, subject to a

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relative entropy budget constraint. The bounds in (1) measure the greatest error the adversary can introduce, subject to this constraint. The portfolio variance example generalizes to any quadratic function V (x) = x ⊤ Ax + B, A > 0. A similar calculation shows that under the worst-case change of measure, X remains normally distributed with ˜ X ∼ N (μ, ˜ ),

˜ = ( −1 − 2θ A)−1 , 

˜ −1 μ. μ˜ = 

The relative entropy associated with this change of measure evaluates to

1 ˜ −1 ) + tr( −1  ˜ − I) log(det(  η(θ ) = 2  ˜ . + (μ − μ) ˜ ⊤  −1 (μ − μ) By inverting the mapping θ → η(θ ), we can find the worstcase θ associated with any relative entropy budget η. In most of our examples, it is easier to evaluate model error at various values of θ and calculate the corresponding value for relative entropy, rather than to specify the level of relative entropy in advance; we return to this point in Section 3.

2.2. Optimization problems and precise conditions As the portfolio variance example illustrates, risk measurement often takes place in the context of an investment or related decision. We, therefore, extend the basic problem of robust evaluation of E[V (X )] to optimization problems of the form inf E[Va (X )],

a∈A

(3)

for some parameter a ranging over a parameter set A. For example, a could be a vector of portfolio weights or a parameter of a hedging strategy. We will introduce conditions on Va and the law of X . We formulate a robust version of the optimization problem (3) as (4) inf sup E[mVa (X )]. a m∈P η

Here, we seek to optimize against the worst-case model error imposed by a hypothetical adversary. The dual to the inner maximization problem is   1 (5) inf inf sup E mVa (X ) − (m log m − η) . a θ >0 m θ Proposition 2.1 Under Assumptions A.1–A.2 introduced in Appendix A, problem (5) is equivalent to   1 inf inf sup E mVa (X ) − (m log m − η) . (6) θ >0 a m θ

∗ ), the corresponding optimal objective For fixed θ ∈ (0, θmax function of inner inf a supm in (6) becomes   η η 1 (7) H (θ ) + := inf sup E mVa (X ) − m log m + a m θ θ θ

η 1 = log E exp(θ Va ∗ (θ ) (X )) + , θ θ where the optimal decision is 1 a ∗ (θ ) = arg inf log E[exp(θ Va (X ))], (8) a θ

and the worst-case change of measure is  

m ∗θ = exp θ Va ∗ (θ ) (X ) /E exp(θ Va ∗ (θ ) (X )) .

(9)

For a fixed value of a, 1 lim log E[exp(θ Va (X ))] = E[Va (X )], θ →0+ θ

corresponding to the nominal case without model uncertainty. To avoid too much technical complication, we only consider a simple case. When θ1 log E[exp(θ Va (X ))] is continuous both in a and θ , we can define the optimal decision and objective function when θ approaches 0 as follows: 1 log E[exp(θ Va (X ))] θ = arg inf E[Va (X )],

a ∗ (0) = lim arg inf θ →0+

a

a

1 log E[exp(θ Va ∗ (0) (X ))] = E[Va ∗ (0) (X )]. θ →0+ θ

H (0) = lim

Because Va (x) is convex in a for any x, the objective function E[Va (X )] is convex in a. Because θ > 0, the objective function in (8) is convex as well. The constrained problem (4) is equivalent to η (10) inf H (θ ) + . θ >0 θ As a consequence of [(Petersen et al. 2000, Theorem 3.1)], when the set of θ > 0 leading to finite H (θ ), is non-empty, (10) has a solution θ > 0 and the optimal value and solution solve the original constraint problem (4). Proposition 2.2 With Assumption A.2 introduced in Appendix A, the objective function in (8) is convex in a. Proof Because Va (x) is convex in a for any x, the objective function E[Va (X )] is convex in a. Because θ > 0, the objective  function in (8) is convex as well. For given η > 0, we can find an optimal θη∗ , with m ∗ and a ∗ (θη∗ ) as optimal solutions, and     (11) η = E m ∗ θη∗ , a ∗ (θη∗ ) log m ∗ θη∗ , a ∗ (θη∗ ) ,

(θη∗ , a ∗ (θη∗ ))

i.e. the uncertainty upper bound is reached at the optimal perturbation. So with θη∗ > 0, and the adversary’s optimal choice as (9), the original constraint problem (4) has the optimal objective     E m ∗ θη∗ , a ∗ θη∗ Va ∗ (θη∗ ) (X )    E Va ∗ (θη∗ ) (X ) exp θη∗ Va ∗ (θη∗ ) (X )    = , (12) E exp θη∗ Va ∗ (θη∗ ) (X )

which differs from the objective function of the penalty version (7) through the constant term. In practice, we may be interested in seeing the relation between the level of uncertainty and the worst-case error, which involves comparing different values of η. In this case, rather than repeat the procedure above multiple times to solve (10), we can work directly with multiple values of θ > 0 and evaluate η(θ ) with each, as in (11). Working with a range of values of θ , this allows us to explore the relationship between η and the worst-case error (and this is the approach we use in our numerical examples). This method requires that η be an

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Robust risk measurement and model risk increasing function of θ , a property we have observed numerically in all of our examples.

2.3. Robustness with heavy tails: extension to α-divergence In order to use relative entropy to describe model uncertainty, we need the tails of the distribution of V (X ) to be exponentially bounded, as in Assumption A.1 introduced in Appendix A. To deal with heavy-tailed distribution, we can use an extension of relative entropy called α-divergence and defined as (see also Rényi (1961) and Tsallis (1988))  1 − f˜α (x) f 1−α (x)d x Dα (m) = Dα ( f, f˜) = α(1 − α) 1 − E[m α ] , = α(1 − α) with m the likelihood ratio f˜/ f , as before, and the expectation on the right taken with respect to f . Relative entropy can be considered a special case of α-divergence, in the sense that R(m) = E[m log m] = limα→1+ Dα (m). With relative entropy replaced by α-divergence, the constraint problem (4) becomes sup E[mVa (X )]. inf a m:D (m)0 supm E mVa (X ) − (Dα (m) − η) θ   1 = inf θ >0 inf a supm E mVa (X ) − (Dα (m) − η) . θ (13) The supremum is taken over valid likelihood ratios— non-negative random variables with mean 1. Dey and Juneja (2010) apply an equivalent polynomial divergence and minimize it subject to linear constraints through a duality argument. We use a similar approach. Proposition 2.3 Suppose Assumption A.3 introduced in Appendix A holds. For any a ∈ A, θ > 0 and α > 1, the pair (m ∗ (θ, α, a), c(θ, α, a)) that solves the following equations with probability 1 is an optimal solution to (13): 1

m ∗ (θ, α, a) = (θ (α − 1)Va (X ) + c(θ, α, a)) α−1 ,

(14)

for some constant c(θ, α, a), such that θ (α − 1)Va (X ) + c(θ, α, a) ≥ 0,

(15)

  1 E (θ (α − 1)Va (X ) + c(θ, α, a)) α−1 = 1.

(16)

and

Proof The objective of (13) is concave in m. Proceeding as in (Dey and Juneja 2010, Proof of Theorem 2), we can construct a new likelihood ratio (1 − t)m ∗ + tm using an arbitrary m; the objective becomes  K (t) := E ((1 − t)m ∗ + tm)Va  η 1 ((1 − t)m ∗ + tm)α + , + θ α(1 − α) θ    1 (m ∗ )α−1 (m − m ∗ ) . (17) K ′ (0) = E Va + θ (1 − α)

In order to have K ′ (0) = 0 for any m, we need the term inside braces in (17) to be constant. By the definition of m ∗ , K ′ (0) = 0  holds, so m ∗ is optimal. If Va (X ) is not bounded from below, then when θ > 0 and α ≥ 0, (15) cannot be satisfied. For the case in which the adversary seeks to minimize the objective function (that is, to get the lower bound of the error interval), we need α < 0 to satisfy (15). A feasible likelihood ratio exists in a neighbourhood of θ = 0, by the following argument. In the nominal case θ = 0, 1 we have m ∗ (0, α, a) = c(0, a) α−1 , so we can always choose c(0, α, a) = 1. By continuity, we can find a set [0, θ0 ) such that c(θ, α, a) satisfying (15) and (16) exists for any θ ∈ [0, θ0 ). Once c(θ, α, a) is found, (14) gives an optimal change of measure (not necessarily unique). The optimal decision becomes  α−1  E θ (α − 1)Va (X ) a ∗ (θ ) = arg min a α   1 c(θ, α, a) . (18) + c(θ, α, a) α−1 Va (X ) + θ α(1 − α) In contrast to the relative entropy case, it is not clear whether the objective in (18) is convex in a. Measuring potential model error through α-divergence focuses uncertainty on the tail decay of the nominal probability density. For example, in the simple scalar case Va (x) = x k , taking α > 1 leads to a worst-case density function f˜X (x) ≈ cx k/(α−1) f X (x),

(19)

for x ≫ 0, where f X is the density function of X under the nominal measure. Incorporating model uncertainty makes the tail heavier, asymptotically, by a factor of x k/(α−1) . As illustrated using relative entropy and α-divergence, our method can potentially be generalized to a much higher level as follows. If we can find a measurement or premetric, with which the worst-case likelihood ratio can be derived easily, then most of our analysis can be carried out without much difficulty. A possible choice of such measurement or premetric has the form E[φ(m)], where the function φ(m) ≥ 0 for any likelihood ratio m. Relative entropy and α-divergence are special cases.

3. Implementation: robust Monte Carlo In this section, we present methods for estimating the model error bounds in practice through what we call robust Monte Carlo. In addition to calculating bounds, we present ways of examining the worst-case model perturbation to identify the greatest model vulnerabilities, and we also show how to constrain the possible sources of model error.

3.1. Estimating the bounds on model error We assume the ability to generate independent replications X 1 , X 2 , . . . of the stochastic input X , recalling that X may be a random variable, a random vector or a path of a stochastic process. A standard Monte Carlo estimator of E[V (X )] is n 1  V (X i ). N i=1

34

P. Glasserman and X. Xu

For any fixed θ and likelihood ratio m θ ∝ exp(θ V (X )), we can estimate the expectation of V (X ) under the change of measure defined by m θ by generating the X i from the original (nominal) measure and forming the estimator N i=1 V (X i ) exp(θ V (X i )) , (20) N i=1 exp(θ V (X i ))

which converges to E[m θ V (X )] as N → ∞. Assuming V ≥ 0, we have E[V (X )] ≤ E[m θ V (X )] if θ > 0 and E[V (X )] ≥ E[m θ V (X )] if θ < 0. Our estimator of these bounds requires virtually no additional computational effort beyond that required to estimate the nominal value E[V (X )]. From the same replications X 1 , . . . , X N , we can estimate the likelihood ratio by setting mˆ θ,i =  N

exp(θ V (X i ))

j=1 exp(θ V (X j ))/N

,

i = 1, . . . , N .

η(θ ˆ )=

1 N

mˆ θ,i log mˆ θ,i .

(21)

i=1

Thus, we can easily estimate (η(θ ), E[m θ V (X )]) across multiple values of θ . Given a relative entropy budget η, we then lookup the smallest and largest values of E[m θ V (X )] estimated with η(θ ˆ ) ≤ η to get the model error bounds at that level of η. We will illustrate this procedure through several examples. Just as importantly, we can use the same simulation to analyse and interpret the worst-case model error. We do this by estimating expectations E[m θ h(X )] of auxiliary functions h(X ) under the change of measure by evaluating estimators of the form N 1  mˆ θ,i h(X i ). (22) N i=1

Through appropriate choice of h, this allows us to estimate probabilities, means and variances of quantities of interest, for example, that provide insight into the effect of the worst-case change in probability law. In some case, we may want to sample from the worst-case law, and not evaluate expectations under the change of measure. If V is bounded, we can achieve this through acceptance– rejection: to simulate under the law defined by θ , we generate candidates X from the original nominal law and accept them with probability exp(θ V (X ))/M, with M chosen so that this ratio is between 0 and 1. If V is unbounded, we need to truncate it at some large value and the sampling procedure then incurs some bias as a result of the truncation. These techniques extend to problems of optimization over a decision parameter a, introduced in Section 2.2, for which a standard estimator is N 1  Va (X i ), min a N i=1

For θ > 0, a worst-case objective function estimator is N i=1 Vaˆ ∗ (X i ) exp(θ Vaˆ ∗ (X i )) , N i=1 exp(θ Vaˆ ∗ (X i ))

N

 exp(θ Va (X )) 1 , aˆ = arg inf log a θ N ∗

i=1

and the estimated optimal likelihood ratio is mˆ ∗θ,i =  N

exp(θ Vaˆ ∗ (X i ))

j=1 exp(θ Vaˆ ∗ (X j ))/N

,

i = 1, . . . , N .

By continuous mapping theorem, for given aˆ ∗N and any θ ∈ [0, θmax ), the averages of both numerator and denominator of (23) are consistent estimators. That is, N    1  Vaˆ ∗ (X i ) exp θ Vaˆ ∗ (X i ) → E Vaˆ ∗ (X ) exp(θ Vaˆ ∗ (X )) , N N N N N i=1

N    1  exp θ Vaˆ ∗ (X i ) → E exp(θ Vaˆ ∗ (X )) . N N N i=1

This in turn allows us to estimate the relative entropy at θ as N 

where the estimated optimal decision parameter is

(23)

Hence, (23) is a consistent estimator for (12) with aˆ ∗ . In the case where E[exp(θ Va (X ))] is continuous in a and the optimal decision a ∗ is unique, it is easy to show that aˆ ∗ converges to a ∗ in distribution. More generalized results can be found in Sample Average Approximate literature, e.g. Shapiro et al. (2009). Similar estimators are available in the α-divergence framework. For given θ > 0, α > 1 and a, we estimate the worst-case likelihood ratio as   1 ˆ α, a) α−1 , mˆ ∗θ,α,a,i = θ (α − 1)Va (X i ) + c(θ, for some constant c(θ, ˆ a), s.t.

θ (α − 1)Va (X i ) + c(θ, ˆ α, a) > 0, for each i with  N   1 1   ˆ α, a) α−1 = 1. θ (α − 1)Va (X i ) + c(θ, N i=1

For given θ > 0 and α > 1, we solve for an optimal a as N  α−1  ∗ aˆ (θ ) = arg min θ (α − 1)Va (X i ) a α i=1   1 c(θ, ˆ α, a) + c(θ, ˆ α, a) α−1 Va (X i ) + . θ α(1 − α)

The robust estimator for the objective becomes N 1  Va (X i )mˆ θ,α,a ∗ (θ ),i . N i=1

3.2. Incorporating expectation constraints When additional information is available about the ‘true’model, we can use it to constrain the worst-case change of measure. Suppose the information available takes the form of constraints on certain expectations. For example, we may want to constrain the mean (or some higher moment) of some variable of a model. We formulate this generically through constraints of the form E[mh i (X )] ≤ ηi or E[mh i (X )] = ηi for some function h i and scalars ηi .

35

Robust risk measurement and model risk Such constraints can be imposed as part of an iterative evaluation of model risk. In (22), we showed how a change of measure selected by an adversary can be analysed through its implications for auxiliary functions. If we find that the change of measure attaches an implausible value to the expectation of some h i (X ), we can further constrain the adversary not just through the relative entropy constraint but through additional constraints on these expectations. This helps ensure the plausibility of the estimated model error and implicitly steers the adversary to allocate the relative entropy budget to other sources of model uncertainty. The adversary’s problem becomes (24)

sup E[mV (X )], m∈PM

where

PM = {m : R(m) ≤ η, E[mh i (X )] ≤ ηi , i = 1, . . . , n M } for some ηi , η ∈ [0, ∞).

Here, we have added n M constraints on the expectations of h i (X ) under the new measure. We can move the constraints into the objective with Lagrange multipliers λi and transform (24) into a penalty problem; the argument in Petersen et al. (2000) still holds as the terms of h i (X ) can be combined with that of V :  1 inf sup E mV (X ) − (m log m − η) θ >0,λi >0 m θ  nM  − λi [mh i (X ) − ηi ] . i=1

When θ and the λi are fixed, the problem can be treated as before in (6).

Proposition 3.1 For fixed θ > 0 and λi > 0, i = 1, . . . , n M , such that     nM  λi h i (X ) < ∞. E exp θ V (X ) − i=1

The worst change of measure is   m ∗θ

∝ exp θ V (X ) −

nM 



λi h i (X )

i=1

.

The optimization over (θ, λi ) becomes     nM  1 inf log E exp θ V (X ) − λi h i (X ) θ >0,λi >0 θ i=1

+

η + θ

nM 

ηi λi .

i=1

For equality constraints, the optimization is over λi ∈ R.

This is a standard result on constraints in exponential families of probability measures. It is used in Avellaneda et al. (2000) and Cont and Tankov (2004), for example, where the constraints calibrate a base model to market prices. Glasserman and Yu (2005) and Szechtman and Glynn (2001) analyse the convergence of Monte Carlo estimators in which constraints are imposed by applying weights to the replications. For an optimization problem as in (3), adding constraints entails solving another layer of optimization. For example, if

the original problem is a minimization problem as in (3), then for given (θ, λi ), the optimal decision becomes a ∗ (θ, λi )     nM  1 λi h i (a, X ) = arg inf log E exp θ Va (X ) − a θ i=1

+

nM 

ηi λi .

i=1

3.3. Restricting sources of model uncertainty In some cases, we want to go beyond imposing constraints on expectations to leave entire distributions unchanged by concerns about model error. We can use this device to focus robustness on parts of the model of particular concern. We will see a different application in Section 8 where we use an exponential random variable to define a default time in a model with a stochastic default intensity. In that setting, we want to allow the default intensity to be subject to model uncertainty, but we want to leave the exponential clock unchanged as part of the definition of the default time. Suppose, then, that the stochastic input has a representation as (X, Y ), for a pair of random variables or vectors X and Y . We want to introduce robustness to model error in the law of X , but we have no uncertainty about the law of Y . For a given θ > 0, we require that E[exp(θ Va (X, Y ))|Y = y] < ∞ for any y, and formulate the penalty problem  inf sup E m(X, Y )Va (X, Y ) a m   1 (25) − m(X, Y ) log m(X, Y ) − η θ s.t. E[m(X, Y )|Y = y] = 1, ∀y (26) m(x, y) ≥ 0

∀x, y.

We have written m(X, Y ) to emphasize that the likelihood ratio may be a function of both inputs even if we want to leave the law of Y unchanged. Proposition 3.2 For problem (25) with θ > 0 and E[exp(θ Va (X, Y ))|Y = y] < ∞ for all y:

(1) Any likelihood ratio that satisfies (26) preserves the law of Y . (2) For any a, the likelihood ratio exp(θ Va (x, y)) , (27) m ∗ (x, y) = E[exp(θ Va (X, Y ))|Y = y] is an optimal solution to the maximization part of problem (25). (3) The corresponding optimal decision becomes 1 a ∗ (θ ) = arg inf E [log E[exp(θ Va (X, Y ))|Y ]] . θ Proof The feasible set of likelihood ratios m is convex, and the objective function is concave in m, so it suffices to check first-order conditions for optimality. Define   1 K¯ (t) = E tm ∗ + (1 − t)m Va (X, Y ) − (tm ∗ θ  + (1 − t)m) log(tm ∗ + (1 − t)m) − η ,

36

P. Glasserman and X. Xu

where m is an arbitrary likelihood ratio satisfying (26). Obviously, m ∗ satisfies (26). Taking the derivative of K¯ at zero and substituting for m ∗ , we get   

1 1 ∗ ∗ ′ ¯ K (0) = E (m − m) Va (X, Y ) − log m − θ θ  

  1 1 =E E Va (X, Y ) − log m ∗ − (m ∗ − m)|Y θ θ  1 =E (log E[exp(θ Va (X, Y ))|Y ] − 1) θ  ∗

×E (m − m)|Y . (28) By constraint (26), for any Y = y, the conditional expectation E[(m ∗ − m)|Y ] in (28) equals zero, so K¯ ′ (0) = 0. Hence m ∗ is an optimal solution satisfying constraint (26). Next, we show that any likelihood ratio satisfying (26) preserves the distribution of Y . Let a tilde indicate the distribution following the change of measure. ˜ ∈ D) = E[m ∗ (θ, X, Y )IY ∈D ] P(Y

= E E[m ∗ (θ, X, Y )IY ∈D |Y ]

= E IY ∈D E[m ∗ (θ, X, Y )|Y ]

for any Y -measurable set D.

Thus, the likelihood ratio for the marginal law of Y is identically equal to 1, indicating that the distribution of Y is unchanged.  To implement (27), we need to generate multiple copies X 1 , . . . , X N for each outcome of y and then form the Monte Carlo counterpart of (27), exp(θ Va (x, y))

i=1 exp(θ Va (X i ,

y))/N

(29)

Robust Monte Carlo Recap: We conclude this section with a brief summary of the implementation tools of this section. By simulating under the nominal model and weighting the results as in (20), we can estimate the worst-case error at each level of θ . We can do this across multiple values of θ at minimal computational cost. By also estimating η(θ ) as in (21), we can plot the worst-case error as a function of relative entropy. • To examine the effect of the change of measure defined by θ , we can estimate moments and the expectations of other auxiliary functions using (22). We can also sample directly from the measure defined by θ using acceptance-rejection—exactly if V is bounded and approximately if not. • We can constrain the worst-case change of measure through constraints on moments or other auxiliary functions using Proposition 3.1. This technique can be used iteratively to constrain the potential model error if the values estimated through (22) appear implausible. • Using Proposition 3.2 and (29), we can constrain the worst-case model to leave certain marginal distributions unchanged. This too can be used iteratively to focus robustness on the most uncertain features of model.

•

The rest of the paper deals with applications of the ideas developed in the previous sections. In Section 2.1, we illustrated the key ideas of robust risk measurement through an application to portfolio variance. Here, we expand on this example.

4.1. Mean-variance optimal portfolio We extend our earlier discussion of portfolio variance to cover the selection of mean-variance optimal portfolios under model uncertainty. For the mean-variance objective, let γ > 0 be a risk-aversion parameter and consider the optimization problem   γ inf −E a ⊤ X − a ⊤ (X − E[X ])(X − E[X ])⊤ a . (30) a 2 As before, a denotes a vector of portfolio weights. To illustrate the method of Section 3.2, we constrain the mean vector and limit uncertainty to the covariance matrix, which leads to the robust problem inf sup E[mVa (X )] a m    γ = inf sup −E m a ⊤ X − a ⊤ (X − μ)(X − μ)⊤ a a m 2 s.t. E[m X ] = μ.

= E[IY ∈D ] = P(Y ∈ D)

mˆ ∗ (x, y) =  N

4. Portfolio variance

Following the argument in Section 3.2, for some a, θ > 0 and λ, the worst-case likelihood ratio is   (31) m ∗ ∝ exp θ Va (X ) − λ⊤ (X − μ)

where λ solves      1 + λ⊤ μ. inf log E exp θ V (X ) − λ⊤ X λ θ Proceeding as in Section 2.1, we find that the worst-case change of measure preserves the normality of X . The term with λ is linear in X and, therefore, affects only the mean of X . Because we have constrained the mean, m ∗ satisfies 

θγ ⊤ a (X − μ)(X − μ)⊤ a , (32) m ∗ ∝ exp 2

Matching (31) and (32), we find that λ = a. For given θ > 0, let A(θ ) = {a :  −1 −θ γ aa ⊤ > 0} denote the set of portfolio vectors a that ensure that the resulting covariance matrix is positive definite. Then, for given (a, θ ) such that θ > 0 and a ∈ A(θ ), the worst-case change of ˜ where  ˜ −1 =  −1 − θ γ aa ⊤ . We measure has X ∼ N (μ, ), can find the optimal a by numerically solving      1 log E exp θ V (X ) − λ⊤ X + λ⊤ μ a ∗ (θ ) = arg inf a∈A(θ ) θ 1 + a ⊤ μ. (33) = arg inf  a∈A(θ ) det(I − θ γ aa ⊤ )

The corresponding relative entropy is 1 ˜ −1 )) + tr( −1  ˜ − I) . log(det(  η(θ ) = 2 To illustrate, we consider an example with 10 assets, where μi = 0.1, σii = 0.3 and ρi j = 0.25 for i = j, i, j = 1, . . . , 10 and γ = 1. We refer to the optimal portfolio at these parameter values as the nominal portfolio (NP). At each

37

Robust risk measurement and model risk NP, worst scenario RP, worst scenario RP, nominal model NP, nominal model

−0.12

NP, worst scenario NP, nominal model ρ∈(0.05,0.45) κ∈(0.72,1.32)

−0.12

−0.14

−0.14

−0.16

−0.16

ρ=0.45

E[Va]

E[Va]

ρ=0.25,κ=1

−0.18

κ=0.72

−0.18

−0.2

−0.2

−0.22

−0.22

−0.24

−0.24

ρ=0.05

κ =1.32

0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

Relative entropy

0.15

0.2

0.25

Relative entropy

Figure 1. Expected performance vs. relative entropy. The left panels shows the performance of the nominal portfolio (NP) and the robust portfolio (RP) under the nominal and worst-case models. The right panel shows the performance of the nominal portfolio under perturbations in model parameters. Higher values on the vertical scale indicate worse performance. Table 1. Realized and forecast variance with model uncertainty. 2002

2008

Realized variance ±2×Std. Err.

0.35 × 10−3 (0.29, 0.42) × 10−3

0.65 × 10−3 (0.53, 0.77) × 10−3

Forecast variance ±(2×Std. Err. + Model Err.) θ = 100 θ = 500 θ = 900

0.21 × 10−3 (0.20, 0.22) × 10−3 (0.21, 0.25) × 10−3 (0.18, 0.32) × 10−3 (0.16, 0.47) × 10−3

0.21 × 10−3 (0.20, 0.22) × 10−3 (0.17, 0.22) × 10−3 (0.14, 0.32) × 10−3 (0.12, 0.58) × 10−3

θ value, we compute the robust portfolio (RP), meaning the one that is optimal under the change of measure defined by θ . In the left panel of figure 1, we plot the performance of the two portfolios (as measured by the mean-variance objective— recall that we are minimizing) against relative entropy (which we also compute at each θ ). The performance of the NP portfolio under the nominal model is simply a horizontal line. The performance of the RP portfolio under the nominal model is always inferior, as it must be since NP is optimal in the nominal model. However, under the worst-case model, the RP values are better than the NP values, as indicated by the upper portion of the figure. In the lower portion of the figure, we see the performance of the nominal portfolio under the best-case model perturbation possible at each level of relative entropy. The vertical gap between the two portions of the NP curve indicate the model risk at each level of relative entropy. One of the themes of this paper is that model error as gauged by relative entropy does not necessarily correspond to a straightforward error in parameters. To illustrate, in the right panel we examine the performance of the nominal portfolio under specific parameter perturbations. We vary the common correlation parameter from ρ = 0.05 (which produces the best performance) to ρ = 0.45 (which produces the worst); the relative entropy first decreases and then increases as ρ moves through this range. We also examine the effect of multiplying the covariance matrix of the assets by κ ∈ (0.72, 1.32). The key point—and one to which we return often—is that the worst-case change of measure results in significantly worse performance than any of these parameter perturbations.

Glasserman and Xu (forthcoming) study a dynamic version of the mean-variance problem with stochastic factors and transaction costs. The analysis results in closed-form solutions for both the investor and adversary. For general multi-period problems, Iyengar (2005) develops a robust version of dynamic programming.

4.2. Empirical example To apply these ideas to data, we use daily returns from the CRSP database on the 126 stocks that were members of the S&P500 index from 1 January, 1990, to 31 December, 2011. We first estimate the mean μ and covariance  of daily return using the first 12 years of data, through the end of 2001. For the covariance matrix, we use the shrinkage method in Ledoit and Wolf (2003). Based on the estimated mean and covariance matrix, we construct the mean-variance optimal portfolio a = (γ )−1 (μ − λI ) ⊤

where λ = (I (γ )

−1

(34) ⊤

μ − 1)/(I (γ )

−1

I)

and γ = 10. We assume a static portfolio with total capital of 1. We take the portfolio variance from the initial time period as a forecast of the future variance for the same portfolio. We compare this forecast with the realized variance in 2002, when the dot-com bubble burst. In the first column of table 1, we see that the realized variance in 2002 is quite large compared to the forecast using the previous 12 years of data. Confidence intervals equal to two

38

P. Glasserman and X. Xu

Table 2. Worst-case portfolio variance at different levels of θ and α. The middle column reports estimates using parameters estimated at α = 2.5, showing first the portfolio variance and then the degrees of freedom parameter (in parentheses) estimated using να = ν + kno − kθ,α and maximum likelihood. θ

α=2

α = 2.5

α = 2.5, worst parameters (DOF)

α=3

α = 3.5

0 0.1 0.4 0.7 1

0.109 0.159 0.308 0.458 0.607

0.109 0.131 0.174 0.210 0.241

0.109 0.130 (3.15,3.65) 0.174 (2.84,3.18) 0.209 (2.77,2.93) 0.238 (2.74,2.84)

0.109 0.125 0.152 0.173 0.190

0.109 0.122 0.143 0.159 0.171

times the standard error of the realized variance and forecast have no overlap. The sampling variability in the initial period is not large enough to explain the realized variance. Next, we introduce error intervals based on relative entropy. We use the portfolio variance as the objective and obtain the worst-case variance at different levels of θ . Let Model Error = |nominal variance-worst variance|.

Now, we can form a new interval by combining both standard error and model error. In the lower part of table 1, the new interval almost reaches the realized variance in 2002 when θ = 500, and it covers the confidence interval of realized variance when θ = 900. By considering both sampling variability and model error, we can cover the 2002 scenario. This gives us a rough sense of the level of robustness needed to capture a sharp change like that in 2002. We now position ourselves at the end of 2007 and undertake a similar analysis. Again, we use the previous 12 years of data to form a forecast, which is 0.21 × 10−3 . We choose θ = 900 as the robustness level, based on the study of 2002, so that the whole confidence interval of 2002 is contained. The model errors for the forecast of 2002 were 0.10 × 10−3 and 0.25 × 10−3 for θ = 500 and 900, respectively, and they change to 0.10 × 10−3 and 0.36 × 10−3 in the forecast of 2008. The forecast with both standard error and model error forms a pretty wide interval, which has a slight overlap with the confidence interval of the realized variance in 2008. Although the crisis in 2008 was more severe than the drop in 2002, the market change in 2002 provides a rough guide of potential model risk. The particular combination we have used of sampling error and model error is somewhat heuristic, but it nevertheless shows one way these ideas can be applied to historical data.

To illustrate, we consider an portfolio with n = 10 assets, ν = 4, μi = 0.1, ii = 0.28 + 0.02 × i and ρi j = 0.25 for i, j = 1, . . . , n and i = j. We use a randomly generated portfolio weight vector  a = 0.0785, 0.1067, 0.1085, 0.1376, 0.0127, 0.2204,  0.0287, 0.1541, 0.1486, 0.0042 , and simulate N = 107 samples to examine the worst-case scenario. Table 2 shows the portfolio variance across various values of θ and α, with θ = 0 corresponding to the baseline nominal model. For fixed α, increasing θ increases the uncertainty level and increases the worst-case variance. The middle column of the table shows results using estimated parameters at α = 2.5; we return to these at the end of this section. We saw in (19) that the choice of α influences the tail of V (X ) under the worst-case change of measure. A smaller α in table 2 yields a heavier tail, but this does not necessarily imply a larger portfolio variance. To contrast the role of α with θ , we can think of choosing α based on an assessment of how heavy the tail might be and then varying θ to get a range of levels of uncertainty. In both cases, some calibration to the context is necessary, as in the empirical example of the previous section and in the discussion below. To understand the influence of the α parameter, we examine the tail of the portfolio excess return, r = a ⊤ X − μ. Figure 2 plots the tail probability of |r| on a log–log scale. Because r has a t distribution, the log of the density of |r|, denoted by f |r | (x), is asymptotically linear log f |r | (x) ≈ −(ν + 1) log x,

for x ≫ 0.

4.3. The heavy-tailed case

1

m ∗ (θ, α) = (θ (α − 1)Va (X ) + c(θ, α)) α−1 (35) with c(θ, α) s.t. E[m ∗ (θ, α)] = 1 where Va (X ) = a ⊤ (X − μ)(X − μ)⊤ a.

nominal worst parameters α = 2.5, θ = 1

2 0

log( PDF of |r| )

To illustrate the use of α-divergence in the heavy-tailed setting, we now suppose that the vector of asset returns is given by X ∼ μ + Z , where Z ∼ tν (, ν) has a multivariate t distribution with ν > 2 degrees of freedom and covariance matrix ν/(ν − 2). Because neither the t-distribution nor a quadratic function of X has a moment generating function, we use αdivergence as an uncertainty measure. With a fixed portfolio weight vector a, Proposition 2.3 yields the worst-case likelihood ratio

−2 −4 −6 −8 −10 −12 −14 −2

−1.5

−1

−0.5

0

0.5

1

1.5

log(|r|)

Figure 2. Tail density of absolute returns |r |.

2

39

Robust risk measurement and model risk Table 3. Difference of slopes kθ,α − kno of the worst-case and nominal densities, as in figure 2. 2/(α − 1) θ = 0.1 θ = 0.4 θ = 0.7 θ =1

α=2

α = 2.5

α=3

α = 3.5

2 1.090 1.631 1.773 1.840

1.333 0.846 1.159 1.231 1.263

1 0.694 0.899 0.943 0.962

0.8 0.590 0.735 0.764 0.777

Using the fact that   log m ∗ (θ, α) =

  1 log θ (α − 1)r 2 + c(θ, α) α−1 2 ≈ log |r|, α−1 we find (as in (19)) that    2 log x, for x ≫ 0 (36) log f˜|r | (x) − log f |r | (x) ≈ α−1 where f˜|r | is the density of |r| under the change of measure. This suggests that the difference of the slopes in figure 2 between the nominal and worst scenario should be roughly 2/(α − 1). Asymptotically, the tail under the worst scenario is similar to a t distribution with degrees of freedom ν−2/(α−1). We fit linear functions to the curves in figure 2 in the region log(|r|) ∈ (0.5, 2) and compare the slopes of nominal kno and worst scenario kθ,α . table 3 lists the differences kθ,α − kno ; as we increase θ , the difference of slopes gets closer to the limit 2/(α − 1) in (36). By reweighting the sample under the nominal model using m ∗ (θ, α), we can estimate model parameters as though the worst-case model were a multivariate t. We estimate the degrees of freedom parameter using να,θ = ν + kno − kθ,α

(37)

and estimate the covariance matrix as   worst covariance = E ma ⊤ (X − μ)(X − μ)⊤ a ≈

1 N

N 

m(X i )a ⊤ (X i − μ)(X i − μ)⊤ a.

i=1

We can then generate a second set of samples from the t distribution with these parameters to see how this compares with the actual change of measure. In the middle of table 2, we show the estimated να,θ using (37) as the first number in parentheses. The second value is a maximum likelihood estimate using m ∗θ,α to weight the nominal samples. The two values are relatively close; we use only (37) in sampling under the worst-case parameter values and in figure 2. The variance results under the parameters estimated at α = 2.5 are very close to those estimated under the worst-case model at α = 2.5, suggesting that the worst case might indeed be close to a t distribution. Interestingly, figure 2 shows that using the parameters from the worst case actually produces a heavier tail; the worst-case change of measure magnifies the variance through relatively more small returns than does the approximating t distribution. In table 4, we see that the α-divergence under the approximating t is much larger. Thus, the adversary has economized the use of α-divergence to mag-

Table 4. Comparison of α-divergence using the worst-case change of measure and the approximating t distribution from the worst case. θ

α = 2.5

Approximating t-dist.

0.001 0.012 0.031 0.058

0.086 0.230 0.287 0.323

0.1 0.4 0.7 1

nify the portfolio variance without making the tail heavier than necessary.

5. Conditional value at risk The next risk measure we consider is conditional value at risk (CVaR), also called expected shortfall. The CVaR at quantile β for a random variable X representing the loss on a portfolio is defined by C V a Rβ = E[X |X > V a Rβ ], where V a Rβ satisfies 1 − β = P(X > V a Rβ ). As in Rockafellar and Uryasev (2002), CVaR also equals to the optimal value of the minimization problem min a

1 E[(X − a)+ ] + a, 1−β

(38)

for which the optimal a is V a Rβ . To put this problem in our general framework, we set Va (X ) = (1 − β)−1 (X − a)+ + a. The main source of model error in measuring CVaR is the distribution of X . As in previous sections, we can introduce robustness to model uncertainty by considering a hypothetical adversary who changes the distribution of X . Of particular concern is the worst-case CVaR subject to a plausibility constraint formulated through relative entropy or α-divergence. Jabbour et al. (2008) and Zhu and Pykhtin (2007) consider robust portfolio optimization problems using CVaR but different types of model uncertainty. To illustrate the general approach, we introduce two specific examples that offer some analytic tractability, one in the relative entropy setting and one using α-divergence.

5.1. Relative entropy uncertainty Suppose X follows a double exponential distribution D E(μ, b) with location parameter μ and scale parameter b, meaning that

40

P. Glasserman and X. Xu   η(θ ) = E m a∗∗ (θ ),θ log m a∗∗ (θ ),θ   E Va ∗ (θ ) exp θ Va ∗ (θ ) (X )   =θ E exp θ Va ∗ (θ ) (X )   − log E exp θ Va ∗ (θ ) (X )

its density function is 

|x − μ| . f (x) ∝ exp − b

Then, for given a and θ > 0, the density function of X under the worst-case change of measure becomes = θ κa′ ∗ (θ ) (θ ) − κ a ∗ (θ ) (θ ). 

θ |x − μ| Figure 4 shows the nominal and worst-case densities starting + (x − a)+ . f˜(x) = m ∗θ,a (x) f (x) ∝ exp − b 1−β from a nominal density that is D E(0, 1), using β = 95% and The values of a and β are connected by P(X > a) = 1 − β θ = 0.03. The nominal 95% VaR is a = 2.30; the worst-case under the nominal distribution. Because θ/(1 − β) > 0, we model error (for CVaR) at θ = 0.03 shifts more mass to the need 1/b > θ/(1 − β) to ensure this density function is well right tail and increases the VaR to 3.19. The CVaR increases defined. The exponent is a piecewise linear function of the from 3.30 to 3.81. The increase in VaR and the corresponding argument x, so f˜ can be considered a generalization of the increase in CVaR reflect the magnitude of underestimation of risk consistent with this level of the uncertainty parameter θ . double exponential distribution. We can find the VaR and CVaR explicitly in this example. First, we evaluate the normalization constant (8): 5.2. The heavy-tailed case E[exp(θ Va (X ))] ⎧  

 If the nominal distribution of the loss random variable X is  μ−a  ⎪ 1 1 ⎪ exp(θa) 1 + , if a > μ − 1 exp ⎪ heavy-tailed, then E[exp(θ Va (X ))] is infinite and the calculaθb ⎪ 2 1− b ⎪ 1−β ⎪ ⎪  

tions in (39) and following do not apply. In this case, we need ⎨  1 1 θ 1 to use α-divergence as the uncertainty measure. With α > 1, exp 1−β (μ − a) (39) = 2 exp(θa) θb + θb 1− 1−β 1+ 1−β ⎪ ⎪ θ > 0 and a fixed, the worst case likelihood ratio now becomes ⎪  

⎪ ⎪  a−μ  ⎪ 1 1 ⎪ , else. exp b + 1− ⎩ θb (40) m ∗ (X ) = (θ (α − 1)Va (X ) + c(θ, α, a)) α−1 , 1+ 1−β

θ,a

Denote the cumulant generating function of Va (X ) by κa (θ ) = log E[exp(θ Va (X ))]; then 1 a ∗ (θ ) = arg min κa (θ ), a θ

To find a ∗ , we observe that the function E[exp(θ Va (X ))] is convex in a and its derivative at a = μ is 



 θb − 1 θ d E exp(θ Va (X ))  2+ exp(θ μ). = da 2 1 − β − θb a=μ

This is positive provided β > 1/2, so we can solve the first order condition for a > μ to get 

2(1 − β − θ b) ∗ , a (θ ) = μ − b log 1 − θb which is the VaR under the worst-case change of measure. The VaR for the nominal model is V a Rβ = μ − b log(2(1 − β)). and the nominal CVaR is C V a Rβ = V a Rβ + b = μ − b log(2(1 − β)) + b. Under the worst-case change of measure at parameter θ , the CVaR becomes C V a Rβ,θ = a ∗ (θ ) +

1 1 b

+

θ 1−β

.

So, here we can see explicitly how the worst-case CVaR increases compared to the nominal CVaR. The corresponding relative entropy is

for some constant c(θ, α, a) satisfying (15) and (16). If the density function of X under the nominal distribution is regularly varying with index ρ, i.e. lim x→∞ f (t x)/ f (x) = t ρ for any t > 0 and some index ρ < 0, then under the worstcase change of measure it is regularly varying with index ρ + 1/(α − 1), as suggested by (19). We require ρ + 1/(α − 1) < 0 to guarantee the new density function is well defined. Because α > 1, the worst index is smaller than the nominal one, meaning that the worst-case distribution has a heavier tail. For purposes of illustration, it is convenient to choose as nominal model a generalized Pareto distribution with density function

 1 ξgp − ξgp −1 1 1+ x , for x ≥ 0, some bgp > 0 f (x) = b bgp and ξgp > 0, or a generalized extreme value distribution with density 

1 1 1 − ξgev −1 − ξgev , f (x) = (1 + ξgev x) exp −(1 + ξgev x) ξgev for x ≥ 0 and ξgev > 0.

These are regularly varying with index −(1 + 1/ξ ), with ξ = ξgp or ξ = ξgev , accordingly. Figure 3 shows two examples—a generalized Pareto density on the left and a generalized extreme value distribution on the right, each shown on a log scale. In each case, the figure compares the nominal distribution and the worst-case distribution with α = 4. As in figure 4, the worst-case model error shifts the VaR to the right and increases the weight of the tail beyond the shifted VaR, increasing the CVaR. A recurring and inevitable question in incorporating robustness into risk measurement is how much uncertainty to allow— in other words, where to set θ or α. If the distribution of X is

41

Robust risk measurement and model risk log PDF of X following Generalized Pareto

log PDF of D following Generalized extreme value

0

−1 nominal θ=0.01 α=4

−1 −2

−3

log(PDF)

log(PDF)

−3 −4 −5 −6 θ

−8

VaR

=4.85

2

4

VaR =5.69

nominal

0

CVaR

θ

6

−5 −6 CVaR =11.71 θ

−8

=8.23

nominal

8

−4

−7

CVaR =11.83

−7

−9

nominal θ=0.01 α=4

−2

10

−9

12

VaRnominal=4.79

0

2

VaRθ=5.64

4

6

x

CVaRnominal=8.20

8

10

12

x

Figure 3. Density of X . The nominal distribution is generalized Pareto (left) or generalized extreme value (right), with parameters bgp = 1 (scale), ξgp = 0.3 (shape), and ξgev = 0.3 (shape). Other parameters are θ = 0.01, α = 4, and β = 95%.

PDF of X

PDF of X

0.5

0.07

nominal θ=0.03

0.45

nominal θ=0.03

0.06

0.4 CVaR

nominal CVaR =3.81

0.3

θ

0.25

CVaRnominal=3.30

0.05

=3.30

PDF

PDF

0.35

0.04 0.03

0.2 0.15

CVaRθ=3.81

0.02

0.1

VaR

0 −10

−5

0

nominal

0.01

VaRθ=3.19 VaR =2.30 nominal

0.05

=2.30

VaRθ=3.19

5

10

0

2

2.5

3

3.5

4

4.5

5

x

x

Figure 4. The dotted red line shows the worst-case density, with β = 95% and θ = 0.03, relative to a D E(0, 1) nominal density (the solid blue line). The right panel gives a magnified view of the right tail.

estimated from historical data, then the precision with which the tail decay of X is estimated (the exponential decay in the light-tailed setting and power decay in the heavy-tailed setting) can provide some guidance on how much uncertainty should be incorporated, as we saw in Section 4.2. Also, the Monte Carlo approach presented in Section 3 illustrates how auxiliary quantities (for example, moments of X ) can be calculated under the worst-case change of measure to gauge its plausibility.

6. Portfolio credit risk In this section, we apply robustness to the problem of portfolio credit risk measurement. We develop the application within the framework of the standard Gaussian copula model; the same techniques are applicable in other models as well.

6.1. The Gaussian copula model We consider a portfolio exposed to n obligors, and we focus on the distribution of losses at a fixed horizon. Let Yi denote the default indicator for ith obligor, meaning that  1, if the ith obligor defaults within the horizon; Yi = 0, otherwise.

A default of obligor i produces a loss of ci , so the total loss from defaults is

L=

n 

ci Yi .

i=1

We are interested in robust measurement of tail probabilities P(L > x) for loss thresholds x. In the Gaussian copula model, each default indicator Yi is represented through the indicator of an event {X i > xi }, where X i has a standard normal distribution, and the threshold xi is chosen so that P(Yi = 1) = P(X i > xi ) = pi , for a given default probability pi . Dependence between default indicators is introduced through correlations between the X i . For simplicity, we focus on a single-factor homogeneous model in which the X i are given by  X i = ρ Z + 1 − ρ 2 ǫi , where Z , ǫ1 , . . . , ǫn are independent standard normal random variables. We interpret Z as a broad risk factor that affects all obligors, whereas ǫi is an idiosyncratic risk associated with the ith obligor only. We have n = 100 obligors, each with a 1% default probability pi , so xi = 2.33. The loss given default is ci ≡ 1 for all i = 1, . . . , n.

42

P. Glasserman and X. Xu 0.18 worst P(L>x) P(L>x) using the worst mean and std

0.16 0.14

P(L>x)

0.12 0.1 0.08 0.06 0.04 0.02

0

0.05

0.1

0.15

Relative entropy

Figure 5. Loss probability as a function relative entropy. The solid blue line shows results under the worst-case change of measure. The dotted red line shows results using parameter values estimated from the worst-case change of measure. The comparison shows that the vulnerability to model error goes well beyond errors in parameters.

6.2. Robustness and model error The Gaussian copula model offers an interesting application because it is both widely used and widely criticized for its shortcomings. Taking the Gaussian copula as a reference model, our interest lies in examining its greatest vulnerabilities to model error—in other words, finding which perturbations of the model (in the sense of relative entropy) produce the greatest error in measuring tail loss probabilities P(L > x). Importantly, we are interested in going beyond parameter sensitivities to understand how the worst-case error changes the structure of the model. Taking our risk measure as P(L > x) means taking V (Z , ǫ1 , . . . , ǫn ) = I L>x , so the worst-case change of measure at parameter θ is m ∗θ ∝ exp(θ I L>x ).  exp(θ ) C P(L ∈ dl) ˜ ∈ dl) = ⇒ P(L 1 C P(L ∈ dl)

if l > x; otherwise.

(41)

Here, C > 1 is a normalization constant. This change of measure lifts the probabilities of losses greater than x and lowers the probability of all other scenarios. Equivalently, we can say that the probability of any outcome of the default indicators (Y1 , . . . , Yn ) is increased by exp(θ )/C if it yields a loss greater than x and is lowered by a factor of C otherwise. We investigate the implications of this transformation to the model through numerical experiments. We take x = 5, which yields P(L > x) = 3.8%. Our results are based on simulation with N = 106 samples. Figure 5 shows how the loss probability varies with relative entropy. The solid blue line shows results under the worst-case change of measure defined by (41). The dotted red line shows results under parameter changes only; these are determined as follows. At each relative entropy level, we simulate results under the worst-case change of measure (41); we estimate all model parameters (the means, standard deviations and correlations for the normal random variables Z , ǫ1 , . . . , ǫn ); we then simulate the Gaussian copula model with these modified parameters.

A comparison of the lines in figure 5 confirms that the worstcase change of measure has an impact that goes well beyond a change in parameter values. If we compare the two curves at the same relative entropy, the worst-case model continues to show a higher loss probability. In other words, focusing on parameter changes only does not fully utilize the relative entropy budget. The changes in parameter values do not maximize the model error at a given relative entropy budget. Table 5 reports parameter estimates obtained under the worstcase model at two values of θ . They indicate, in particular, that the parameters of the ǫi are affected very little by the change in distribution. Indeed, with 95% confidence, Jarque-Bera and Anderson-Darling test reject normality of Z at θ ≥ 1 but fail to reject normality of the ǫi even at θ = 2. The model is more vulnerable to errors in the dependence structure introduced by Z than to errors in the distribution of the idiosyncratic terms. To gain further insight into the worst-case change of distribution, we examine contour plots in figure 6 of the joint density function of ǫ100 and Z . The joint density function is derived by using the original joint density function and the likelihood ratio m ∗θ . The leftmost figure shows θ = 0.5, and the next two correspond to θ = 2. The increase in θ shifts probability mass of Z to the right but leaves the joint distribution of the ǫi essentially unchanged. This shift in Z changes the dependence structure in the copula and produces the lift in the probability mass function of L described by (41). In the middle panel of figure 6, we see a slight asymmetry in the upper right corner, reflecting the fact that defaults are more likely when both the ǫi and Z are increased. The left panel of figure 7 shows contours of the joint density of (X 99 , X 100 ) under the worst-case change of measure, which distorts the upper-right corner, reflecting the increased probability of joint defaults. The right panel shows the ratio of the worst-case density to the nominal density. Figure 8 shows the nominal and worst-case marginal distributions of Z and L. The worst case makes Z bimodal and inflates the distribution of L beyond the threshold of 5. In particular, the greatest vulnerability to model error takes us outside the Gaussian copula model, creating greater dependence between obligors in the direction of more likely defaults, rather than just through a change of parameters within the Gaussian copula framework. Next, we illustrate the effect of imposing constraints on Z , using the method of Section 3.2. We constrain the first moment to equal 0 or the first two moments to equal 0 and 1; one might take these values to be part of the definition of Z . To

Table 5. Statistics of ǫ j and Z under the worst-case change of measure.

max(ρǫi ǫ j , ρǫi ,Z ) min(ρǫi ǫ j , ρǫi ,Z ) average(|ρǫi ǫ j |, |ρǫi ,Z |) average(μǫ j ) average(σǫ j ) average(skewǫ j ) average(excess kur tosisǫ j ) mean of Z standard deviation of Z

θ = 0.5

θ =2

4.3 × 10−3 −3.4 × 10−3 5.6 × 10−3 7.6 × 10−4 1.00 1.7 × 10−3 8.2 × 10−4 0.047 1.04

0.013 −4.7 × 10−3 6.4 × 10−3 6.8 × 10−3 1.01 0.013 0.017 0.39 1.23

43

Robust risk measurement and model risk θ=2

θ=2 3

3

2

2

2

1

1

1

0

(1.86,0)

ε100

ε100

ε100

θ=0.5 3

0

0

−1

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−2

−2

−2

−3 −3

−2

−1

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3

−2

−1

0

z

1

2

−3 −3

3

−2

−1

0

1

2

3

ε99

z

Figure 6. Contours of joint densities of (Z , ǫ100 ) with θ = 0.5 (left) and θ = 2 (middle), and joint density of (ǫ99 , ǫ100 ) at θ = 2 (right).

PDFθ/PDFnominal 4

3

3

2

2

1

1

X99

X99

PDFθ 4

0

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−1

−2

−2

−3

−3

−4 −4

−2

0

2

−4 −4

4

−2

0

2

4

X100

X100

Figure 7. Contours of the joint density of (X 99 , X 100 ) under the worst scenario θ = 2 (left), and the ratio of the worst-case joint density to the nominal density (right).

1

0.4 nominal θ=2 θ=0.8

0.35

−1

0.3

−2

0.25

log(PDF of L)

PDF of Z, θ=2

nominal θ=2 θ=0.8

0

0.2 0.15

−3 −4 −5 −6 −7

0.1

−8 0.05

−9

0 −4

−3

−2

−1

0

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2

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4

z

−10

0

5

10

15

20

25

30

L

Figure 8. Marginal density of Z and L under worst scenario with θ = 0.8 and θ = 2 vs. nominal model.

match relative entropy values, we find that an unconstrained value of θ = 2 corresponds to constrained values θ = 2.7 (with one constraint) and θ = 3.7 (with two constraints); see Table 6. Figure 9 compares the marginal distribution of Z

under the constrained and unconstrained worst-case changes of measure. The constraints lower the height of the second peak in the bimodal distribution of Z . Not surprisingly, the worst-case value of P(L > x) decreases as we add constraints.

44

P. Glasserman and X. Xu

Table 6. Default probability for unconstrained and constrained cases. The values of θ for the constrained cases are chosen to keep the relative entropy fixed across all three cases. P(L > x) Nominal, θ = 0 Unconstrained, θ = 2 Constraint on 1st moment of Z , θ = 2.7 Constraint on 1st and 2nd moments of Z , θ = 3.7 Constraint on marginal distribution of Z , θ = 4

0.037 0.221 0.186 0.153 0.152

distribution at a fixed point in time. A hypothetical adversary can change the dynamics of the underlying asset and will do so in a way that maximizes hedging error subject to a relative entropy constraint. Our objectives are to quantify the potential hedging error, develop a hedging strategy that is robust to model error and to identify the greatest sources of vulnerability to model error in the nominal model.

7.1. Delta hedging: nominal model 0.45 nominal unconstrained constrained Z, 1st mnt constrained Z, 1st & 2nd mnt

0.4 0.35

PDF of Z

0.3 0.25 0.2 0.15 0.1 0.05 0 −4

−3

−2

−1

0

1

2

3

4

Z

Figure 9. Density of Z under the nominal, unconstrained worst-case and constrained worst-case measures.

We can further restrict the marginal distribution of Z through the method of Section 3.3. Such a restriction is important if one indeed takes Z as an overall market risk factor and not simply a tool for constructing a copula. Using 103 samples for Z and 104 samples of ǫ for each realization of Z , we report the resulting probability in the last row of table 6, taking θ = 4 to make the relative entropy roughly equal to that in the unconstrained case with θ = 2. The default probability is slightly smaller than the case with constraints on first and second moments. Figure 10 shows the distribution of ǫ under the worst scenario, taking θ = 9 to make the effect more pronounced. With the marginal distribution of Z held fixed, the potential model error moves to the idiosyncratic terms. The worst-case joint density of (ǫ99 , ǫ100 ) puts greater weight on large values of either ǫ99 or ǫ100 . The worst-case marginal density of ǫ100 changes in a way similar to the marginal density of Z in figures 8 and 9.

7. Delta hedging error In our next application, we take hedging error as our measure of risk. This application goes beyond our previous examples by adding model dynamics to the robust risk measurement framework. The nominal model specifies dynamics for the evolution of an underlying asset, which leads to a hedging strategy for options written on the underlying asset. Model risk in this context can take the form of misspecification of the dynamics of the underlying asset, rather than just a marginal

For simplicity, we take the nominal model to be the Black–Scholes framework. The risk-neutral dynamics of the underlying asset are given by d St = rn dt + σn d Wt , St and the drift under the physical measure is μn . The risk-neutral drift enters in the option delta, but hedging error is generated under the physical measure so the physical drift is also relevant. The subscript n indicates that these parameters apply to the nominal model. We consider the problem of discrete hedging of a European call option with strike K and maturity T : the interval [0, T ] is divided into N T equal periods, and the hedging portfolio is rebalanced at the start of each period. With discrete rebalancing, we introduce hedging error even under the nominal model. We consider a discrete-time implementation of a selffinancing delta hedging strategy. At time t = 0, the proceeds of the sale of the option (at price C(0, T, S0 )) are used to form a portfolio of stock and cash, with rn the interest rate for holding or borrowing cash. We denote by δσn (t, St ) the number of shares of stock held at time t. At time 0, the portfolio’s cash and stock values are given by cash(0) = C(0, T, S0 ) − S0 δσn (0, S0 ),

stock(0) = S0 δσn (0, S0 ).

After the rebalancing at time kT /N T = kt, they are given by  cash(k) = ern t cash(k − 1) − Skt δσn (kt, Skt )  −δσn (k − 1)t, S(k−1)t , stock(k) = Skt δσn (kt, Skt ).

At maturity, the option pays (ST − K )+ , resulting in a hedging error is He = (ST − K )+ − cash(N T ) − stock(N T ).

For our measure of hedging performance, we use E[|He |], the expected absolute hedging error. A hypothetical adversary seeks to perturb the dynamics of S to magnify this hedging error. In our general formulation, we would take X to be the discrete path of the underlying asset and V (X ) = |He |. Alternative approaches to related problems include the uncertain volatility formulation ofAvellaneda et al. (1995), where the volatility is assumed to lie within a closed interval but is otherwise unknown. In Mykland (2000), uncertainty is defined more generally through bounds on integrals of coefficients. Tankov and Voltchkova (2009) study the best volatility parameter to use for delta hedging to minimize expected squared

45

Robust risk measurement and model risk θ=9 0.4

3

0.35

2

0.3

PDF of ε100 θ=9

4

ε100

1 0 −1

0.25 0.2 0.15

−2

0.1

−3

0.05

−4 −4

−2

0

2

4

nominal θ=9

0 −4

−3

−2

−1

0

ε99

1

2

3

4

ε100

Figure 10. The marginal distribution of Z is fixed. The left figure is the joint density of (ǫ99 , ǫ100 ) under the worst scenario, and the right figure is the marginal density of ǫ100 under the worst scenario. Both figures have θ = 9.

7.2. Model error and hedging error Now, we take a dynamic perspective on hedging error. We use simulation to investigate the vulnerability of discrete delta hedging to model error and to examine the worst-case change of measure that leads to hedging errors. We continue to use the Black–Scholes model as the nominal model with the same parameters as before. Our simulation results use 108 paths. From simulated sample paths and (9), we can estimate the optimal likelihood ratio m ∗θ for each path (we use θ = 0.5 for most results), which enables us to estimate the density function of |He | under the worst-case change of measure. The density is illustrated in figure 12, where we can see that the change of measure makes the right tail heavier. In figure 12, the tail

1 nominal robust θ=0.5

0.9 0.8 0.7

δ(0,S0)

0.6 0.5 0.4 0.3 0.2 0.1 0 60

70

80

90

100

110

120

130

140

S0

Figure 11. Optimal delta vs. S0 with θ = 0.5.

PDF of abs(He)

1

nominal worst θ=0.5

0 −1 −2

log(PDF)

hedging error under a jump-diffusion model for the underlying asset. Bertsimas et al. (2000) analyse asymptotics of the delta hedging error as N T → ∞. In delta hedging, the volatility is unknown and is typically extracted from option prices. If the nominal model holds, then the minimizer of hedging error is indeed the nominal volatility σn . Under our formulation of robustness with discrete delta hedging, we can calculate a robust value of this input σn in the sense of minimizing the maximum value of the hedging error E[|He |] at given value of θ . The result is illustrated in figure 11 for an example with an initial stock price of S0 = 100, strike K = 100, maturity T = 1, nominal volatility σn = 0.2, risk-free rate rn = 0.05, resulting in a Black– Scholes call price of 10.45 at t = 0. The drift under the physical measure is μn = 0.1 The figure shows the nominal and robust values of delta as functions of the underlying asset; the robust σn is optimized against the worst-case change of measure at θ = 0.5. The robust delta is slightly larger out-of-the-money and smaller in-the-money. Figure 11 suggests that if we are restricted to delta-hedging but are allowed to use different values for volatility, then the nominal value is almost the best we can do. Branger et al. (2011), among others, also find that Black–Scholes delta hedging performs surprsingly well, even when its underlying assumptions are not satisfied.

−3 −4 −5 −6 −7 −8 0

0.5

1

1.5

2

2.5

3

3.5

abs(Hedging error)

Figure 12. Density of absolute hedging error under nominal and worst scenario, with θ = 0.5.

is fit through a non-parametric method, using the ‘ksdensity’ command in MATLAB with a normal kernel and bandwidth 0.1. To investigate the dynamics of the underlying asset under the worst-case change of measure—in other words, to investigate the adversary’s strategy—we generate paths conditional on

46

0.08

0

0.06

−5

Theta

Gamma

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reaching points (t, St ). For every t = T (2 + 8k/N T ), for k = 1, . . . , 12, and every St = 70 + 6l for l = 1, . . . , 10, we simulate N sample paths conditioned to pass through (t, St ) by using Brownian bridge sampling. If we use pathi to denote the ith simulated path, then the conditional likelihood given St = x is f˜(pathi |St = x) m ∗ (pathi |St = x) = f (pathi |St = x) f˜(pathi ) f (St ∈ (x, x + d x)) = f (pathi ) f˜(St ∈ (x, x + d x)) f˜(pathi ) = m ∗ (pathi ) ∝ f (pathi )

Because the expectation of the conditional likelihood ratio should be 1, we apply the normalization m ∗ (pathi |St = x) =  N

m ∗ (pathi )

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∗ (path

j )/N

across the N simulated paths. As a point of comparison for the simulation results, it is useful to consider potential sources of hedging error. With discrete rebalancing, we would expect a large move in the underlying asset to produce a large hedging error. Figure 13 plots the option gamma and the time-decay theta, and these suggest that the hedging error is particularly vulnerable close to maturity when the underlying is near the strike. (Time in

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Robust risk measurement and model risk AR(1) for random source, with ρ∈ (−0.15,0.15)

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the figure runs from left to right, with time 1 indicating option maturity.) Indeed, the gamma at the strike becomes infinite at maturity.

In figure 14, we use the simulation results to plot contours of the worst drift (upper left) and worst volatility (lower left) of the Brownian increment in the step immediately following

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the conditional value at (t, St ). The conditional worst drift is highest close to maturity and just below the strike and it is lowest close to maturity and just above the strike, as if the adversary were trying to push the underlying toward the strike near maturity to magnify the hedging error. In fact, at every step t, the worst-case drift has an S−shape centred near the strike. The worst-case volatility is also largest near the strike and near maturity, consistent with the view that this is where the model is most vulnerable. If the underlying is far from the strike, large hedging errors are likely to have been generated already, so the adversary does not need to consume relative entropy to generate further hedging errors. The contours of relative entropy show that the adversary expends the greatest effort near the strike and maturity. There is a slight asymmetry in the relative entropy and worst-case volatility below the strike near inception. This may reflect the asymmetry in gamma around the strike, which is greater far from maturity. It should also be noted that the adversary’s strategy is pathdependent, so figure 14 does not provide a complete description. In particular, at any (t, St ), we would expect the adversary to expend greater relative entropy—applying a greater distortion to the dynamics of the underlying—if the accumulated hedging error thus far is small than if it is large. The contours in the figure implicitly average over these cases in conditioning only on (t, St ). To generate figure 14, we used kernel smoothing. The smoothed value at (s, t) is a weighted average of results at (si , ti ), i = 1, . . . , n, using a kernel K (., .) > 0, n f (si , ti )K ((si , ti ), (s, t)) n f smooth (s, t) = i=1 . i=1 K ((si , ti ), (s, t))

In particular, we used K ((s ′ , t ′ ), (s, t)) = φ((s ′ , t ′ ) −(s, t)/a), with φ the density of the standard normal distribution and   a scaled Euclidean normal under which the distance between adjacent corners in the grid is 1. That is, (60, 1) − (60, 0) = 1, (60, 1) − (140, 1) = 1 and so on. The constant a is chosen so that for any neighbouring nodes (s, t) and (s ′ , t ′ ) on the grid, (s, t) − (s ′ , t ′ )/a = 1.

7.3. Comparison with specific model errors In this section, we examine specific types of model errors and compare them against the worst case. In each example, we replace the Black–Scholes dynamics of the underlying asset with an alternative model. For each alternative, we evaluate the hedging error and the relative entropy relative to the nominal model. By controlling for the level of relative entropy, we are able to compare different types of model error, including the worst case, on a consistent basis. In each plot in figure 15, the horizontal axis shows the relative entropy of the perturbed model (with respect to the nominal model) and the vertical axis is the absolute hedging error estimated from simulation. We take values of θ in [0, 0.23]. In panel (a) of figure 15, we perturb the nominal model through serial correlation: we replace the i.i.d. Brownian increments with AR(1) dynamics. The perturbed model thus has W˜ t = ρW˜ t−1 + 1 − ρ 2 ǫt and W˜ 1 = ǫ1 , where ǫt are

independent and normally distributed with mean 0 and variance t. With ρ ∈ (−0.15, 0.15), the relative entropy reaches a minimum near ρ = 0. The expected hedging error seems to be robust with respect to serial dependence, never getting close to the worst case error except near the origin. The second plot in (a) suggests that a larger ρ leads to smaller hedging error. For larger ρ > 0, W˜ is more mean reverting, which may explain the smaller hedging error. In panel (b) of figure 15, we use Merton’s jump-diffusion model, d St = (rn − λE[exp(Yi ) − 1]))dt + σn d Wt + d Jt St−  Nt where J is a compound Poisson, Jt = i=1 exp(Yi ), with Nt a Poisson process with intensity λ, and Yi i.i.d. N (0, σ J ), with σ J = 1. When increasing σ J from 0 to 1, or the jump intensity λ from 0 to 0.2, both the relative entropy and the expected hedging error increase almost linearly, with similar slope. Panels (c) and (d) of figure 15 test the Heston stochastic volatility model, in which the square of volatility vt = σ 2 follows the dynamics √ dvt = κ(βσ − vt )dt + σσ vt d Wtv , (42) where Wtv is a Brownian motion, ρ = corr (Wtv , Wt ). We pick κ = 5, βσ = σn2 = 0.04, ρ = −0.2 and σσ = 0.05. When discretized to dates ti = it, i = 1, . . . , N T , the likelihood ratio for the price process becomes m(st1 , . . . , st NT ) = =

f˜(st1 , . . . , st NT ) f (st1 , . . . , st NT ) E v [ f˜(st1 , . . . , st NT |vt1 , . . . , vt N )] f (st1 , . . . , st NT )

where f and f˜ are the joint density functions of prices under the nominal and Heston models, respectively. In the second equality, f˜(.|.) denotes the conditional density of prices given the variance process, and the expectation is taken over the variance process. The conditional expectation is approximated using 1000 sample paths of v. As the speed of mean-reversion κ changes from 3 to 20, the relative entropy and the expected hedging error decrease. As κ becomes larger, the expected hedging error gets closer to the nominal value, while relative entropy appears to converge to some positive value. With a large κ, any deviation from the nominal variance decays quickly, leaving only a short-term deviation introduced by the diffusion term of (42). As the long-run limit βσ varies from 0.036 to 0.044, relative entropy and expected hedging error attain their lowest values near 0.04, which is the nominal value of squared volatility. Holding fixed the level of relative entropy, the expected hedging error is very similar when βσ < 0.04 and βσ > 0.04. As the volatility of volatility σσ varies from 0 to 0.13, both relative entropy and expected hedging error increase. As σσ gets closer to zero, the volatility behaves more like a constant, which is the nominal model. And, as the correlation ρ between the two Brownian motions varies from −0.5 to 0.7, the change in hedging error is very small, with the maximum hedging error obtained when ρ is close to nominal value −0.2. The relative entropy reaches the minimum value when ρ equals the nominal value −0.2.

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For our last comparison, we use the variance-gamma model of Madan et al. (1998), St = S0 exp((μ + ω)t + X t )

where X t = βvg γ (t; 1, νvg ) + σ Wγ (t;1,νvg ) 

1 2 1 log 1 − βvg νvg − σ νvg ω = νvg 2 where γ (t; 1, νvg ) is the gamma process with unit mean rate. Parameter βvg controls the skewness of return and νvg controls the kurtosis; see panels (b) and (c) of figure 16. The figure suggests that skewness and kurtosis have limited impact on hedging error.

It is noteworthy that in most of the examples in figures 15 and 16, the observed hedging error is significantly smaller than that of the worst-case achievable at the same level of relative entropy. As our final test, we add constraints on the evolution of the underlying asset, thus limiting the adversary’s potential impact. First, we constrain the moments of the realized mean and realized variance of the returns of the underlying asset. Let NT W = i=1 Wi /N T be the average of the Brownian increments Wi along a path. We constrain ]=0  NT the mean E[mW and the realized variance E[m i=1 (Wi − W )2 /(N T − 1)] = t. Figure 17(a) shows that this has only a minor effect on the worst-case hedging error. In figure 17(b), we constrain

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Figure 17. The blue dots are for constraint cases, and the red dots are for the unconstrained case. Table 7. Worst-case results and parameters for CVA example. θ Nominal ρ −0.3 0

CVA ×104

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4.80 (0.43%) 6.27 (0.56%) 7.82 (0.70%) 1.68 2.42 3.18 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 −1.06 −1.06 −1.47 −0.39 −0.39 −0.80

5.86 (0.53%) 7.78 (0.70%) 9.81 (0.88%) 0.89 1.30 1.71 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 −0.79 −0.79 −1.13 −0.27 −0.27 −0.60

7.26 (0.65%) 9.82 (0.88%) 12.49 (1.13%) 0.27 0.39 0.52 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 −0.43 −0.43 −0.67 −0.12 −0.12 −0.34

9.14 (0.82%) 12.60 (1.14%) 16.17 (1.46%) 0.00 0.00 0.00 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 0.05 0.05 −0.06 0.07 0.07 0.014

11.67 (1.05%) 16.45 (1.48%) 21.25 (1.91%) 0.40 0.61 0.80 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 0.70 0.70 0.76 0.32 0.32 0.49

15.14 (1.36%) 21.84 (1.97%) 28.31 (2.55%) 1.99 3.08 4.04 −0.299 0.000 0.299 0.200 0.200 0.200 0.201 0.201 0.201 1.59 1.59 1.89 0.66 0.66 1.12

19.90 (1.79%) 29.45 (2.65%) 38.07 (3.42%) 5.60 8.86 11.44 −0.299 0.000 0.299 0.200 0.201 0.201 0.201 0.201 0.202 2.83 2.83 3.43 1.13 1.13 1.99

26.46 (2.38%) 40.23 (3.62%) 51.36 (4.63%) 12.53 20.27 25.49 −0.299 0.000 0.299 0.201 0.201 0.201 0.201 0.202 0.202 4.56 4.56 5.49 1.78 1.78 3.12

the mean and variance of the realized variance as a way of constraining total volatility. Here, the reduction in the worstcase hedging error is more pronounced. The overall conclusion from figure 17 is that even with constraints on the first two moments of the underlying asset returns, the worst-case hedging error generally remains larger than the hedging errors we see in figures 15 and 16 under specific alternatives. To put it another way, the hypothetical adversary shows much more creativity in undermining the Black–Scholes delta-hedging strategy than is reflected in these models. Indeed, the alternatives are all time-homogeneous, whereas a key feature of figure 14 is that the greatest vulnerabilities occur close to maturity and, to a lesser extent, at inception.

8. Credit valuation adjustment Our final application of the robust risk measurement framework examines CVA, which has emerged as a key tool for quantifying counterparty risk among both market participants and regulators. 8.1. Background on CVA CVAmeasures the cost of a counterparty’s default on a portfolio of derivatives. Rather than model each derivative individually, we will work with a simplified model of the aggregated exposure between two parties. We model this aggregated exposure as an Ornstein–Uhlenbeck process X t ,

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Robust risk measurement and model risk d X t = κx (μx − X t )dt + σx d Wtx .

(43)

This allows the aggregated exposure to be positive for either party (and thus negative for the other); we can think of the two parties as having an ongoing trading relationship so that new swaps are added to their portfolio as old swaps mature, keeping the dynamics stationary. Alternatively, we can take X as a model of the exposure for a forward contract on a commodity or FX product where the underlying asset price is mean-reverting. The time-to-default for the counterparty is modelled through a stochastic default intensity λt , which follows a CIR-jump process  dλt = κλ (μλ − λt )dt + σλ λt d Wtλ + d Jt , where W x and W λ are Brownian motions with correlation ρ, and Jt is a compound Poisson process with jump intensity ν j and jump sizes following an exponential distribution with mean 1/γ . The long-run limit of X matches the initial value, X 0 = μx , and similarly λ0 = μλ . As in Zhu and Pykhtin (2007), the CIR-jump model guarantees that λt ≥ 0. Given the default intensity process, the time of default τ is  t −1 λs ds and ξ ∼ E x p(1), τ =  (ξ ), where(t) = 0

(44) meaning that ξ has a unit-mean exponential distribution and is independent of everything else. The CVA for a time horizon T is then given by C V A = (1 − R)E[e−r τ Iτ 0, then g (θ, a) ↑ ∞ as θ ↑ θmax (a), where θmax (a) := sup{θ : g (θ, a) < ∞}; if P(Va (X ) < 0) > 0, then g (θ, a) ↑ ∞ as θ ↓ θmin (a) where θmin (a) := inf {θ : g (θ, a) < ∞}.

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Part (1) of the assumption ensures feasibility of the optimization problem. (For a maximization problem, we would require that Va (x) be concave in a.) Part (2) ensures the finiteness of Fˆa (θ ) and its derivative, so that the corresponding exponential change of measure is well-defined. We denote by (θmin (a), θmax (a)) the interval (possibly infinite) in which Fˆa (θ ) is finite and thus an exponential change of measure defined by exp(θ Va (X )) is well defined. For any θ > 0 and decision parameter a, if part (2) of Assumption A.1 is satisfied, the optimal change of measure for the adversary is described by the likelihood ratio m ∗θ,a = exp(θ Va (X ))/E[exp(θ Va (X ))],

(46)

where we need θ ∈ (0, θmax (a)) . By substituting (46) into (5), we get η 1 (47) inf inf log E[exp(θ Va (X ))] + . a θ >0 θ θ If θmax (a) < ∞, then as θ ↑ θmax (a), the objective function in (47) goes to infinity, so the infimum over θ will automatically make the optimal θ smaller than θmax . That is, we can safely

consider θ < ∞ instead of θ ∈ (0, θmax ). This allows us to change the order of inf a and inf θ in (5), whereas θmax (a) depends on the decision a. Now, we can relax the constraints for θ in both (5) and (47) to θ > 0. Assumption A.1 is relevant to the inf a and inf θ ordered as (47). To swap the order, we need the following assumption. Assumption A.2 (1) (2)

∗ ) for some If (θ ∗ , a ∗ , m ∗ ) solves (5), then θ ∗ ∈ [0, θmax ∗ ∗ ), the set θmax ∈ [0, ∞] such that for any θ ∈ [0, θmax {a ∈ A : E[exp(θ Va (X ))] < ∞} is compact. ∗ ), E[exp(θ V (X ))] is lower semiFor any θ ∈ [0, θmax a continuous in a.

Because E[exp(θ Va (X ))] is not necessarily continuous in a, the lower semi-contintuity condition in Assumption A.2 is needed to guarantees that the infimum in (8) can be attained. 0 almost surely under Assumption A.3 For any a, Va (X ) > α the nominal measure, and E[Va (X ) α−1 ] < ∞.

Quantitative Finance, Vol. 10, No. 6, June–July 2010, 593–606

Robustness and sensitivity analysis of risk measurement procedures RAMA CONT*yz, ROMAIN DEGUESTy and GIACOMO SCANDOLOx yIEOR Department, Columbia University, New York, NY, USA zLaboratoire de Probabilite´s et Mode`les Ale´atoires, CNRS, Universite´ de Paris VI, Paris, France xDipartimento di Matematica per le Decisioni, Universita` di Firenze, Firenze, Italia (Received 18 December 2008; in final form 1 February 2010) Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property. Keywords: Risk management; Risk measurement; Coherent risk measures; Law invariant risk measures; Value-at-Risk; Expected shortfall

1. Introduction One of the main purposes of quantitative modeling in finance is to quantify the risk of financial portfolios. In connection with the widespread use of Value-at-Risk and related risk measurement methodologies and the Basel committee guidelines for risk-based requirements for regulatory capital, methodologies for measuring of the risk of financial portfolios have been the focus of recent attention and have generated a considerable theoretical literature (Artzner et al. 1999, Acerbi 2002, 2007, Fo¨llmer and Schied 2002, 2004, Frittelli and Rosazza Gianin 2002). In this theoretical approach to risk measurement, a risk measure is represented as a map assigning a number (a measure of risk) to each random payoff. The focus of this literature has been on the properties of such maps and requirements for the risk measurement procedure to be coherent, in a static or dynamic setting. Since most risk measures such as Value-at-Risk or Expected Shortfall are defined as functionals of the

*Corresponding author. Email: [email protected]

portfolio loss distribution, an implicit starting point is the knowledge of the loss distribution. In applications, however, this probability distribution is unknown and should be estimated from (historical) data as part of the risk measurement procedure. Thus, in practice, measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a risk measure that summarizes the risk of this loss distribution. While these two steps have been considered and studied separately, they are intertwined in applications and an important criterion in the choice of a risk measure is the availability of a method for accurately estimating it. Estimation or mis-specification errors in the portfolio loss distribution can have a considerable impact on risk measures, and it is important to examine the sensitivity of risk measures to these errors (Gourieroux et al. 2000, Gourieroux and Liu 2006).

1.1. A motivating example Consider the following example, based on a data set of 1000 loss scenarios for a derivatives portfolio

Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2010 Taylor & Francis http://www.informaworld.com DOI: 10.1080/14697681003685597

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Figure 1. Empirical sensitivity (in percentage) of the historical VaR at 99% and historical ES at 99%.

Figure 2. Empirical sensitivity (in percentage) of the ES at 99% estimated with diverse methods.

incorporating hundreds of different risk factors.y The historical Value-at-Risk (VaR) i.e. the quantile of the empirical loss distribution, and the Expected Shortfall (Acerbi 2002) of the empirical loss distribution, computed at the 99% level are, respectively, 8.887 M$ and 9.291 M$. To examine the sensitivity of these estimators to the addition of a single observation in the data set, we compute the (relative) change (in %) in the estimators when a new observation is added to the data set. Figure 1 displays this measure of sensitivity as a function of the size of the additional observation. While the levels of the two risk measures are not very different, they display quite different sensitivities to a change in the data set, the Expected Shortfall being much more sensitive to large observations while VaR has a bounded sensitivity. While Expected Shortfall has the advantage of being a coherent risk measure (Artzner et al. 1999, Acerbi 2002), it appears to lack robustness with respect to small changes in the data set. Another point, which has been left out of most studies on risk measures (with the notable exception of Gourieroux and Liu 2006), is the impact of the estimation method on these sensitivity properties. A risk measure such as Expected Shortfall (ES) can be estimated in different ways: either directly from the empirical loss distribution (‘historical ES’) or by first estimating a parametric model (Gaussian, Laplace, etc.) from the observed sample and computing the Expected Shortfall using the estimated distribution. Figure 2 shows the sensitivity of the Expected Shortfall for the same portfolio as above, but estimated using three different methods. We observe that different estimators for the same risk measure exhibit very different sensitivities to an additional observation (or outlier). These examples motivate the need for assessing the sensitivity and robustness properties of risk measures in conjunction with the estimation method being used to compute them. In order to study the interplay of a risk

measure and its estimation method used for computing it, we define the notion of risk measurement procedure as a two-step procedure that associates with a payoff X and a data set x ¼ (x1, . . . , xn) of size n a risk estimate b  ðxÞ based on the data set x. This estimator of the ‘theoretical’ risk measure (X ) is said to be robust if small variations in the loss distribution—resulting either from estimation or mis-specification errors—result in small variations in the estimator.

yData courtesy of Socie´te´ Ge´ne´rale Risk Management unit.

1.2. Contribution of the present work In the present work, we propose a rigorous approach for examining how estimation issues can affect the computation of risk measures, with a particular focus on robustness and sensitivity analysis of risk measurement procedures, using tools from robust statistics (Huber 1981, Hampel et al. 1986). In contrast to the considerable literature on risk measures (Artzner et al. 1999, Kusuoka 2001, Acerbi 2002, 2007, Fo¨llmer and Schied 2002, Rockafellar and Uryasev 2002, Tasche 2002), which does not discuss estimation issues, we argue that the choice of the estimation method and the risk measure should be considered jointly using the notion of risk estimator. We introduce a qualitative notion of ‘robustness’ for a risk measurement procedure and a way of quantifying it via sensitivity functions. Using these tools we show that there is a conflict between coherence (more precisely, the subadditivity) of a risk measure and the robustness, in the statistical sense, of its commonly used estimators. This consideration goes against the traditional arguments for the use of coherent risk measures and therefore merits discussion. We complement this abstract result by computing measures of sensitivity, which allow us to quantify the robustness of various risk measures with respect to the data set used to compute them. In particular, we show that the same ‘risk measure’ may exhibit quite different sensitivities depending on the

Robustness and sensitivity analysis of risk measurement procedures estimation procedure used. These properties are studied in detail for some well-known examples of risk measures: Value-at-Risk, Expected Shortfall/CVaR (Acerbi 2002, Rockafellar and Uryasev 2002, Tasche 2002) and the class of spectral risk measures introduced by Acerbi (2007). Our results illustrate, in particular, that historical Value-at-Risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as Expected shortfall. Statistical estimation and sensitivity analysis of risk measures have also been studied by Gourieroux et al. (2000) and Gourieroux and Liu (2006). In particular, Gourieroux and Liu (2006) consider non-parametric estimators of distortion risk measures (which includes the class studied in this paper) and focus on the asymptotic distribution of these estimators. By contrast, we study their robustness and sensitivity using tools from robust statistics. Methods of robust statistics are known to be relevant in quantitative finance. Czellar et al. (2007) discuss robust methods for estimation of interest-rate models. Dell’Aquila and Embrechts (2006) discuss robust estimation in the context of extreme value theory. Heyde et al. (2007), which appeared simultaneously with the first version of this paper, discuss some ideas similar to those discussed here, but in a finite data set (i.e. non-asymptotic) framework. We show that, using appropriate definitions of consistency and robustness, the discussion can be extended to a large-sample/asymptotic framework which is the usual setting for discussion of estimators. Our asymptotic framework allows us to establish a clear link, absent in Heyde et al. (2007), between properties of risk estimators and those of risk measures.

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2. Estimation of risk measures Let ( , F , P) be a given probability space representing market outcomes and L0 be the space of all random variables. We denote by D the (convex) set of cumulative distribution functions (cdf ) on R. The distribution of a random variable X is denoted FX 2 D, and we write X  F if FX ¼ F. The Le´vy distance (Huber 1981) between two cdfs F, G 2 D is 4

d ðF, GÞ ¼ inff" 4 0 : F ðx  "Þ  "  GðxÞ  F ðx þ "Þ þ ", 8x 2 Rg: The upper and lower quantiles of F 2 D of order  2 (0, 1) are defined, respectively, by 4

 qþ  ðF Þ ¼ inffx 2 R : F ðxÞ 4 g  q ðF Þ 4

¼ inffx 2 R : F ðxÞ  g:  Abusing notation, we denote q  ðX Þ ¼ q ðFX Þ. For p  1 p we denote by D the set of distributions having a finite pth moment, i.e. Z jxjp dF ðxÞ 5 1, R

Dp

the set of distributions whose left tail has a and by finite p moment. We denote (F ) the mean of F 2 D1 and  2(F ) the variance of F 2 D2. For any n  1 and any x ¼ (x1, . . . , xn) 2 Rn, 4

Fxemp ðxÞ ¼

n 1X Ifxxi g n i¼1

denotes the empirical distribution of the data set x; Demp will denote the set of all empirical distributions.

1.3. Outline Section 2 recalls some basic notions on distribution-based risk measures and establishes the distinction between a risk measure and a risk measurement procedure. We show that a risk measurement procedure applied to a data set can be viewed as the application of an effective risk measure to the empirical distribution obtained from this data and give examples of effective risk measures associated with various risk estimators. Section 3 defines the notion of robustness for a risk measurement procedure and examines whether this property holds for commonly used risk measurement procedures. We show, in particular, that there exists a conflict between the subadditivity of a risk measure and the robustness of its estimation procedure. In section 4 we define the notion of sensitivity function for a risk measure and compute sensitivity functions for some commonly used risk measurement procedures. In particular, we show that, while historical VaR has a bounded sensitivity to a change in the underlying data set, the sensitivity of Expected Shortfall estimators is unbounded. We discuss in section 5 some implications of our findings for the design of risk measurement procedures in finance.

2.1. Risk measures The ‘Profit and Loss’ (P&L) or payoff of a portfolio over a specified horizon may be represented as a random variable X 2 L  L0( , F , P), where negative values for X correspond to losses. The set L of such payoffs is assumed to be a convex cone containing all constants. A risk measure on L is a map  : L ! R assigning to each P&L X 2 L a number representing its degree of riskiness. Artzner et al. (1999) advocated the use of coherent risk measures, defined as follows. Definition 2.1 (coherent risk measure (Artzner et al. 1999)): A risk measure  : L ! R is coherent if it is (1) monotone (decreasing): (X )  (Y ) provided X  Y; (2) cash-additive (additive with respect to cash reserves): (X þ c) ¼ (X )  c for any c 2 R; (3) positive homogeneous: (X ) ¼ (X ) for any   0; (4) sub-additive: (X þ Y )  (X ) þ (Y ). The vast majority of risk measures used in finance are statistical, or distribution-based risk measures,

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i.e. they depend on X only through its distribution FX:

As a consequence, we recover the well-known facts that ES is a coherent risk measure, while VaR is not.

FX ¼ FY ) ðX Þ ¼ ðY Þ: In this case,  can be represented as a map on the set of probability distributions, which we still denote by . Therefore, by setting 4

ðFX Þ ¼ ðX Þ, we can view  as a map defined on (a subset of ) the set of probability distributions D. We focus on the following class of distribution-based risk measures, introduced by Acerbi (2002) and Kusuoka (2001), which contains all examples used in the literature: Z1 ð1Þ m ðX Þ ¼  q u ðX ÞmðduÞ, 0

where m is a probability measure on (0, 1). Let Dm be the set of distributions of r.v. for which the above risk measure is finite. m can then be viewed as a map m : Dm } R. Notice that if the support of m does not contain 0 or 1, then Dm ¼ D. Three cases deserve particular attention. . Value at Risk (VaR). This is the risk measure that is frequently used in practice and corresponds to the choice m ¼  for a fixed  2 (0, 1) (usually   10%), that is 4

VaR ðF Þ ¼  q  ðF Þ:

ð2Þ

Its domain of definition is all D. . Expected shortfall (ES). This corresponds to choosing m as the uniform distribution over (0, ), where  2 (0, 1) is fixed: Z 4 1 ES ðF Þ ¼ VaRu ðF Þdu: ð3Þ  0 In this case, Dm ¼ D1 , the set of distributions having an integrable left tail. . Spectral risk measures (Acerbi 2002, 2007). This class of risk measures generalizes ES and corresponds to choosing m(du) ¼ (u)du, where  : [0, 1] ! [0, þ1) is a density on [0, 1] and u } (u) is decreasing. Therefore, Z1 4  ðF Þ ¼ VaRu ðF ÞðuÞdu: ð4Þ 0

q

If  2 L (0, 1) (but not in Lqþ") and   0 around 1, then Dp  Dm , where p1 þ q1 ¼ 1. For any choice of the weight m, m defined in (1) is monotone, additive with respect to cash and positive homogeneous. The subadditivity of such risk measures has been characterized as follows (Kusuoka 2001, Fo¨llmer and Schied 2002, Acerbi 2007). Proposition 2.2 (Kusuoka 2001, Fo¨llmer and Schied 2002, Acerbi 2007): The risk measure m defined in (1) is sub-additive (hence coherent) on Dm if and only if it is a spectral risk measure.

2.2. Estimation of risk measures Once a (distribution-based) risk measure  has been chosen, in practice one has first to estimate the P&L distribution of the portfolio from available data and then apply the risk measure  to this distribution. This can be viewed as a two-step procedure. (1) Estimation of the loss distribution FX: one can use either an empirical distribution obtained from a historical or simulated sample or a parametric form whose parameters are estimated from available data. This step can be formalized as a function from X ¼ [n1Rn, the collection of all possible data bx the sets, to D; if x 2 X is a data set, we denote F corresponding estimate of FX. (2) Application of the risk measure  to the estimated bx , which yields an estimator P&L distribution F 4 b b  ðxÞ ¼ ðFx Þ for (X ).

We call the combination of these two steps a risk measurement procedure.

Definition 2.3 (risk measurement procedure): A risk measurement procedure (RMP) is a couple (M, ), where  : D ! R is a risk measure and M : X ! D an estimator for the loss distribution. The outcome of this procedure is a risk estimator b  : X ! R defined as 4 bx Þ, x °b  ðxÞ ¼ ðF

that estimates (X ) given the data x (see diagram).

R =

◦

2.2.1. Historical risk estimators. The historical estimator b  h associated with a risk measure  is the estimator obtained by applying  to the empirical P&L distribution bx ¼ F emp : (sample cdf ) F x b  h ðxÞ ¼ ðFxemp Þ:

For a risk measure m, as in (1), n X b mh ðxÞ ¼ m ðFxemp Þ ¼  wn,i xðiÞ , i¼1

x 2 Rn ,

where x(k) is the kth least element of the set {xi}in, and the weights are equal to   i1 i 4 , for i ¼ 1, . . . , n  1, wn,i ¼ m n n   n1 ,1 : wn,n ¼ m n

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Robustness and sensitivity analysis of risk measurement procedures Historical estimators are L-estimators in the sense of Huber (1981).

Let  be a positively homogeneous risk measure, then we have ðF ð j ÞÞ ¼ ðF Þ:

Example 2.4: Historical VaR is given by d h ðxÞ ¼ xðbncþ1Þ , VaR 

ð5Þ

where bac denotes the integer part of a 2 R. Example 2.5: given by c h ðxÞ ES 

Therefore, if the scale parameter is estimated by maximum likelihood, the associated risk estimator of  is then given by

The historical expected shortfall ES is

! bnc 1 X ¼ xðiÞ þ xðbncþ1Þ ðn  bncÞ : n i¼1

ð6Þ

Example 2.6: The historical estimator of the spectral risk measure  associated with  is given by Z i=n n X b h ðxÞ ¼  wn,i xðiÞ , where wn,i ¼ ðuÞdu: ð7Þ ði1Þ=n

i¼1

2.2.2. Maximum likelihood estimators. In the parametric approach to loss distribution modeling, a parametric model is assumed for FX and parameters are estimated from data using, for instance, maximum likelihood. We call the risk estimator obtained the ‘maximum likelihood risk estimator’ (MLRE). We discuss these estimators for scale families of distributions, which include as a special case (although a multidimensional one) the common variance–covariance method for VaR estimation. Let F be a centered distribution. The scale family generated by the reference distribution F is defined by  x 4 4 DF ¼ fF ð j Þ :  4 0g, where F ðx j Þ ¼ F :  If F 2 Dp ( p  1), then DF  Dp and it is common to choose F with location 0 and scale 1, so that F ( |) has location 0 and scale  2. In line with common practice in short-term risk management we assume that the risk factor changes have location equal to zero. Two examples of scale families of distributions that we will study are: . the Gaussian family where F has density  2 1 x f ðxÞ ¼ pffiffiffiffiffiffi exp  , 2 2p

b  ðxÞ ¼ cb  mle ðxÞ:

Example 2.7 (MLRE for a Gaussian family): The MLE of the scale parameter in the Gaussian scale family is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X ð10Þ b  ðxÞ ¼ x2 : n i¼1 i

The resulting risk estimators are given by b  ðxÞ ¼ cb  ðxÞ, where, depending on the risk measure considered, c is given by c ¼ VaR ðF Þ ¼ z ,

expfz2 =2g pffiffiffiffiffiffi ,  2p Z1 zu ðuÞdu, c ¼  ðF Þ ¼ 

c ¼ ES ðF Þ ¼

0

where z is the -quantile of a standard normal distribution. Example 2.8 (ML risk estimators for Laplace distributions): The MLE of the scale parameter in the Laplace scale family is 1 b ðxÞ ¼ n

n X i¼1

jxi j:

ð11Þ

Note that this scale estimator is not the standard deviation but the Mean Absolute Deviation (MAD). The resulting risk estimator is b  ðxÞ ¼ cb ðxÞ,

ð12Þ

where c takes the following values, depending on the risk measure considered (we assume   0.5): c ¼ VaR ðF Þ ¼  lnð2Þ,

. the Laplace or double exponential family where F has density

c ¼ ES ðF Þ ¼ 1  lnð2Þ, Z 1=2 Z c ¼  ðF Þ ¼  lnð2uÞðuÞdu þ 0

1 1=2

lnð2  2uÞðuÞdu:

1 f ðxÞ ¼ expðjxjÞ: 2 The Maximum Likelihood b ¼b  mle ðxÞ of  is defined by b  ¼ arg:max40

n X i¼1

Estimator

ln f ðxi j Þ,

(MLE)

ð8Þ

and solves the following nonlinear equation: n X i¼1

xi

f 0 ðxi =b Þ ¼ nb : Þ f ðxi =b

ð9Þ

2.3. Effective risk measures In all of the above examples we observe that the risk estimator b  ðxÞ, computed from a data set x ¼ (x1, . . . , xn), can be expressed in terms of the empirical distribution Femp x ; in other words, there exists a risk measure eff such that, for any data set x ¼ (x1, . . . , xn), the risk estimator b  ðxÞ is equal to the new risk measure eff applied to the empirical distribution 4

 ðxÞ: eff ðFxemp Þ ¼ b

ð13Þ

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We will call eff the effective risk measure associated with the risk estimator b . In other words, while  is the risk measure we are interested in computing, the effective risk measure eff is the risk measure that the procedure defined in definition 2.3 actually computes. So far, the effective risk measure eff is defined for all empirical distributions by (13). Consider now a risk estimator b  that is consistent with the risk measure  at F 2 D, that is

with b h coincides with . The same remains true even if the density  is not decreasing, so that  is not a spectral risk measure.

for any i.i.d. sequence Xi  F. Consistency of a risk estimator for a class of distributions of interest is a minimal requirement to ask for. If b  is consistent with the risk measure  for F 2 Deff  D, we can extend eff to Deff as follows: for any sequence (xi)i1 such that

Example 2.13 (Laplace ML risk estimator): Consider the risk estimators introduced in example 2.8. The associated effective risk measure is defined on D1 and given by Z eff ðF Þ ¼ cðF Þ, where ðF Þ ¼ jxjdF ðxÞ:

n!1

b  ðX1 , . . . , Xn Þ ! ðF Þ a.s.,

n 1X d If xi g ! F ð Þ 2 Deff , n i¼1

Notice that, in both of these examples, the effective risk measure eff is different from the original risk measure .

4

eff ðF Þ ¼ lim b  ðx1 , . . . , xn Þ:

ð14Þ

n!1

3. Qualitative robustness

Consistency guarantees that eff (F ) is independent of the chosen sequence. Definition 2.9 (effective risk measure): Let b  : X ! R be a consistent risk estimator of a risk measure  for a class Deff of distributions. There is a unique risk measure eff : Deff } R such that for any x ¼ (x1, . . . , xn) 2 X; 4  ðx1 , . . . , xn Þ . eff ðF Þ ¼ limn!1 b sequence (xi)i1 such that

.

R

R

we define

4 eff ðFxemp Þ ¼ b  ðxÞ

Example 2.12 (Gaussian ML risk estimator): Consider the risk estimator introduced in example 2.7. The associated effective risk measure is defined on D2 and given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z eff ðF Þ ¼ cðF Þ, where ðF Þ ¼ x2 dF ðxÞ:

data for

set any

n 1X d If xi g ! F ð Þ 2 Deff : n i¼1

Equation (13), defining the effective risk measure, allows us in most examples to characterize eff explicitly. As shown in the examples below, eff may be quite different from  and lacks many of the properties  was initially chosen for. Example 2.10 (historical VaR): The empirical quantile d h is a consistent estimator of VaR for any F 2 D such VaR   dh that qþ  ðF Þ ¼ q ðF Þ. Otherwise, VaR ðX1 , . . . , Xn Þ may not have a limit as n ! 1. Therefore, the effective risk d h is VaR restricted to measure associated with VaR  the set  Deff ¼ fF 2 D : qþ  ðF Þ ¼ q ðF Þg:

Example 2.11 (historical estimator of ES and spectral risk measures): A general result on L-estimators by Van Zwet (1980) implies that the historical estimator of any spectral risk measure  (in particular of the ES) is consistent with  at any F where the risk measure is defined. Therefore, the effective risk measure associated

We now define the notion of qualitative robustness of a risk estimator and use it to examine the robustness of the various risk estimators considered above. 3.1. C-Robustness of a risk estimator Fix a set C D representing the set of ‘plausible’ loss distributions, containing all the empirical distributions: Demp C. In most examples the class C is specified via an integrability condition (e.g., existence of moments) in which case it automatically contains all empirical distributions. Consider an interior element F 2 C, i.e. for any 40, there exists G 2 C, with G 6¼ F, such that d(G, F )  . We call a (risk) estimator C-robust at F if the law of the estimator is continuous with respect to a change in F (remaining in C ) uniformly in the size n of the data set. Definition 3.1: A risk estimator b  is C-robust at F if, for any "40, there exist 40 and n0  1 such that, for all G 2 C, d ðG, F Þ   ¼) d ðLn ðb , GÞÞ, Ln ðb , F ÞÞ  ",

8n  n0 ,

where d is the Le´vy distance. When C ¼ D, i.e. when any perturbation of the distribution is allowed, the previous definition corresponds to the notion of qualitative robustness (also called asymptotic robustness) as outlined by Huber (1981). This case is not generally interesting in econometric or financial applications since requiring robustness against all perturbations of the model F is quite restrictive and excludes even estimators such as the sample mean. Obviously, the larger the set of perturbations C, the harder it is for a risk estimator to be C-robust. In the

Robustness and sensitivity analysis of risk measurement procedures remainder of this section we will assess whether the risk estimators previously introduced are C-robust w.r.t. a suitable set of perturbations C. 3.2. Qualitative robustness of historical risk estimators

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Proposition 3.5: The historical estimator of VaR is C-robust at any F 2 C, where 4

 C ¼ fF 2 D : qþ  ðF Þ ¼ q ðF Þg:

In other words, if the quantile of the (true) loss distribution is uniquely determined, then the empirical quantile is a robust estimator.

The following generalization of a result of Hampel et al. (1986) is crucial for the analysis of robustness of historical risk estimators. Proposition 3.2: Let  be a risk measure and F 2 C D. If b  h , the historical estimator of , is consistent with  at every G 2 C, then the following are equivalent: (1) the restriction of  to C is continuous (w.r.t. the Le´vy distance) at F; (2) b  h is C-robust at F.

A proof of proposition 3.2 is given in appendix A. From this proposition, we obtain the following corollary that provides a sufficient condition on the risk measure to ensure that the corresponding historical/empirical estimator is robust. Corollary 3.3: at any F 2 C.

bh

If  is continuous in C, then  is C-robust

Proof: Fix G 2 C and let (Xn)n1 be an i.i.d. sequence distributed as G. Then, by the Glivenko–Cantelli Theorem we have, for almost all !, n!1

emp d ðFXð!Þ , GÞ ! 0,

X ¼ ðX1 , . . . , Xn Þ:

By continuity of  at G it holds that, again for almost all !, emp b  ðXð!ÞÞ ¼ ðFXð!Þ Þ ! ðGÞ, and therefore b  is consistent with  at G. A simple application of proposition 3.2 concludes. œ

Our analysis will use the following important result adapted from Huber (1981, theorem 3.1). For a measure m on [0, 1] let 4

Am ¼ f 2 ½0, 1 : mðfgÞ 4 0g

be the set of values where m puts a positive mass. We remark that Am is empty for a continuous m (as in the definition of spectral risk measures). Theorem 3.4: Let m be a risk measure of the form (1). If the support of m does not contain 0 or 1, then m is  continuous at any F 2 D such that qþ  ðF Þ ¼ q ðF Þ for any  2 Am. Otherwise, m is not continuous at any F 2 D. In other words, a risk measure of the form (1) can be continuous at some F if and only if its computation does not involve any extreme quantile (close to 0 or 1.) In this case, continuity is ensured provided F is continuous at all points where m has a point mass.

3.2.1. Historical VaR. In this case, Am ¼ {}, so combining corollary 3.3 and theorem 3.4 we have the following.

3.2.2. Historical estimator of ES and spectral risk measures. Let  defined in (4) be in terms of a density  in Lq(0, 1), so that D ¼ Dp ( p and q are conjugate.) However, here we do not assume that  is decreasing, so that  need not be a spectral risk measure, although it is still in the form (1). Proposition 3.6: For any F 2 Dp, the historical estimator of  is Dp-robust at F if and only if, for some "40, ðuÞ ¼ 0,

8u 2 ð0, "Þ [ ð1  ", 1Þ,

ð15Þ

i.e.  vanishes in the neighborhood of 0 and 1. Proof: We have seen in section 2.3 that b h is consistent with  at any F 2 D. If (15) holds for some ", then the support of m (recall that m(du) ¼ (u)du) does not contain 0 or 1. As Am is empty, theorem 3.4 yields continuity of  at any distribution in D. Hence, we have D-robustness of b  at F thanks to corollary 3.3. On the contrary, if (15) does not hold for any ", then 0 or 1 (or both) are in the support of m and therefore  is not continuous at any distribution in D, in particular at F. Therefore, by proposition 3.2 we conclude that b  is not D-robust at F. œ

An immediate, but important consequence is the following.

Corollary 3.7: The historical estimator of any spectral risk measure  defined on Dp is not Dp-robust at any F 2 Dp. In particular, the historical estimator of ES is not D1-robust at any F 2 D1. Proof: It is sufficient to observe that, for a spectral risk measure, the density  is decreasing and therefore it cannot vanish around 0, otherwise it would vanish on the entire interval [0, 1]. œ Proposition 3.6 illustrates a conflict between subadditivity and robustness: as soon as we require a (distribution-based) risk measure m to be coherent, its historical estimator fails to be robust (at least when all possible perturbations are considered). 3.2.3. A robust family of risk estimators. We have just seen that ES has a non-robust historical estimator. However, we can remove this drawback by slightly modifying its definition. Consider 0515251 and define the risk measure Z 2 1 VaRu ðF Þdu: 2   1  1

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This is simply the average of VaR levels across a range of loss probabilities. As 1 I 5u52 ðuÞ ¼ 2   1 1 vanishes around 0 and 1, proposition 3.6 shows that the historical (i.e. using the empirical distribution directly) risk estimator of this risk measure is D1-robust. Of course, the corresponding risk measure is not coherent since  is not decreasing. Note that, for 151/n, where n is the sample size, this risk estimator is indistinguishable from the historical Expected Shortfall! Yet, unlike Expected Shortfall estimators, it has good robustness properties as n ! 1. One can also consider a discrete version of the above risk measure: k 1X VaRuj ðF Þ, 0 5 u1 5 5 uk 5 1, k j¼1

The following result exhibits conditions on the function under which the MLE of the scale parameter is weakly continuous on D . Theorem 3.10 (weak continuity of the scale MLE): Let DF be the scale family associated with the distribution F 2 D and assume that (F ) ¼ 0, (F ) ¼ 1 and F admits a density f. Suppose now that , defined as in (16), is even, increasing on Rþ, and takes values of both signs. Then, the following two assertions are equivalent: .  mle : D } Rþ, defined as in (17), is weakly continuous at F 2 D ; R . is bounded and (, F ) X (x/)F (dx) ¼ 0 has a unique solution  ¼  mle(F ) for all F 2 D . A proof is given in appendix A. Using the above result, we can now study the qualitative robustness of parametric risk estimators for Gaussian or Laplace scale families.

which enjoys similar robustness properties.

3.3. Qualitative robustness of the maximum likelihood risk estimator We now discuss the qualitative robustness for MLRE in a scale family of a risk measure m defined as in (1). First, we generalize the definition of the scale Maximum 4 Likelihood Estimator b  mle ðxÞ ¼  mle ðFxemp Þ given in equations (8) and (9) for distributions that do not belong to Demp. Let DF be the scale family associated with the distribution F 2 D and assume that (F ) ¼ 0, (F ) ¼ 1 and F admits a density f. Define the function   

@ 1 x f 0 ðxÞ 4 ¼ 1  x ln f , x 2 R: ðxÞ ¼ @   f ðxÞ ¼1 ð16Þ

mle

The ML estimator  (G ) of the scale parameter (G ) for G 2 DF corresponds to the unique  that solves Z   x 4 ð, GÞ ¼ ð17Þ GðdxÞ ¼ 0, for G 2 DF :  R By defining D ¼ {G 2 D : j (x)jG(dx)51}, we can extend the definition of  mle(G ) to all G 2 D . Note that when G 2 = DF,  mle(G ) may exist if G 2 D but does not correspond to the ML estimator of the scale parameter of G. Moreover, from definition (17), we notice that if we compute the ML estimator of the scale parameter for a distribution Gxemp 2 Demp we recover the MLE b  mle ðxÞ introduced in equations (8) and (9). In the examples below, we have computed the function for the Gaussian and Laplace scale families. Example 3.8 (Gaussian scale family): The function for the Gaussian scale family is g

ðxÞ ¼ 1 þ x2 ,

2

and we immediately conclude that D g ¼ D . Example 3.9 (Laplace scale family): The function the Laplace scale family is equal to l

ðxÞ ¼ 1 þ jxj,

and we obtain D l ¼ D1.

g

ð18Þ l

for

ð19Þ

Proposition 3.11 (non-robustness of Gaussian and Laplace MLRE): Gaussian (respectively Laplace) MLRE of cash-additive and homogeneous risk measures are not D2-robust (respectively D1-robust) at any F in D2 (respectively in D1). Proof: We detail the proof for the Gaussian scale family. The same arguments hold for the Laplace scale family. Let us consider a Gaussian MLRE of a translation invariant and homogeneous risk measure, denoted by b  ðxÞ ¼ c b  mle ðxÞ. First of all we notice that the function g associated with the MLE of the scale parameter of a distribution belonging to the Gaussian scale family is even, and increasing on Rþ. Moreover, it takes values of both signs. Secondly, we recall that the effective risk measure associated with the Gaussian ML risk estimator is eff (F ) ¼ c mle(F ) for all F 2 Deff ¼ D g ¼ D2. Therefore, as g is unbounded, by using theorem 3.10, we know that eff is not continuous at any F 2 D2. As the h Gaussian MLRE considered b  verifies b  ðxÞ ¼ c eff ðxÞ, 2 and is consistent with eff at all F 2 D by construction, we can apply proposition 3.2 to conclude that, for F 2 D2, b  is not D2-robust at F. œ 4. Sensitivity analysis In order to quantify the degree of robustness of a risk estimator, we now introduce the concept of the sensitivity function. Definition 4.1 (sensitivity function of a risk estimator): The sensitivity function of a risk estimator at F 2 Deff is the function defined by eff ð"z þ ð1  "ÞF Þ  eff ðF Þ 4 , SðzÞ ¼ Sðz; F Þ ¼ limþ "!0 " for any z 2 R such that the limit exists. S(z; F ) measures the sensitivity of the risk estimator to the addition of a new data point in a large sample. It corresponds to the directional derivative of the effective risk measure eff at F in the direction z 2 D. In the

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Robustness and sensitivity analysis of risk measurement procedures language of robust statistics, S(z; F ) is the influence function (Costa and Deshayes 1977, Deniau et al. 1977, Huber 1981, Hampel et al. 1986) of the effective risk measure eff and is related to the asymptotic variance of the historical estimator of  (Huber 1981, Hampel et al. 1986). Remark 1: If D is convex and contains all empirical distributions, then "z þ (1  ")F 2 D for any " 2 [0, 1], z 2 R and F 2 D. These conditions hold for all the risk measures we are considering.

The case q()5z is handled in a very similar way. Finally, if z ¼ q(), then, again by (21), we have q(F") ¼ z for any " 2 [0, 1). Hence, SðzÞ ¼ 

d q ðF" Þj"¼0 ¼ 0, d" œ

and we conclude.

This example shows that the historical VaR has a bounded sensitivity to a change in the data set, which means that this risk estimator is not very sensitive to a small change in the data set.

4.1. Historical VaR We have seen above that the effective risk measure d h is the restriction of VaR to associated with VaR   Deff ¼ C ¼ fF 2 D : qþ  ðF Þ ¼ q ðF Þg:

The sensitivity function of the historical VaR has the following simple explicit form. Proposition 4.2: If F 2 D admits a strictly positive density f, then the sensitivity function at F of the historical VaR is 8 1 if z 5 q ðF Þ, > < f ðq ðF ÞÞ , if z ¼ q ðF Þ, ð20Þ SðzÞ ¼ 0, > :   , if z 4 q ðF Þ:  f ðq ðF ÞÞ

Proof: First we observe that the map u } q(u) X qu(F ) is the inverse of F and so it is differentiable at any u 2 (0, 1) and we have q0 ðuÞ ¼

1 1 : ¼ F 0 ðqðuÞÞ f ðqu ðF ÞÞ

Fix z 2 R and set, for " 2 [0, 1), F" ¼ "z þ (1  ")F, so that F  F0. For "40, the distribution F" is differentiable at any x 6¼ z, with F"0 ðxÞ ¼ ð1  "Þ f ðxÞ 4 0, and has a jump (of size ") at the point x ¼ z. Hence, for any u 2 (0, 1) and 4 þ " 2 [0, 1), F" 2 Cu, i.e. q In u ðF" Þ ¼ qu ðF" Þ ¼ qu ðF" Þ. particular, 8   < q1" for  5 ð1  "ÞF ðzÞ, , ð21Þ q ðF" Þ ¼ q " , for   ð1  "ÞF ðzÞ þ ", 1" : z, otherwise. Assume now that z4q(), i.e. F(z)4; from (21) it follows that     q ðF" Þ ¼ q , for " 5 1  : 1" F ðzÞ

As a consequence,

VaR ðF" Þ  VaR ðF0 Þ d ¼  q ðF" Þj"¼0 " d"   d   

1  ¼ q

¼ d" 1  " "¼0 f ðqð=ð1  "ÞÞÞ ð1  "Þ2 "¼0  : ¼ f ðq ðF ÞÞ

SðzÞ ¼ limþ "!0

4.2. Historical estimators of Expected Shortfall and spectral risk measures We now consider a spectral risk measure  defined by a weight function  as in (4). Proposition 4.3: Consider a distribution F with a density f 40. Assume the following. (1) Z

1

u ðuÞdu 5 1: f ðqu ðF ÞÞ

0

(2) The risk measure is only sensitive to the left tail (losses) 9u0 2 ð0, 1Þ,

8u 4 u0 ,

ðuÞ ¼ 0:

(3) The density f is increasing in the left tail: 9x050, f increasing on (1, x0). The sensitivity function at F 2 D of the historical estimator of  is then given by Z F ðzÞ Z1 u 1u SðzÞ ¼  ðuÞdu þ ðuÞdu: f ðqu ðF ÞÞ 0 F ðzÞ f ðqu ðF ÞÞ We note that the assumptions in the proposition are verified for Expected Shortfall and for all commonly used parametric distributions in finance: Gaussian, Student-t, double-exponential, Pareto (with exponent 41), etc. Proof: Using the notation introduced in the proof of proposition 4.2 we have SðzÞ ¼ limþ "!0

Z

0

1

VaRu ðF" Þ  VaRu ðF Þ ðuÞdu: "

We will now show that the order of the integral and the limit ! 0 can be interchanged using a dominated convergence argument. First note that lim"!0þ "1(VaRu(F"))  VaRu(F )) exists, and is finite for all u 2 (0, 1). Define u ¼ inf{F(x0), u0} and consider   "):

040. By the mean value theorem, for any u  u(1 8  0 ,

9 2 ð0, Þ,

VaRu ðF" Þ ¼ VaRu ðF0 Þ  "

u : ð1  Þ2 f ðqðu=ð1  ÞÞÞ

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Therefore, we have

VaRu ðF" Þ  VaRu ðF Þ



" u  ð1  Þ2 f ðqðu=ð1  ÞÞÞ u  , 2 ð1  "0 Þ f ðqðu=ð1  ÞÞÞ u  2 L1 ðÞ, ð1  "0 Þ2 fðqðuÞÞ

As a consequence,  pffiffiffiffiffiffiffiffiffiffiffiffiffi   d c  z 2  2 ðF" Þ ¼ 1 : 0 ð0Þ ¼ c d" 2  "¼0  5 "  "0 ,

œ

4.4. ML risk estimators for Laplace distributions

f, q increasing,

so that we can apply dominated convergence Z1 VaRu ðF" Þ  VaRu ðF Þ SðzÞ ¼ limþ ðuÞdu " 0 "!0  Z1  d qu ðF" Þ ðuÞdu  ¼ d" 0 "¼0 Z1 Z F ðzÞ u 1u ðuÞdu þ ðuÞdu, ¼ f ðqu ðF ÞÞ 0 F ðzÞ f ðqu ðF ÞÞ

We have seen that the effective risk measure of the Laplace MLRE of VaR, ES, or any spectral risk measure is eff ðF Þ ¼ cðF Þ,

where c ¼ (G ), G is the distribution with density R g(x) ¼ ejxj/2, and (F ) ¼ Rjxj dF(x).

Proposition 4.6: Let  be a positively homogeneous risk measure. The sensitivity function at F 2 D1 of its Laplace MLRE is SðzÞ ¼ cðjzj  ðF ÞÞ:

œ

thanks to proposition 4.2.

F 2 D1 ,

Since the effective risk measure associated with historical ES R is ES itself, defined on D1 ¼ fF 2 D : x F ðdxÞ 5 1g, an immediate consequence of the previous proposition is the following.

Proof: As usual, we have, for z 2 R, S(z) ¼ 0 (0), where (") ¼ c(F"), F" ¼ (1  ")F þ "z and c is defined above. We have

Corollary 4.4: The sensitivity function at F 2 for historical ES is if z  q ðF Þ,  z þ 1  q ðF Þ  ES ðF Þ, SðzÞ ¼ if z  q ðF Þ. q ðF Þ  ES ðF Þ,

and we conclude that

D1

This result shows that the sensitivity of historical ES is linear in z, and thus unbounded. This means that this risk measurement procedure is less robust than the historical VaR. 4.3. ML risk estimators for Gaussian distributions We have seen that the effective risk measure associated with Gaussian maximum likelihood estimators of VaR, ES, or any spectral risk measure is eff ðF Þ ¼ cðF Þ,

2

F 2 Deff ¼ D ,

where c ¼ (Z ), Z  N(0, 1), is a constant depending only on the risk measure  (we are not interested in its explicit value here). Proposition 4.5: The sensitivity function at F 2 D2 of the Gaussian ML risk estimator of a positively homogeneous risk measure  is   c  z 2 SðzÞ ¼ 1 : 2  Proof: Let, for simplicity,  ¼ (F ). Fix z 2 R and set, as usual, F" ¼ (1  ")F þ "z (" 2 [0, 1)); observe that F" 2 D2 for any ". If we set (") X c(F"), with c ¼ (N(0, 1)), then we have S(z) ¼ 0 (0). It is immediate to compute Z Z  2 ðF" Þ ¼ x2 F" ðdxÞ ¼ ð1  "Þ x2 F ðdxÞ þ "z2  "z2

ð"Þ ¼ cð1  "ÞðF Þ þ c"jzj, 0 ð0Þ ¼ cjzj  cðF Þ: œ This proposition shows that the sensitivity of the Laplace MLRE at any F 2 D1 is not bounded, but linear in z. Nonetheless, the sensitivity of the Gaussian MLRE is quadratic at any F 2 D2, which indicates a greater sensitivity to outliers in the data set. Table 1. Behavior of sensitivity functions for some risk estimators. Risk estimator

Dependence in z of S(z)

Historical VaR Gaussian ML for VaR Laplace ML for VaR Historical Expected Shortfall Gaussian ML for Expected Shortfall Laplace ML for Expected Shortfall

Bounded Quadratic Linear Linear Quadratic Linear

4.5. Finite-sample effects The sensitivity functions computed above are valid for (asymptotically) large samples. In order to assess the finite-sample relevance accuracy of risk estimator sensitivities, we compare them with the finite-sample sensitivity SN ðz; F Þ ¼

b  ðX1 , . . . , XN , zÞ  b  ðX1 , . . . , XN Þ : 1=ðN þ 1Þ

Figure 3 compares the empirical sensitivities of historical, Gaussian, and Laplace VaR and historical, Gaussian, and Laplace ES with their theoretical (large sample) counterR R parts. We have used the same set of N ¼ 1000 loss ¼ ð1  "Þð 2 Þ þ "z2  "2 z2 ¼  2 þ "½z2   2   "2  2 : scenarios as in section 1.1.

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Robustness and sensitivity analysis of risk measurement procedures x 107

12 Empirical Sensitivity of historical VaR Theorical Sensitivity of historical VaR

5 4 3 2 1 0 −1 −10

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Figure 3. Empirical versus theoretical sensitivity functions of risk estimators for  ¼ 1% at a 1 day horizon. Historical VaR (upper left), historical ES (upper right), Gaussian VaR (left), Gaussian ES (right), Laplace VaR (lower left), Laplace ES (lower right).

The asymptotic and empirical sensitivities coincide for all risk estimators except for historical risk measurement procedures. For the historical ES, the theoretical sensitivity is very close to the empirical one. Nonetheless, we note that the empirical sensitivity of the historical VaR can be equal to 0 because it is strongly dependent on the integer part of N, where N is the number of scenarios and  the quantile level. This dependency disappears asymptotically for large samples. The excellent agreement shown in these examples illustrates that the expressions derived above for theoretical sensitivity functions are useful for evaluating the sensitivity of risk estimators for realistic sample sizes.

This is useful since theoretical sensitivity functions are analytically computable, whereas empirical sensitivities require perturbating the data sets and recomputing the risk measures.

5. Discussion 5.1. Summary of main results Let us now summarize the contributions and main conclusions of this study.

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First, we have argued that when the estimation step is explicitly taken into account in a risk measurement procedure, issues like robustness and sensitivity to the data set are important and need to be accounted for with at least the same attention as the coherence properties set forth by Artzner et al. (1999). Indeed, we do think that it is crucial for regulators and end-users to understand the robustness and sensitivity properties of the risk estimators they use or design to assess the capital requirement, or manage their portfolio. Indeed, an unstable/non-robust risk estimator, be it related to a coherent measure of risk, is of little use in practice. Second, we have shown that the choice of the estimation method matters when discussing the robustness of risk measurement procedures: our examples show that different estimation methods coupled with the same risk measure lead to very different properties in terms of robustness and sensitivity. Historical VaR is a qualitatively robust estimation procedure, whereas the proposed examples of coherent (distribution-based) risk measures do not pass the test of qualitative robustness and show high sensitivity to ‘outliers’. This explains perhaps why many practitioners have been reluctant to adopt ‘coherent’ risk measures. Also, most parametric estimation procedures for VaR and ES lead to non-robust estimators. On the other hand, weighted averages of historical VaR such as Z 2 1 VaRu ðF Þdu, 2  1 1 with 1424140, have robust empirical estimators.

5.2. Re-examining subadditivity The conflict we have noted between robustness of a risk measurement procedure and the subadditivity of the risk measure shows that one cannot achieve robust estimation in this framework while preserving subadditivity. While a strict adherence to the coherence axioms of Artzner et al. (1999) would lead to choosing subadditivity over robustness, several recent studies (Danielsson et al. 2005, Heyde et al. 2007, Ibragimov and Walden 2007, Dhaene et al. 2008) have provided reasons for not doing so. Danielsson et al. (2005) explore the potential for violations of VaR subadditivity and report that, for most practical applications, VaR is sub-additive. They conclude that, in practical situations, there is no reason to choose a more complicated risk measure than VaR, solely for reasons of subadditivity. Arguing in a different direction, Ibragimov and Walden (2007) show that, for very ‘heavy-tailed’ risks defined in a very general sense, diversification does not necessarily decrease tail risk but actually can increase it, in which case requiring subadditivity would in fact be unnatural. Finally, Finally, Heyde et al. (2007) argue against subadditivity from an axiomatic viewpoint and propose to replace it by a weaker property of co-monotonic subadditivity. All these objections to the subadditivity axiom deserve serious consideration and further support the choice of robust risk

measurement procedures over non-robust procedures for the sole reason of saving subadditivity.

5.3. Beyond distribution-based risk measures While the ‘axiomatic’ approach to risk measurement in principle embodies a much wider class of risk measures than distribution-based (or ‘law-invariant’) risk measures, research has almost exclusively focused on this rather restrictive class of risk measures. Our result, that coherence and robustness cannot coexist within this class, can also be seen as an argument for going beyond distribution-based risk measures. This also makes sense in the context of the ongoing discussion on systemic risk: evaluating exposure to systemic risk requires considering the joint distribution of a portfolio’s losses with other, external, risk factors, not just the marginal distribution of its losses. In fact, risk measures that are not distribution-based are routinely used in practice: the Standard Portfolio Analysis of Risk (SPAN) margin system, and cited as the original motivation by Artzner et al. (1999), is a well-known example of such a method used by many clearinghouses and exchanges. We hope to have convinced the reader that there is more to risk measurement than the choice of a ‘risk measure’: statistical robustness, and not only ‘coherence’, should be a concern for regulators and end-users when choosing or designing risk measurement procedures. The design of robust risk estimation procedures requires the explicit inclusion of the statistical estimation step in the analysis of the risk measurement procedure. We hope this work will stimulate further discussion on these important issues.

Acknowledgements We thank Hans Fo¨llmer, Peter Bank, Paul Embrechts, Gerhard Stahl and seminar participants at QMF 2006 (Sydney), Humboldt University (Berlin), Torino, Lecce, INFORMS Applied Probability Days (Eindhoven), the Cornell ORIE Seminar, the Harvard Statistics Seminar and EDHEC for helpful comments. This project has benefited from partial funding by the European Research Network ‘Advanced Mathematical Methods for Finance’ (AMAMEF).

References Acerbi, C., Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance, 2002, 26, 1505–1518. Acerbi, C., Coherent measures of risk in everyday market practice. Quant. Finance, 2007, 7, 359–364. Artzner, P., Delbaen, F., Eber, J. and Heath, D., Coherent measures of risk. Math. Finance, 1999, 9, 203–228. Costa, V. and Deshayes, J., Comparaison des RLM estimateurs. Theorie de la robustesse et estimation d’un parametre. Asterisque, 1977, 43–44.

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Czellar, V., Karolyi, G.A. and Ronchetti, E., Indirect robust Then, by using that jC is continuous at F, there exists estimation of the short-term interest rate process. J. Empir.  4 0 such that, for each G 2 C that satisfies d ðF, G Þ  2,  Finance, 2007, 14, 546–563. Danielsson, J., Jorgensen, B., Samorodnitsky, G., Sarma, M. we have d((F ), (G )) ¼ j(F )  (G )j5"/2. As Demp C, and de Vries, C., Subadditivity re-examined: the case for we obtain that   Value-at-Risk. Preprint, London School of Economics, 2005.   P jðF Þ  ðG emp Þj  " : Pðd ðF, Gxemp Þ  2Þ x Dell’Aquila, R. and Embrechts, P., Extremes and robustness: 2 a contradiction? Financial Mkts Portfol. Mgmt, 2006, 20, emp   Therefore, it suffices to show that Pðd ðF, Gx Þ  2Þ 103–118. Deniau, C., Oppenheim, G. and Viano, C., Courbe d’influence 1  ð"=2Þ. et sensibilite. Theorie de la robustesse et estimation d’un Now, using that Glivenko–Cantelli convergence is parametre. Asterisque, 1977, 43–44. uniform in G, we have for each "40 and 40 the Dhaene, J., Laeven, R., Vanduffel, S., Darkiewicz, G. and existence of n0  1 such that, for all G 2 C, Goovaerts, M., Can a coherent risk measure be too subadditive? J. Risk Insurance, 2008, 75, 365–386. " Pðd ðG, Gxemp Þ  Þ  1  , 8n  n0 : Fo¨llmer, H. and Schied, A., Convex measures of risk and 2 trading constraints. Finance Stochast., 2002, 6, 429–447. Fo¨llmer, H. and Schied, A., Stochastic Finance: An Introduction Therefore, the C-robustness follows from the triangular in Discrete Time, 2004 (Walter de Gruyter: Berlin). inequality d ðF, Gxemp Þ  d ðF, G Þ þ d ðG, Gxemp Þ and by Frittelli, M. and Rosazza Gianin, E., Putting order in risk  taking   . measures. J. Bank. Finance, 2002, 26, 1473–1486. Gourieroux, C., Laurent, J. and Scaillet, O., Sensitivity analysis ‘2. ) 1’. Conversely, assume that b  h is C-robust at F and of values at risk. J. Empir. Finance, 2000, 7, 225–245. fix "40. Then there exists 40 and n0  1 such that, for Gourieroux, C. and Liu, W., Sensitivity analysis of distortion all G 2 C, risk measures. Working Paper, 2006. Hampel, F., Ronchetti, E., Rousseeuw, P. and Stahel, W., "  h , F Þ, Ln ðb  h , G ÞÞ 5 , 8n  n0 : d ðF, G Þ 5  ) d ðLn ðb Robust Statistics: The Approach Based on Influence Functions, 3 1986 (Wiley: New York). Heyde, C., Kou, S. and Peng, X., What is a good risk measure: As a consequence, from the triangular inequality bridging the gaps between data, coherent risk measures, and insurance risk measures. Preprint, Columbia University, jðFÞ  ðG Þj ¼ dððFÞ ,ðGÞ Þ 2007.  h , FÞÞ þ dðLn ðb  h ,F Þ, Ln ðb  h ,G ÞÞ  d ððFÞ , Ln ðb Huber, P., Robust Statistics, 1981 (Wiley: New York). Ibragimov, R. and Walden, J., The limits of diversification when  h ,G Þ,ðG Þ Þ, þ dðLn ðb losses may be large. J. Bank. Finance, 2007, 31, 2551–2569. Kusuoka, S., On law invariant coherent risk measures. and the consistence of b  h with  at F and any G 2 C, Adv. Math. Econ., 2001, 3, 83–95. Rockafellar, R. and Uryasev, S., Conditional Value-at-Risk for it follows that jC is continuous at F. general distributions. J. Bank. Finance, 2002, 26, 1443–1471. Tasche, D., Expected shortfall and beyond. In Statistical Data Analysis Based on the L1-Norm and Related Methods, edited A.2. Proof of theorem 3.10 by Y. Dodge, pp. 109–123, 2002 (Birkha¨user: Boston, Basel, Berlin). We will show that the continuity problem of the ML scale Van Zwet, W., A strong law for linear functions of order function  mle : D ! Rþ of portfolios X can be reduced to statistics. Ann. Probab., 1980, 8, 986–990.

Appendix A: Proofs A.1. Proof of proposition 3.2 ‘1. ) 2’. Assume that jC is continuous at F and fix "40. Since for all G 2 C  h , F Þ, Ln ðb  h , G ÞÞ d ðLn ðb

 h , FÞ, ðF Þ Þ þ d ððF Þ , Ln ðb  h , G ÞÞ,  d ðLn ðb

and b  h is consistent with  at F, it suffices to prove that there exists 40 and n0  1 such that, for all G 2 C, "  h , G ÞÞ  , d ðG, F Þ   ¼) d ððF Þ , Ln ðb 2

8n  n0 :

the continuity (on a properly defined space) issue of the ML location function of portfolios Y ¼ ln(X2). The change of variable here is made to use the results of Huber (1981) concerning the weak continuity of location parameters. The distribution F can be seen as the distribution of a portfolio X0 with (X0) ¼ 0 and (X0) ¼ 1. Then, by setting Y0 ¼ lnðX20 Þ, and denoting by G the distribution of Y0 we have Gð yÞ ¼ PðY0  yÞ ¼ PðX20  e y Þ ¼ F ðe y=2 Þ  F ðe y=2 Þ, gð yÞ ¼ G0 ð yÞ ¼ e y=2 f ðe y=2 Þ:

Moreover, by introducing the function

Z g0 ð yÞ 4 ’ð yÞ ¼  , D’ ¼ G : ’ð yÞGðdyÞ 5 1 , gð yÞ

we can define, as in Huber (1981), the ML location function mle(H ) for any distribution H 2 Du as the solution of the following implicit relation Z ’ð y  ÞHðdyÞ ¼ 0: ðA1Þ

Note that Strassen’s Theorem (Huber 1981, theorem 3.7) gives us the following sufficient condition to obtain the desired result:  " " " Now we consider the distribution FX 2 D of the random  h , G ÞÞ  : P jðF Þ  ðGxemp Þj   1  ¼) dððFÞ , Ln ðb 2 2 2 variable X representing the P&L of a portfolio and

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assume that FX has density fX and that the solution to R (x/)FX(dx) ¼ 0 has a unique solution  ¼  mle(FX). Denoting by FY the distribution of Y ¼ ln(X2), it is easy to check that FY 2 Du since, for y ¼ ln(x2), we have ’ð yÞ ¼ 

g0 ð yÞ gð yÞ

e ð yÞ=2 f ðe ð yÞ=2 Þ þ 12 e y f 0 ðe ð yÞ=2 Þ e ð yÞ=2 f ðe ð yÞ=2 Þ   0 ð yÞ=2 1 Þ ð yÞ=2 f ðe ¼ 1þe 2 f ðe ð yÞ=2 Þ   1 f 0 ðxÞ 1 ðxÞ: ¼ 1þx ¼ 2 f ðxÞ 2 1

¼ 2

ðA2Þ

Noticing that FY ðdyÞ ¼ fY ð yÞdy ¼ xfX ðxÞdðlnðx2 ÞÞ ¼ 2fX ðxÞdx ¼ 2FX ðdxÞ, we immediately obtain from equations (17), (A1) and (A2) that  mle(FX) ¼ mle(FY) when Y ¼ 2 ln(X ). We have therefore shown that a scale function characterized by the function can also be interpreted as a location function characterized by the function u. From equation (A2), we see that, for all x 2 R, 2u(x) ¼ [ex/2]. Therefore, as is assumed to be even and increasing on Rþ, it implies that u is increasing on R. Moreover, as takes values of both signs it is also true for u. To conclude, we apply theorem 2.6 of Huber (1981), which states that a location function associated with u is weakly continuous at G if and only if u is bounded and the location function computed at G is unique.

Journal of Risk Research, 2014 Vol. 17, No. 3, 317–335, http://dx.doi.org/10.1080/13669877.2013.808688

Specificity of reinforcement for risk behaviors of the Balloon Analog Risk Task using math models of performance Nicole Prausea* and Steven Lawyerb a Department of Psychiatry, University of California, Los Angeles, CA, USA; bDepartment of Psychology, Idaho State University, Pocatello, ID, USA

(Received 17 October 2012; final version received 5 April 2013) Risky decision-making has been studied using multitrial behavioral tasks. Concordance of such tasks to risky behaviors could be improved by: (1) mathematically modeling the components of decision change and (2) providing reinforcement specific to the risk behavior studied. Men completed two Balloon Analog Risk Tasks (BART). One provided financial reinforcement (money) and the other provided sexual reinforcement (seconds of erotic film viewing). Parameters of a mathematical model of BART performance were fit to each individual. Correlations between the model parameters and four risk categories (financial, sexual, antisociality, and substance use) demonstrated predictive utility for the same behaviors regardless of task reinforcement, providing little evidence of reinforcement specificity. A reward sensitivity parameter was uniquely related to sexual risk behavior. Additional analyses explored parameter stability fit to fewer trials. Keywords: Balloon Analog Risk Task; sexual risk taking; impulsivity; decisionmaking; HIV

Risky sexual behaviors are rarely directly observed for research purposes, but the decision processes related to sexual risk behaviors are accessible in the laboratory. Most tasks tapping decision processes rely on behavioral analogs of sexual decision-making to assess sexual risk likelihood. Laboratory operationalizations of sexual risk have included face-valid questions about sexual risk intentions (e.g. Ariely and Loewenstein 2006; Fromme, Katz, and D’Amico 1997) and stop-response vignettes (e.g. MacDonald, Zanna, and Fong 1996; Marx and Gross 1995; Murphy, Monahan, and Miller 1998; Testa, Livingston, and Collins 2000; Wilson, Calhoun, and Bernat 1999). Such overt questions about intentions to engage in sexual behaviors in the laboratory do change in response to experimental manipulation (Norris et al. 2009). Intentions also are likely to correspond to real-world behaviors, particularly if the intentions are strong (Webb and Sheeran 2006). Correspondence between laboratory decision-making tasks and real-world sexual behaviors might be increased by identifying reinforcers more specific to sexual risk behaviors. The current study investigated whether introducing a sexual, instead of monetary, reward in the Balloon Analog Risk Task (BART) would result in stronger relationships between the BART and real-world sexual risk behaviors. *Corresponding author. Email: [email protected] Ó 2013 Taylor & Francis

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The notion of a global impulsivity trait that leads to taking risks across multiple domains has been challenged. This has implications for prevention strategies. For example, decreasing impulsive behavior broadly could inhibit a broad range of health risk behaviors (Peters et al. 2009). However, other suggest that impulsivity is specific to the domains that an individual values, such as recreational bungee jumpers (Hanoch, Johnson, and Wilke 2006; Weber, Blais, and Betz 2002). Similarly, those who take sexual risks do not necessarily engage in risky behaviors in other domains. Even within sexual behaviors, health risks appear separable from sexual behaviors that risk later mate choice (Kruger, Wang, and Wilke 2007). Even when domains of personality predict sexual interactions or sexual risk taking, the explanatory power tends to be very small (e.g. Big Five personality domains predicting number of sexual partners by age 20, R2 = 0.03; number of sex acts without a condom, R2 = 0.01; Miller et al. 2004; Smith et al. 2007). Characterizing risk in the laboratory Many tasks assessing various aspects of risky decision-making have been developed, including the Iowa Gambling Task (Bechara et al. 1994) and delay and probability discounting (Green and Myerson 2004; Mitchell 1999). The Wisconsin Card Sorting Task (Grant and Berg 1948), relatedly, explores faulty decision-making related to executive impairment. Citing concerns that such tasks still lacked strong relationships to real-world risk behaviors, Lejuez and colleagues (2002) developed the BART. In the BART, participants have the option to ‘pump’ a balloon on a computer and receive money for each pump. A participant may choose to bank this money (add it to their total) at any time. However, if the balloon bursts before the money is banked, all of the money for that trial is lost. The BART predicts health risk behaviors above and beyond demographic and questionnaire correlates of risk (Lejuez et al. 2002), including MDMA consumption in high-risk youth (Hopko et al. 2006) and smoking behavior (Lejuez, Aklin, Jones, et al. 2003). The task exhibits acceptable test–retest reliability (White, Lejuez, and de Wit 2008) and appears robust to the influence of socially desirable responses (Ronay and DeYeong 2006). BART scores moderate the relationship of personality factors and alcohol consumption (Skeel et al. 2008). Data from physiological methods further support the sensitivity of the BART to changes in risk behaviors, including differences in error-related negativity (from event-related potentials) when the balloon bursts in those with a positive family history of alcoholism (Fein and Chang 2008) and the ability to manipulate BART performance within session using transcranial direct current stimulation (Fecteau et al. 2007). Scores appear most sensitive to variability in impulsive decision-making, rather than impulsive disinhibition, which may be better captured by tasks like Go/No-Go (Reynolds et al. 2006). Performance on the BART appears dependent on executive functions as evinced by involvement of frontal lobe activation. Risk taking in the BART broadly employs mesolimbic-frontal regions, while decision-making in the task specifically engages dorsolateral prefrontal cortex (DLPFC, Rao et al. 2008). In a BART-like task designed for rats, the functional DLPFC equivalent was required for constraining responses while the orbitofrontal cortex (OFC) was active in motivating responses (Jentsch et al. 2010). This is consistent with highly impulsive individuals exhibiting greater OFC activation in other risky gains tasks (Lee et al. 2008).

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Findings are consistent with the involvement of frontal regions in other behavioral tasks require inhibition of prepotent responses (e.g. Liddle, Kiehl, and Smith 2001). Such deficits could characterize a number of problem behaviors, suggesting the BART might be sensitive to general, not money specific, behavioral regulation problems. Several modifications of the BART to better characterize its correspondence to health risk-relevant personality and behavioral measures have been attempted. When each balloon pump was an opportunity to lose money, rather than the typical opportunity to gain money, the task with opportunities to gain converged uniquely with a measure of sensation seeking (Benjamin and Robbins 2007). BART scores have long been adjusted to account for additional balloon pumps that participants were unable to make due to a balloon explosion (Lejuez et al. 2002), but an additional test providing participants information about pump limits and requiring them to designate the desired number of pumps a priori did not change the relationship of the BART scores to sensation seeking, impulsivity, anxiety, or depression measures (Pleskac et al. 2008). Behavioral tendencies literature suggests that the use of money as the primary reinforcement may help explain these discrepancies. Risk specificity Financially and sexually risky behaviors tend to covary and may have a common etiology. For example, prefrontal cortices are involved in inhibition of both reflexive (Ploner et al. 2005) and more controlled behavioral (Aron, Robbins, and Poldrack 2004) performance on tasks, while transcranial stimulation further supports the causative role of frontal lobe dysfunction in response inhibition failure (Chambers et al. 2006). Despite having no explicit questions about gambling or sexual risk taking, a questionnaire measure of sensation seeking (Zuckerman et al. 1964) correlates with both gambling (Bonnaire, Lejoyeux, and Dardennes 2004) and sexual (Schroth 1996) risks. Similarly, a questionnaire measure of impulsivity (Patton, Stanford, and Barratt 1995) is related to both problematic gambling (Castellani and Rugle 1995) and sexual risk taking (Semple et al. 2006).1 Clinically, persons with substance use problems who report concomitant gambling problems are more likely to have had more than 50 lifetime sexual partners (Petry 2000), while financial and sexual risk taking co-occurs particularly in male ‘pathological’ gamblers (Martins et al. 2004). On the other hand, the more specific behavioral intention assessments can be to the actual behavior one is trying to characterize, the more likely that the intentions will be consistent with the behavior (Ajzen and Fishbein 1973). Currently, the BART might be best suited to identify those with risky financial behaviors, such as problem gambling. There is a surprising lack of use of behavioral financial risk tasks, particularly the BART, to study problem gambling (Potenza 2013), although such tasks have been associated with aspects of gambling, such as age of starting to gamble (Betancourt et al. 2012). The BART is not consistently related to sexually risky behaviors (Aklin et al. 2005), which may be due to the financial, nonsexual nature of the reinforcement in that task. For example, a measure of sexual sensation seeking better predicts viewing of internet erotica than a general sensation seeking measure (Kalichman et al. 1994; Perry, Accordino, and Hewes 2007). The logic seems tautological: of course questions more specific to the behavior being predicted should correspond more closely to self-reports of that same behavior.

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However, the semblance of laboratory task behaviors and real-world behaviors being predicted is desirable when the goal of the laboratory investigation is to understand how decisions made in the laboratory changed to generate those highly similar reports. Modeling task performance Math models of risk tasks offer the possibility of more precisely describing what changed in decision processes, rather than merely recording the fact that decisions changed. A total score indicates whether decisions are more or less risky, but a math model could indicate whether risk increased due to differences in recall of the outcomes of immediately previous trials, increased estimated likelihood of possible gains due to a recent large win, or something else. Individual differences in these coefficients can be meaningfully characterized (Neufeld et al. 2002). For example, such modeling allowed researchers to specifically attribute riskier decisions under the influence of alcohol as due to increased weighting of rewards and decreased weighting of punishments and not due to changes in memory for previous trial outcome or sensitivity to those rewards or punishments (Lane, Yechiam, and Busemeyer 2006). Several math models recently were fit to data from the BART (Wallsten, Pleskac, and Lejuez 2005). The best-fitting model had four free parameters and instantiated certain decision-making strategies across participants, such as stationarity in the decision-maker’s subjective probability, governed by a balloon-trialdependent updating process, that the balloon would burst at each pump and the decreased probability of any decision-maker pumping with each successive pump opportunity. The estimated parameters for this model include a0, m0, γ+, and β. a0 represents the likelihood that the participant believes the balloon will not burst on their first pump (↑a0 = increased belief balloon will not burst before any pumps made), and m0 represents their certainty about that estimate (↑m0 = increased certainty of a0 before any pumps made). γ + is the individual’s value of potential gains on a trial. The models tested used prospect theory assumptions for predicting probabilistic distributions of consecutive choices, and β represented the participant’s use of the decision rule that estimated the likelihood of the balloon bursting given what they had amassed to that point (money they stood to lose) as compared to the likelihood of the balloon not bursting considering how much they valued the potential gain.2 The BART total score correlated with drug categories tried, unprotected sex, reckless driving, and stealing, but the parameters differentiated the different risk types predicted. Participants who compared their loss risks and values at each pump (β) reported fewer drug categories tried, but this factor was not related to other risk categories. Relatedly, those whose performance on the BART consistent with strongly valuing rewards (high γ+) was related to more risk in every category (except reckless driving). Finally, differences at baseline in predicted risk of balloon burst/loss and certainty of that expected loss (a0, m0) were not related to task parameters. In the current study, 75 men completed two, counterbalanced BARTs each. Participants were limited to men since women’s risk taking may occur through different neural mechanisms (e.g. Lee et al. 2009) not being investigated. One BART task used the usual monetary reinforcement (BART-M); a second BART task used sexual reinforcement (BART-S). It was expected that the total BART score for each

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task would be most strongly related to proximal risk behaviors (e.g. BART-S would predict the number of sexual partners). The task parameters are compared in an exploratory fashion exploring: (1) whether they better predict financial or sexual risk behaviors depending on the (financial or sexual) BART reinforcement used, (2) the reliability of the coefficients following multiple estimations, and (3) stability of the task coefficients across reduced trials. Methods Participants Adult men were recruited from the community through newspaper, campus, and classroom advertisements, which included the opportunity for monetary payment for participation. Study volunteers phoned the laboratory where further details of the study, including the possibility of viewing sexual images, were provided. This is consistent with other studies examining the reinforcing properties of sexual stimuli (Archer et al. 2006) to increase the likelihood that volunteers would experience the sexual reward (seconds of access to erotica) as rewarding. Individuals who agreed to participate in the study arranged a time to complete study procedures. The study was approved by the University’s Human Subjects Committee. See ‘Participants’ in Results section for additional performance information. Measures Questionnaires Participants completed questionnaires reflecting the same general risk-taking behaviors and constructs measured in Lejuez et al.’s (2002) study with adolescents, and to also represent behaviors and constructs associated with sexuality-related outcomes, including sexual risk taking. For the purposes of the present study, eight behavioral risk questions were examined representing four different domains. A primary and secondary indicator represented each domain to test specificity. Sexual risk was assessed primarily by the number of lifetime female sexual partners and secondarily as the frequency of viewing adult websites. Finance-related risk was assessed primarily by the frequency with which the person gambled and secondarily by the count of stealing behaviors. Antisociality was assessed primarily by the frequency of getting in physical fights and secondarily by frequency of carrying and weapon. Substance use was assessed primarily by the frequency of driving under the influence of alcohol and secondarily by the number of cigarettes consumed daily. Viewing erotic films is not typically presented as a sexual risk variable, but was used in this study for two reasons. First, gambling is a risk variable that is very proximal to the money-reinforcement task, so viewing sexual images would provide a more parallel, proximal test of the erotica-reinforcement task. Second, there was very little variability in behavioral sexual risk in this sample (e.g. anal sex, unprotected vaginal intercourse). Sexual motivation has been shown to increase sexual risk intentions (Ariely and Loewenstein 2006; George et al. 2009), so viewing sexually motivating stimuli also could reasonably be expected to increase sexual risk. People who view more visual sexual stimuli also have been shown to engage in more risky sexual behaviors (Häggström-Nordin, Hanson, and Tydén 2005; Larsson and Tydén 2006).

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Behavioral measures Balloon Analog Risk Task (money; BART-M) This task is a slightly modified version of the original BART task (Lejuez et al. 2002). In this task, participants sat in front of a computer on which they saw: (1) a small balloon accompanied by a balloon ‘pump’ button; (2) a reset button labeled Collect $$$; (3) a trial display showing the money amassed according to the number of pumps made; and (4) a Total Earned display. During the task, participants clicked on the balloon pump button, which inflated the balloon incrementally and earned the participant money ($0.01) in a temporary bank. When a balloon was pumped past its individual explosion point (established at random by the computer) the balloon exploded, all the money in the temporary bank is lost, and another uninflated balloon appeared on the screen. At any point before the balloon exploded, participants could stop pumping the balloon click the ‘Collect $$$’ button, which transferred all the money from the temporary bank to the permanent bank. Each task consisted of 30 balloons and participants were given no specific information regarding the probability of explosion. Participants could earn up to approximately $10 on this task, depending on their performance. Balloon Analog Risk Task (sexual; BART-S) The BART-S task was the same as the BART-M task except that each button press earned participants 1 s of viewing time of erotica of their choice (instead of $0.01). Participants were told that they could watch the number of seconds of erotica earned. Participants were provided a description of the clips, which showed adult heterosexual activity and female homosexual activity, including sexual foreplay (e.g. kissing and undressing), oral-genital contact (e.g. fellatio, cunnilingus, or both), and intercourse (i.e. vaginal penetration, or tribadism in the lesbian film) from commercially available adult films. They were told that the videos contained no violence or low base rate (e.g. anal sex, Woodard et al. 2008) sexual acts and depicted only consensual sex among adults to reduce the risk for expecting negative (e.g. ‘disgust’) emotions. Similar films are used commonly in sexuality-related research (Lohr, Adams, and Davis 1997) to generate sexual arousal. Participants could earn up to about 10 min of viewing time on this task, depending on their performance. Using visual sexual stimuli as a reinforcer is novel. The choice to provide time to access erotic films as the sexual reward offered several advantages over alternatives that were considered. Ethical and legal considerations considerably limit the scope of sexual rewards available for laboratory study. For example, actual food can be offered to study its reinforcement value in those struggling with weight problems (Epstein et al. 2012), but offering access to a new sexual partner is not legal in laboratory settings in the USA. Vibratory stimulation has been used in laboratory settings (Rowland, Keeney, and Slob 2004). However, the rewarding nature of such stimulation has only recently been investigated (Prause and Siegle, 2013). While vibrator use is relatively common in women (Herbenick et al. 2009, 2010), men rarely report using them for their own sexual pleasure (Reece et al. 2009). Methods for equating vibratory parameters (Prause et al. 2011) for reinforcement similar to monetary parameters also is unclear. Seconds of visual sexual stimuli offered a few advantages, in addition to being legal to provide as reinforcement. First, novelty in general (Bevins et al. 2002), and novel films in particular (Blatter

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and Schultz 2006), are reinforcing. Novel erotic films offer increased pleasant feelings over repeated erotic films for men (Kelley and Musialowski 1986), suggesting that the promise of a new film should be reinforcing. Second, an unusually high reinforcement value of widely available novel visual sexual stimuli has been proposed as a primary mechanism escalating the viewing of online erotica (D’Orlando 2011). Third, erotic films alone have been repeatedly demonstrated to activate areas of the brain consistent with pleasure in both men and women (Georgiadis and Kringelbach 2012). Finally, as a practical matter, films could be quantified by length of time viewed to parallel (at least ordinally) money reinforcement. Ultimately, BART performance in the current study informs whether access to erotica is an effective reinforcement and, in particular, whether using it as reinforcement might increase the ability to predict real-life sexual behaviors beyond money reinforcement. Procedure Upon arrival to the lab and providing informed consent, participants completed all questionnaire and behavioral measures. The order of the measures was counterbalanced such that half the participants completed the questionnaires first and half completed the behavioral measures first. In addition, the order of the two behavioral tasks (regardless of when the questionnaires were completed) also was counterbalanced. At the end of the session, participants were paid $10 in cash for their participation plus up to $20 more based on their performance on the money-related decision-making tasks used in the study. Participants also were given the opportunity to view erotica for the amount of time earned on the BART-erotica task. For sexual reinforcement, they were provided a selection of 20 films based on a single image representative of the sexual scene portrayed. They could view these in a private viewing area. Data analysis Total adjusted scores were included in analyses consistent with Lejuez et al. (2002). These were calculated as the average number of pumps excluding balloons that exploded (i.e. the average number of pumps on each balloon prior to reward collection). ‘Adjusted’ total scores are used because the number of pumps on balloons that explode are necessarily constrained, which reduces between subjects variability in the absolute averages. For model parameter estimates, Model 3 from Wallsten, Pleskac, and Lejuez (2005) was estimated using the algorithm developed for that study. Specifically, a computerized search for MLE was conducted for each individual subject using the Nelder and Mead algorithm (1965) modified to reduce the chance of local reaching local minima. Parameter estimation and analyses were conducted in MatLab (v. 7.8). After visual inspection of scatterplots for nonlinearity, bivariate correlations were conducted. To test if reinforcement specific to the risk behavior of interest better predicted that risk behavior, model parameters were correlated with individual risk behaviors. To examine specificity predictions, the total adjusted BART score (1 value) and model parameters (4 different values) were correlated with the reported risk behaviors. Data were visually inspected for nonlinearity. To specifically compare the tasks, a simple difference score between the money and sexual reinforcement tasks was calculated for each parameter. These also were correlated with risk behaviors. Given the

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four self-reported risk behaviors (gambling, stealing, adult web use, and number of partners) and five scores that were the focus of the analyses, results are discussed in the context of the multiple tests conducted. A number of alternative statistical approaches might have been justified, such as regression, but correlations offer ease of interpretation, few statistical assumptions, and was the approach that allowed comparison with previous studies that also reported correlations (e.g. Pleskac et al. 2008). Given the limited use of trial-by-trial math models of clinical questions to date, basic tests of the psychometric properties were included. To test the reliability of the parameter estimates, the model parameters were estimated a second time and correlated with the parameter values from the first model estimates. Since parameter estimates begin with an algorithm-generated seed (see above), it is possible for these parameters to fluctuate from estimation to estimation. Estimating the model parameters twice provides an estimation of the amount of this fluctuation. Finally, the parameters were estimated based on reduced forms of the task to determine whether 30 trials are necessary to reach similar parameter estimates. The reduced form included 20 trials and 10 trials of the (originally) 30 trial task. These were correlated with parameter estimates from the first full-task parameter estimations to estimate the degree to which these reduced forms matched the estimates from the full, 30 trial task. Results Participants Seventy-eight male participants between the ages of 18 and 57 (M = 26.5, SD = 9.4) were recruited for participation in the study. Data for 12 participants were excluded from analysis. Of these, nine participants were excluded for task performance inconsistent with instructions on the sexual reinforcement task. Specifically, six participants did not burst the balloon on any trial, typically ‘pumping’ only once before advancing trials, while three burst the balloon on every trial (no one exhibited either of these response patterns during the money reinforcement task). One participant refused to complete the BART-S task and data for two participants were excluded due to procedural error. Sixty-six participants remained in the final sample.3 Additional considerations of the implications of this decision are provided in the discussion. A majority of those remaining reporting viewing a sexually explicit film in the past (N = 61, 88.4%), although some reported that they did not watch any sexual films in a typical month (N = 19, 27.5%). Most had visited an adult website in the last month (N = 52, 82%). A majority of participants reported having sex with none (N = 23, 33.3%) or one female sexual partner (N = 29, 42.0%) in their lifetime, which left participants well below the national average for the number of female sexual partners in this age cohort (Laumann et al. 1994). Reinforcement specificity The two tasks performed similarly in their predictions of the eight risk behaviors predicted (see Table 1). The total score of both the BART-M and BART-S were related to self-reported gambling, number of sexual partners, and carrying a weapon, while the total score of the BART-S was additionally related to stealing. The BART-M corresponded more with gambling behavior than did the BART-S. The BART-M also covaried more strongly with the number of sexual partners

p < 0.01, ⁄p < 0.05.

0.22 0.12 0.15 0.04 0.04

Erotica (adj) a0 m0 γ+ β

⁄⁄

0.33 0.05 0.13 0.06 0.16

⁄⁄

Gambling

Money (adj) a0 m0 γ+ β

Parameters 0.17 0.19 0.11 0.10 0.05 0.19 0.02 0.01 0.16 0.05

0.34⁄⁄ 0.04 0.03 0.11 0.07

0.26⁄⁄ 0.01 0.01 0.18 0.17

0.33 0.12 0.01 0.28⁄⁄ 0.21

⁄⁄

No. partners

Sexual Adult web use

0.15 0.21 0.01 0.10 0.11

Stealing

Financial

0.03 0.12 0.04 0.11 0.05

0.13 0.01 0.03 0.19 0.09

Physical fights

0.47⁄⁄ 0.14 0.18 0.18 0.06

0.31 0.28⁄ 0.01 0.18 0.10

⁄

Carry weapon

Antisocial

Risk behaviors

Table 1. Correlations of model parameter estimates with self-reported risk behaviors by reinforcement.

0.01 0.11 0.12 0.04 0.04

0.02 0.01 0.01 0.03 0.12

Driving intoxicated

0.21 0.03 0.04 0.14 0.08

0.19 0.09 0.09 0.05 0.03

Daily cigarettes

Substance

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Table 2. Correlations of model parameter differences with self-reported risk behaviors. Risk behaviors Financial Difference in parameters

a

Gambling

Total score (adj) a0 m0 γ+ β a

Difference = erotica–money;

0.11 0.07 0.20 0.01 0.14 ⁄⁄

Sexual

Stealing ⁄

0.27 0.10 0.02 0.03 0.00

Adult web use

No. partners

0.05 0.11 0.09 0.07 0.02

0.25⁄ 0.07 0.43⁄⁄ 0.04 0.06

p < 0.01, ⁄p < 0.05.

than the BART-S, and the BART-S covaried more strongly with stealing behaviors than the money-reinforcement BART-M. For both versions of the BART, γ+ exhibited the strongest relationship with the number of sexual partners amongst the four parameters. For the BART-M only, a0 was the only estimated parameter significantly related to carrying a weapon. The differences also were calculated between the two tasks for parallel parameters (see Table 2). The difference between tasks in the total BART score was related to stealing and the number of sexual partners. Also, the difference on the m0 parameter was inversely related to the number of sexual partners. Coefficient reliability The model was fit a second time to the same data to examine the stability of the parameter estimates. Correlations between each fitted parameter of the model for both erotica-reinforcement and money-reinforcement BART were significant with moderately large or large effect sizes (see Table 3). Reduced task trials Fitting the model to progressively fewer trials resulted in some parameter instability (see Table 4). When reduced to fit to only the first 20 trials, parameters were significantly correlated to the parameters estimated using all 30 trials. There were Table 3. Reliability (linear correlation) of free parameter coefficients. Free parameters Reinforcement

a0

m0

Money a0 m0 γ+ β

0.62⁄⁄ 0.30⁄ 0.05 0.25⁄

0.77⁄⁄ 0.01 0.21

0.94⁄⁄ 0.03

0.98⁄⁄

Erotica a0 m0 γ+ β

0.81⁄⁄ 0.22 0.05 0.03

0.84⁄⁄ 0.12 0.08

0.95⁄⁄ 0.39⁄

0.995⁄⁄

⁄⁄

p < 0.01, ⁄p < 0.05.

γ+

β

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Table 4. Convergence of free parameter estimates with reduced trials. Parameters 30 trials a0 Twenty trials Money a0 m0 γ+ β

m0

γ+

β

0.39⁄⁄ 0.17 0.38⁄⁄ 0.24

0.53⁄⁄ 0.12 0.09

0.81⁄⁄ 0.1

0.97⁄⁄

Erotica a0 m0 γ+ β Ten trials

0.71⁄⁄ 0.24 0.07 0.04

0.69⁄⁄ 0.09 0.11

0.88⁄⁄ 0.41⁄⁄

0.99⁄⁄

Money a0 m0 γ+ β

0.35⁄⁄ 0.2 0.15 0.12

0.39⁄⁄ 0.05 0.07

0.46⁄⁄ 0.1

0.00

Erotica a0 m0 γ+ β

0.38⁄⁄ 0.25⁄ 0.18 0.09

0.30⁄ 0.05 0.02

0.48⁄⁄ 0.02

0.24

⁄⁄

⁄

p < 0.01, p < 0.05.

reductions in stability of the a0 and m0 parameters in both BART-M and BART-S, although the stability appeared lower for those parameters in the BART-M. These remained significantly correlated. When only the first 10 trials of the task for each participant were used to fit the parameters, every parameter’s correspondence with the fit to the full 30-trial task declined. Most notably, the β parameter of the money-reinforcement task was no longer significantly related to the β parameter fit to the full model. Discussion In this study, the ability of the BART using financial reinforcement (BART-M) to predict financial risk taking and the BART using viewing time access for erotica reinforcement (BART-S) to predict sexual risk taking were compared. Consistent with the specificity hypothesis, the BART-M corresponded more with gambling behavior than did the BART-S. Inconsistent with the specificity hypothesis, the BART-M also covaried more strongly with the number of sexual partners than the BART-S, and the BART-S covaried more strongly with stealing behaviors than the BART-M. The BART-M was no more specific than the erotica-reward version of the BART for financial risks (gambling or stealing). Difference scores did suggest that those whose expectations that the balloons would burst were greater at the start of the BART-S than the BART-M also reported fewer sexual partners. The

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BART appears to reflect general decision-making related to risky behaviors minimally influenced by reward type. For both versions of the BART, γ+ exhibited the strongest relationship with the number of sexual partners amongst the four parameters. This suggested that decisions to acquire new sexual partners are driven by the high value of the potential rewards of the sexual reinforcement. According to this model, decisions to pursue sexual partners may not be influenced by consideration of losing potential amassed gains or estimation or certainty of the potential for loss. Extrapolating to a sexual opportunity, this may mean that potential loss in some other amassed gain, such as a friendship with a potential sexual partner, may not effect sexual decisions as strongly as how highly the individual values the reinforcing aspects of the immediate gratification from sexual interaction with that partner. For the BART-M only, a0 was the only estimated parameter significantly related to any variable: the likelihood of carrying a weapon. This suggests that decisions to carry a weapon are best understood as a belief that negative consequences of carrying a weapon are unlikely to occur. Persons who carry weapons might not feel certain that there will be no consequences, particularly value positive reinforcements to be gained by carrying a weapon, or consider that carrying a weapon may risk losing their own health, for instance. Difference scores between the parameters for the two tasks suggested similar mixed support for specificity. The total BART score was related to stealing and the number of sexual partners, suggesting that those who were willing to take more risks to earn seconds of access to erotica than money also were more likely to risk consequences associated with stealing and new sexual partners. The only estimated model parameter difference was an inverse relationship between the m0 and the number of sexual partners. Participants who were more certain that the balloons would not burst at the start of the BART-S reported fewer sexual partners. Extrapolating to behaviors, this could reflect those who anticipated repercussions for pursuing sexual rewards reducing their sexual partner number preemptively. Task reward specificity could be impacted by the reporting biases unique to sexual behavior. The type of strategy used to recall sexual risk behaviors affects the number of behaviors that participants report (Bogart et al. 2007). Reports of risky behavior, especially risky sexual behavior, also are affected by simple social desirability (Alexander and Fisher 2003). It is notoriously difficult to quantify sexual risk, because the number of sexual partners, or a sum of multiple sexual risk behaviors, does not often reflect true risk (Catania et al. 2002). For example, true risk in the area in which the study was conducted appears especially low for many sexual ‘risk’ behaviors with respect to HIV/AIDS risk (CDC 2008). Also, sexual behaviors introduce nonlinear risk, which prohibits the calculation of simple composite indices (Catania et al. 2005), and requires different statistical approaches to assessing individual risk behaviors (e.g. count v. categorical in Schroder, Carey, and Vanable 2005). For example, consistency of condom use may belie the consistency of correct condom use (Crosby et al. 2003). The low rates and variability of sexual risk behaviors reported reduces the range that might optimize parameter estimates. In other words, it is difficult to predict differences in risk behaviors if the sample only provides a small range of the possible sample of behaviors. Generally, men in the USA tend to report a higher numbers of sexual partners than women than they actually were likely to have had. Taken together with the knowledge that the actual risk is likely to be low for multiple

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partners in this area (see above), it is surprising that the reported sexual risk behaviors in this study were so infrequent (cf. Laumann et al. 1994). The low sexual risk behaviors in this sample may limit its generalizability. Some researchers have attempted to create additive or latent variables of multiple risk categories (e.g. Stein, Newcomb, and Bentler 1994). Since true risk is not simply additive, as suggested by these approaches, some have recommended using individual counted behaviors to better reflect risk behaviors (Schroder, Carey, and Vanable 2005). In addition to examining individual behaviors separately, another sexual ‘risk’ variable, amount of erotica viewed, with greater variability also was examined. The rationale for characterizing erotica viewing as risk was provided above. There clearly were different correlates of these two indicators, so the best resolution would be to investigate these in a sample that also varies more in their other behavioral indicators of sexual risk. Participants were selected using a necessary approach that may limit the generalizability of the findings. Not everyone values viewing erotic films, and those who do value it are likely to differ in systematic ways from those who do not. However, it was necessary to select individuals who would value a realistic sexual reinforcement for the task to be comparable to the money reinforcement. Money could be described as universally valued, since money is a token able to provide a variety of secondary reinforcers. This clearly may limit the usefulness of the BART-S for laboratory risk assessment in large samples. For example, those whose sexual behavior preferences or fantasies are different from what is described in the promised erotic film (e.g. homosexual preferences) would devalue the film reinforcement. However, it is unlikely that a ‘real’ sexual reinforcement would improve the correspondence of task and real-world behaviors based on comparisons of faux and real money reinforcement (Bowman and Turnbull 2003). Films portraying risky sexual behavior might have corresponded more closely with actual sex behaviors of those seeking risky sexual encounters. Similarly, many participants who came to the laboratory still failed to complete the BART-S as instructed. This appears likely to be related to the variability in the reinforcing properties of visual erotica, since these same participants did complete the BART-M as instructed. Analyses were run retaining the individuals who completed the task, just without bursting balloons or bursting every balloon. The same pattern of results emerged, although generally effect sizes were smaller (e.g. adjusted total BART score on money reinforcement correlated with number of female sexual partners 0.31 including everyone and 0.33 excluding outlier performers). Those who regularly view visual erotica may not necessarily value prolonged exposure to a sexual stimulus in a laboratory setting with an experimenter present. For example, prolonged exposure to visual erotica without orgasm in men may result in reports of pain rather than pleasure, although physiological evidence of the pain has not been clearly documented (for discussion, see Hite 2006). The question arises: can the sexual reinforcement be modified or replaced with a different sexual reinforcement that is more widely valued and appropriate for laboratory use? The lack of relationships between the BART and substance use adds to the mixed results seen in other studies. Some failed to find relationships between BART scores and cigarettes (Dean et al. 2011; Moallem and Ray 2012), while others found relationships with alcohol (Aklin et al. 2005). Relationships between the BART and substance use variables are inconsistent. This is surprising given the large literature on the importance of constructs like reward sensitivity and decreased

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inhibition in the misuse of many substances. Indeed, it was exactly γ+ and β parameters that were related to the number of different categories of drugs tried in the original Wallsten, Pleskac, and Lejuez (2005) model. Perhaps, measures of substance use that measure the breadth of substances used (e.g. Pleskac et al. 2008) are more indicative of risky behavior as captured in the BART than the level of use of individual substances as in this study. Visual erotica viewing time may not be the optimal or the most specific sexual reinforcement. The laboratory setting and ethics constrain the field of appropriate reinforcements available. The optimal delay to reinforcement (view sexual stimuli now or later) or intensity of reinforcement (view sexual photograph or sexual film for time won) also could affect the efficacy of the sexual reinforcement. Constructing private viewing areas, providing portable media for the visual erotica to be viewed in the home, and reducing interaction with the experimenter to access the reinforcement could improve the correspondence with sexual risk behaviors by modifying the BART-S. Alternatively, different sexual reinforcement could be provided, such as automated vibratory stimulation (Rowland and Slob 1992), access to an established sexual partner, or access to a placebo with varying dose described to participants as pharmacologically enhancing sexual response. The current limitations and lack of specificity observed in the BART-S suggest the BART-M may be sufficient to study a variety of risk behaviors, but a number of creative alterations could be explored further. Notes 1. Some studies conceptually treat sensation seeking and impulsivity constructs as overlapping (McDaniel and Zuckerman 2003). 2. The alternative model was that participants decided at the start of each balloon how many times they would pump and then would simply pump towards that goal number without additional evaluation during that trial. 3. Sample sizes vary slightly for some analyses where individuals did not respond to every question.

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Scandinavian Actuarial Journal, 2014 Vol. 2014, No. 6, 483–509, http://dx.doi.org/10.1080/03461238.2012.724442

Original Article

Stochastic modelling of mortality and financial markets HELENA AROa* and TEEMU PENNANENb a

Department of Mathematics and Systems Analysis, Aalto University, Aalto, Finland; b Department of Mathematics, King’s College London, London, UK (Accepted August 2012)

The uncertain future development of mortality and financial markets affects every life insurer. In particular, the joint distribution of mortality and investment returns is crucial in determining capital requirements as well as in pricing and hedging of mortality-linked securities and other life insurance products. This paper proposes simple stochastic models that are well suited for numerical analysis of mortality-linked cash flows. The models are calibrated with a data set covering six countries and 56 years. Statistical analysis supports the known dependence of old-age mortality on GDP which, in turn, is connected to many sectors of financial markets. Our models allow for a simple quantitative description of such connections. Particular attention is paid to the long-term development of mortality rates, which is an important issue in life insurance. Keywords: mortality risk; market risk; stochastic modeling

1. Introduction General mortality has fallen unexpectedly fast across all age groups over the past decades, with considerable fluctuations in the rate of improvement (Cairns et al. (2008)). Pension providers and national security systems are incurring the costs of unpredictably improved longevity, as they need to pay out benefits longer than was anticipated. An ageing population and uncertainty in mortality improvements have created an acute need for efficient quantitative risk management methods for mortality-linked cash flows. Various mortality-linked instruments have been proposed for hedging longevity risk. These include mortality bonds, mortality swaps and q-forwards whose cash flows are linked to mortality developments in a specified population [see e.g. Blake et al. (2006a), Blake and Burrows (2001), Biffis and Blake (2009), Dowd et al. (2006), Lin and Cox (2005)]. It has been shown in Blake et al. (2006b), Cairns (2011) and Li and Hardy (2011) how such instruments can be used to hedge mortality risk exposures in pension liabilities. There seems, indeed, to be a strong demand for mortality-linked instruments but the supply side is less clear. The supply of mortality-linked instruments would likely increase if their cash flows could be hedged by appropriate trading in more traditional assets for which liquid markets already exist. Such a development has been seen for example in *Corresponding author. E-mail: [email protected] # 2012 Taylor & Francis

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options markets which boomed after the development of hedging strategies that could produce an option’s cash flows by appropriately trading the underlying asset [see Scholes (1998)]. While the cash flows of simple stock options are directly linked to traded assets, the cash flows of mortality-linked instruments seem to have much less to do with existing financial markets. It follows that their cash flows cannot be perfectly hedged. In other words, the markets are incomplete and the seller of a mortality-linked instrument always retains some risk. Nevertheless, any connection between mortality and financial markets may help sellers to better adjust their investments to the cash flows of the sold instruments and thus to reduce their mortality risk exposure. Such connections would be valuable also for pension or life insurers who seek to adjust their investment strategies so that their returns conform to the cash flows of the insurance liabilities as well as possible. This paper introduces simple stochastic models that allow for description of statistical links between mortality and financial markets. Using a data set covering six countries and 56 years, we look for consistent patterns in the long-term development of relevant risk factors. Based on the statistical analysis, we propose a model that incorporates the following features: . . . .

eventual stabilization of mortality rates, long-term link of old-age mortality to GDP, short-term connection between mortality and GDP, connection of GDP to interest rates.

While all these features may be questioned, there is both logical and statistical evidence in their favour. Until now human mortality has generally decreased over time, but it is not clear how long mortality rates of different age groups will continue to diminish. It was conjectured already in Wicksell (1926) that, in the long run, mortality rates will tend to stabilize. Some experts suggest that lifestyle factors, such as obesity, may soon hinder further mortality improvements; see Olshansky et al. (2005). There is also some indication that the decline in coronary heart disease mortality is levelling out in some age groups in the UK and the Netherlands Allender et al. (2008), Vaartjes et al. (2011). A recent survey of mortality trends in Europe can be found in Leon (2011). There is evidence of a connection between mortality and economic cycles, usually represented by GDP or unemployment. Some studies suggest that in the long run, higher economic output results in lower mortality (Preston 1975, 2007). The long-term connection between longevity risk and the economy is also discussed in Barrieu et al. (2012). Others report a more immediate link between the phases of the economic cycle and mortality: Ruhm (2000) and Tapia Granados (2005a, 2005b) discovered that mortality rates increase during economic expansions. The connection of GDP to both short-term and long-term mortality in six OECD countries has recently been studied in Hanewald et al. (2011) and Hanewald (2011). They found that short-term mortality and

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macroeconomic fluctuations are closely linked, and on the other hand, the mortalities of various age groups display signs of co-integration with GDP. GDP, in turn, is connected with various sectors of financial markets; see Stock and Watson (2003) for an extensive review. For example, there is strong evidence of a link between economic activity and the term spread of interest rates [Estrella and Mishkin (1998), Haubrich and Dombrosky (1996), Harvey (1988, 1989), Estrella and Hardouvelis (1991), Davis and Fagan (1997), Plosser and Rouwenhorst (1994)], although the connection may have weakened since the mid-80s as suggested by Wheelock and Wohar (2009). Another connection exists between credit spreads (the difference between corporate and government bond yields) and GDP, as discussed for example in Friedman and Kuttner (1992), Bernanke (1990) and Duffie and Singleton (2003). Although the connection between stock markets and economic cycle is not unambiguous, there is evidence that such a link also exists [Mitchell and Burns (1938), Fischer and Merton (1985), Barro (1990), Bosworth et al. (1975), Cole et al. (2008) and Estrella and Mishkin (1998)]. There is also some evidence of a direct link between asset returns and population age structure [Poterba (2001) and DellaVigna and Pollet (2007)]. The stochastic model presented in this paper describes the relevant risk factors and their connections in terms of linear stochastic difference equations. Due to its simplicity, the model is easy to study both analytically and numerically. This makes it well-suited for the analysis of mortality-linked cash flows and associated investment strategies. The underlying risk factors in our model have natural interpretations, so its behaviour is easily judged by the user. It is easy to calibrate to both the historical data and user’s expectations about the future development of mortality and the economy. This is a useful feature, as the historical data do not always provide the best description of future development of mortality. The rest of this paper is organized as follows. Section 2 briefly recalls the mortality model introduced in Aro and Pennanen (2011) and fits the model to the historical data of six countries. Section 3 presents a statistical analysis of the risk factors. Based on the results of Section 3, Section 4 builds a stochastic model of the risk factors. Section 5 presents illustrative simulations.

2. Parameterization of mortality rates Stochastic mortality modelling has attracted steady attention in literature ever since the seminal work of Lee and Carter (1992), see for example Cairns et al. (2006, 2007), Renshaw and Haberman (2003), Brouhns et al. (2002), Lee and Miller (2001) and Booth et al. (2002) and their references. Most of the proposed models describe age-dependent mortality rates by parametric functions of the age. Indeed, the age dependency displays certain regularities that can be used to reduce the dimensionality of a model. This idea was used already in the classical Gompertz model as well as in the stochastic model proposed in Aro and Pennanen (2011), which we now briefly recall.

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4 486

Let Ex,t be the number of individuals aged [x,x1) years at the beginning of year t in a given population. The number of deaths Dx,t among the Ex,t individuals during year [t,t1) can be described by the binomial distribution: Dx;t
BinðEx;t ; qx;t Þ;

(1)

where qx,t is the probability that an x year-old individual randomly selected at the beginning of year t dies during [t,t1). As in Aro and Pennanen (2011), we model the survival probabilities px,t 1qx,t with the formula: px;t

 Pn i i  exp i¼1 tt / ðxÞ ; ¼ P n 1 þ expð i¼1 tit /i ðxÞÞ

(2)

where fi are user-defined basis functions and tit are stochastic risk factors that may vary over time. In other words, the yearly logistic survival probability of an x year old is given by ! n X px;t ¼ (3) tit /i ðxÞ: logitpx;t :¼ ln 1  px;t i¼1 The logistic transformation implies that the probabilities px,t and qx,t 1px,t remain in the interval (0,1). By an appropriate choice of the functions fi(x) one can incorporate certain desired features into the model. For example, the basis functions can be chosen so that the survival probabilities px,t have a regular dependence on the age x like, for example, in the classical Gompertz model. As in Aro and Pennanen (2011), we will use the three piecewise linear basis functions given by  1  x18 for x  50 32 /1 ðxÞ ¼ 0 for x  50; 1 ðx  18Þ for x  50 /2 ðxÞ ¼ 32 x 2 50 for x  50;  0 for x  50 /3 ðxÞ ¼ x  1 for x  50: 50 The linear combination

3 P i¼1

tit /i ðxÞ will then be piecewise linear and continuous as a

function of the age x; see Figure 1. The risk factors tit now represent points on logistic survival probability curve: t1t ¼ logit p18;t ; t2t ¼ logit p50;t ; t3t ¼ logit p100;t : It is to be noted that this is just one possible choice of basis functions. Another set of basis functions would result in another set of risk factors with different interpretations. In particular, the Cairns
Blake
Dowd model as described in Cairns et al. (2008, Section 4.2)

Stochastic modelling

4875

Figure 1. Three piecewise linear basis functions and their linear combination.

, where x  is the mean over all ages. In this corresponds to /1 ðxÞ 1 and /2 ðxÞ ¼ x  x 1 case, the parameter y describes the general level of mortality, while y2 determines how mortality rates change with age. We will use the three-parameter model described above mainly because of its simple interpretation in the economic context. Once the basis functions fi are fixed, the realized values of the corresponding risk factors tit can be easily calculated from the historical data using standard max-likelihood estimation. The log-likelihood function can expressed as

lt ðtÞ ¼

X x2X

"

Dx;t

X

P

ti /i ðxÞ  Ex;t lnð1 þ e

i

i

ti /i ðxÞ

#

Þ þ ct

where ct is a constant; see Aro and Pennanen (2011). The maximization of lt is greatly facilitated by the fact that lt is a convex function of y (see Aro and Pennanen 2011, Proposition 3). We fit Equation (3) into the mortality data from six countries, obtained from the Human Mortality Database.1 Plots of the historical values of the risk factors are presented in Figures 2
7.

3. Statistical analysis of the risk factors This section presents a statistical analysis of mortality and economic risk factors. In particular, we address the long-term development of the mortality risk factors, and study the connection of mortality with the economy. In order to get a comprehensive view of mortality dynamics in developed countries, we investigate the mortality dynamics of six large countries: Australia, Canada, France, Japan, the UK and the US. As one of our aims is to investigate whether mortality rates might level out in the future, we have chosen countries where life expectancy is already relatively high. 1

www.mortality.org

H. Aro & T. Pennanen

6 488 υ1 AU

υ1 CAN

υ1 FR

8.4 8.2

8

8.2

7.8

8

8

7.6

7.8

7.8

7.4

7.6

7.6

7.2

7.4

7

7.4 1960

1980 υ1

2000

1960

1980 υ1

JP

2000

1980 υ1

UK

2000

US

8

8.5

8

1960

7.8

7.5 8 7

7.6

6.5

7.5

7.4

6 1960

1980

2000

1960

1980

2000

1960

1980

2000

Figure 2. Historical values for risk factor y1, females. Note the different scales.

3.1. Risk factor y1 The risk factor y1 described above gives the annual logit survival probability of 18-year olds. Overall, mortality of young adults is generally low in developed countries, and υ1 CAN

υ1 AU

υ1 FR

7.6

7.4

7.6 7.4

7.3

7.4 7.2

7.2 7.2

7.1

7

7

7

6.8

6.9 1960

1980

2000

1960

υ1 JP

1980

2000

1960

υ1 UK

7.5

1980

2000

υ1 US

7.8

7

7.6

6.9

7 6.5

6.8

7.4

6

6.7 1960

1980

2000

1960 1

1980

2000

Figure 3. Historical values for risk factor y , males. Note the different scales.

1960

1980

2000

Stochastic modelling υ2 AU

7 489

υ2 CAN

υ2 FR

6.5

6.2

6.5

6 5.8

6

6

5.6 5.5

5.4 1960

1980

2000

5.2

5.5 1960

υ2 JP

1980

2000

1960

υ2 UK

6.2 6.5

1980

2000

υ2 US

5.8

6 5.6

6

5.8

5.5

5.6

5.4 5.2

5.4

5 1960

1980

2000

1960

1980

2000

1960

1980

2000

2

Figure 4. Historical values for risk factor y , females. Note the different scales.

deaths are most commonly caused by accidents, assaults and suicide as reported by National Vital Statistics (2010). Such contributors may be unlikely to be efficiently eliminated by medical advances, regulatory interventions or lifestyle changes on a national level. There is also some evidence that the decline in coronary heart disease mortality is slowing down or even reversing in young adult age groups in the UK and the Netherlands [Allender et al. (2008) and Vaartjes et al. (2011)]. As observed in Figures 2 and 3, in some countries the historical values y1 have flattened out in recent years. This is more notable in populations where the yearly survival probabilities have reached higher levels, namely for females, especially in Japan and the υ2 AU

υ2 FR

υ2 CAN

6

5.4 5.6 5.2

5.4

5.5

5

5.2

5

5

4.8

4.8 1960

1980

2000

4.6 1960

υ2 JP

1980

2000

1960

υ2 UK

5.6

5.6

5.4

5.2

5.2

5

5

4.8

4.8

4.6

4.6 1960

1980

2000

2000

5.2

5.4

4.8

1980 υ2 US

5

1960 2

1980

2000

Figure 5. Historical values for risk factor y , males. Note the different scales.

1960

1980

2000

H. Aro & T. Pennanen

8 490 υ3 AU

υ3 CAN

υ3 FR

0.7

0.8

0.8

0.6

0.7

0.6

0.5

0.6

0.4

0.5

0.3

0.4

0.2

0.3

1950 1960 1970 1980 1990 2000

0.4 0.2 0

1950 1960 1970 1980 1990 2000

υ3 JP

1950 1960 1970 1980 1990 2000

υ3 UK

1

υ3 US

0.6

0.9

0.8 0.4

0.6

0.8

0.4

0.7

0.2 0.2

0.6

0

0

1950 1960 1970 1980 1990 2000

1950 1960 1970 1980 1990 2000

0.5 1950 1960 1970 1980 1990 2000

Figure 6. Historical values for risk factor y3, females. Note the different scales.

UK. This observation suggests that mortality rates may start to stabilize when they reach certain levels, a phenomenon suggested already in Wicksell (1926). In order to analyse Wicksell’s conjecture, we fit the equation Dt1t ¼ b þ at1t1

(4)

by Ordinary Least Squares (OLS) regression into the historical values of y1 for each of the six countries. When a
(0,2), Equation (4) corresponds to mean-reverting behaviour with υ3 AU

υ3 CAN

0.5

υ3 FR

0.5

0.6

0.4

0.4

0.4 0.3 0.2

0.2

0.3

0.1

0

0 1960

1980

2000

1960

υ3 JP

1980

2000

1960

υ3 UK

0.6

1980

2000

υ3 US 0.8

0.4

0.4

0.7 0.2

0.2

0.6

0

0

0.5

−0.2 0.4

−0.2 1960

1980

2000

1960 3

1980

2000

Figure 7. Historical values for risk factor y , males. Note the different scales.

1960

1980

2000

Stochastic modelling

9 491

equilibrium level t ¼  ba. For t1  t, the right-hand side of the equation is positive, and it tends to zero as y1 approaches t. While Equation (4) with a
(0,2) corresponds to a stationary time series in the long run, it is doubtful if any of the series in Figures 2 and 3 is stationary over the observed period. This was confirmed by statistical tests (augmented Dickey
Fuller test for unit roots). With a null hypothesis of a unit root, all of the p values were larger than 0.2, and most above 0.5. This illustrates the general challenge in mortality modelling, where one is trying to model the long-range behaviour of uncertain data obtained from a system in a state of transition. There is a reason to believe that the characteristics of the underlying process will change in the future but such information is not necessarily contained in the historical data. Estimates for parameters a and b along with regression diagnostics are presented in Table 1. For women the parameter estimates of a are negative in all countries, with p values of the respective t-test statistics below 0.05. In the case of males, aB0 for all countries except for France. For Australia, Japan and the UK the p values for the regression coefficients are below 0.08. Table 2 reports results for the residual tests. The basic assumptions of linear regression [independence, homoscedasticity, normality; see e.g. Hamilton (1994)] are fulfilled, apart from some indication of serial correlation. Stationarity and unit root tests indicate stationarity and absence of unit roots in the residual time series. Overall, Equation (4) fits well into the historical data. It may thus serve as a reasonable model for survival probability of young adults if one shares Wicksell’s views on the stabilization of mortality rates in the long run. The better fit for females may be explained by the fact that female mortality is generally lower than male mortality, and may thus be closer already to its equilibrium level. Table 1. Parameter estimates, t values and summary statistics for Dt1t ¼ b þ at1t1. AU

CAN

F

JP

UK

USA

Female b p value (t-statistic) a p-value (t-statistic) R2 Adj. R2 p value (F-statistic)

0.778 0.017 0.095 0.020 0.097 0.080 0.020

0.446 0.028 0.053 0.036 0.079 0.062 0.036

0.569 0.020 0.072 0.026 0.089 0.072 0.026

0.614 0.000 0.074 0.000 0.627 0.620 0.000

0.902 0.000 0.107 0.000 0.461 0.451 0.000

0.437 0.015 0.055 0.018 0.100 0.083 0.018

Male b p value (t-statistic) a p value (t-statistic) R2 Adj. R2 p value (F-statistic)

1.149 0.075 0.160 0.077 0.057 0.039 0.077

0.101 0.686 0.012 0.721 0.002 0.016 0.721

0.182 0.549 0.028 0.513 0.008 0.010 0.512

0.518 0.000 0.065 0.000 0.417 0.406 0.000

1.475 0.000 0.189 0.000 0.273 0.259 0.000

0.439 0.207 0.063 0.209 0.029 0.011 0.210

Note: The table reports coefficient estimates for the regression, and the p values of their t-statistics. A low p value supports rejection of the null hypothesis that the coefficient is equal to zero. The F-statistic tests for the joint significance of regressors. A low p value of the F-statistic supports rejection of the null hypothesis that all equation coefficients are equal to zero.

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Table 2. Residual test statistics for the regression Dt1t ¼ b þ at1t1 . AU

CAN

F

JP

UK

USA

Female Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF t-statistic p value KPSS level p value

0.329 0.450 0.546 9.191 B0.01 0.219 0.1

0.533 0.364 0.307 8.374 B0.01 0.164 0.1

0.123 0.170 0.074 8.105 B0.01 0.394 0.080

0.731 0.484 0.013 7.11 B0.01 0.086 0.1

0.084 0.692 0.029 3.722 B0.01 0.255 0.1

0.044 0.481 0.263 4.474 B0.01 0.139 0.1

Male Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF statistic p value KPSS statistic p value

0.364 0.920 0.781 7.008 B0.01 0.219 0.1

0.551 0.221 0.151 8.443 B0.01 0.221 0.1

0.146 0.595 0.519 5.802 B0.01 0.267 0.1

0.349 0.001 0.001 6.945 B0.01 0.134 0.1

0.167 0.543 0.697 5.155 B0.01 0.843 0.1

0.000 0.113 0.311 4.6 B0.01 0.128 0.1

Note: For serial correlation, normality and heteroscedasticity tests, the numbers are p values of the statistics in the first column. Serial correlation is tested with the Breusch
Godfrey tests for up to fourth-order serial correlation. A small p value suggests lack of serial correlation. Normality is tested with the Jarque
Bera test, with null hypothesis of normality. A large p value supports the normality assumption. The Breusch
Pagan test tests against heteroscedasticity, with the null hypothesis of homoscedasticity. A large p value indicates absence of heteroscedasticity. Stationarity of the residuals is tested with the augmented Dickey
Fuller (ADF) and Kwiatkowski
Phillips
Schmidt
Shin (KPSS) tests. The ADF unit root regressions do not include deterministic terms, and the number of lags was selected by their significance (10% level) with a maximum number of lags (10), determined by the method suggested by Schwert (1989). For KPSS test, the null hypothesis was level stationarity.

3.2. Risk factor y2 The significant reduction in coronary disease is a key contributor to the rapid increase in the survival probability of the middle-aged population during recent decades. This effect is also reflected in risk factor y2 (see Figures 4 and 5) corresponding to the survival probability of 50-year olds. In Ford et al. (2007) and Preston and Wang (2006), approximately half of the improvement is attributed to medical treatments, and the remaining half to reductions in other factors, such as blood pressure, cholesterol and smoking prevalence. In the future, both novel medical innovations and wider availability of treatment, as well as possible further reductions in smoking, are likely to contribute to the improvement of y2. On the other hand, these advances may to some extent be outweighed by the detrimental effects of obesity and other lifestyle-related factors [Schroeder (2007) and Olshansky et al. (2005)]. Consequently, it is possible that y2 will level out in the future. As the historical values are generally not showing signs of stabilizing, we simply fit a trend into y2 in order to quantify the rate of improvement of the historical values of y2: Dt2 ¼ b: Results are presented in Tables 3 and 4. The p values of the t-tests for parameter b in all sample countries are 0.001 or smaller. In some cases the test statistics for residuals

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11 493

Table 3. Parameter estimates and p values for Dt2t ¼ b.

Female p value Male p value

AU

CAN

F

JP

UK

USA

0.026 0.000 0.024 0.000

0.020 0.000 0.018 0.000

0.024 0.000 0.015 0.001

0.037 0.000 0.023 0.000

0.017 0.000 0.017 0.000

0.014 0.000 0.014 0.000

indicates some serial correlation and non-normality, but stationarity and absence of unit roots are well supported. 3.3. Risk factor y3 Pension benefits and other mortality-linked securities are often connected with old-age mortality, which has received wide attention in literature [e.g. Thatcher (1999), Mesle and Vallin (2006), Kannisto et al. (1994) and Vallin and Mesle (2004)]. In our model, risk factor y3 describes the mortality of the 100 year old. Figures 6 and 7 show that its value has generally improved in the past, with some strong fluctuations. A striking feature is that, in particular for US and Canadian males and to some extent US females, the development has been very rapid in the twenty-first century. This phenomenon was also recognized in Rau et al. (2008). Reductions in mortality in general and old-age mortality in particular are generally linked with reductions in smoking [Cairns and Kleinow (2011), Doll et al. (2004) and Preston and Wang (2006)]. Another contributor suggested by Ho and Preston (2010) is the intensive deployment of life-saving technologies by the US health care system at very old ages, which has contributed to the mortality improvements of the old, particularly in Table 4. Residual test statistics for Dt2t ¼ b2. AU

CAN

F

JP

UK

USA

Female Serial correlation (BG) Normality (JB) Heteroscedasticity (GQ) ADF t-statistic p value KPSS level p value

0.406 0.803 0.887 6.376 B0.01 0.459 0.052

0.121 0.502 0.999 10.236 B0.01 0.059 0.1

0.017 0.000 0.998 10.067 B0.01 0.227 0.1

0.144 0.023 0.372 4.863 B0.01 0.752 0.1

0.079 0.000 1 2.109 0.037 0.772 0.1

0.422 0.662 1 7.276 B0.01 0.333 0.1

Male Serial correlation (BG) Normality (JB) Heteroscedasticity (GQ) ADF t-statistic p value KPSS level p value

0.006 0.497 0.998 2.564 0.012 1.199 0.1

0.016 0.018 0.998 2.106 0.037 1.268 0.1

0.025 0.000 1 10.376 B0.01 0.384 0.084

0.336 0.640 0.977 7.076 B0.01 0.395 0.080

0.043 0.000 1 1.688 0.089 1.456 0.1

0.150 0.437 0.997 6.13 B0.01 0.507 0.040

Note: See Table 2 for descriptions of the test statistics. The Breusch
Pagan test, which is considered more robust, was not applicable here as it regresses squared residuals on the independent variables. It was replaced by the Goldfeld
Quandt test. As the more robust Breusch
Pagan test regresses the squared residuals on the independent variables which do not exist here, it was not applicable, and was replaced by the Goldfeld
Quandt test.

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12 494

comparison to other countries. These factors could also explain the recent rapid development in the risk factor y3 of US males. Preston (2007) has investigated mechanisms that link mortality to the level of national income. The rationale behind such a connection is that a higher income facilitates increased consumption of goods and services with health benefits. The long-term connection between national income and mortality has also been discussed for example in Barrieu et al. (2012), Cutler et al. (2006), Hanewald (2011), Preston (2007) and Rodgers (1979). Figure 8 presents the logarithms of real GDP per capita in the sample countries. The GDP data were obtained from Maddison (2011). Similarities in the general shape of the plots for log GDP per capita and y3, especially for those of females, suggest that the long-term movements of y3 and log-GDP per capita may indeed be connected. We analyse the dependence of y3 on GDP with the regression: Dt3t ¼ b þ a1 t3t1 þ a2 gt1 ; where gt is the logarithm of GDP per capita. The interpretation is that the drift of t3t depends on its relation to gt. If a1 B0 and a2 0, the drift increases if t3t lags behind gt. Tables 5 and 6 present the regression results. Coefficients a1 are negative and a2 positive for all regressions except for the US males for the entire observation period. Because of the exceptionally rapid recent growth in y3 of US males, we performed the regressions for an additional truncated period of 1950
2000, in order to inspect the dependence between y3 and GDP before this unusual development. For the truncated period a1 is negative and a2 positive, like for other regressions. The p values for all regression coefficients, but those of Japanese females, are below 0.05. Residual test statistics indicates that, in general, the basic assumptions of linear regression are fulfilled, apart from some indication for heteroscedasticity and non-normality. All in all, the results support the hypothesis that the long-term increases in GDP have a positive effect on old-age mortality. log−GDP AU

log−GDP CAN

10

10

9.5

9.5

log−GDP FR 9.8 9.6 9.4 9.2 9 8.8 8.6

9

9 1960

1980

2000

1960

1980

2000

1960

log−GDP UK

log−GDP JP 10

1980

2000

log−GDP US

10

10.2

9.5

10

9

9.5

9.8

8.5

9.6

8

9.4

9

9.2 1960

1980

2000

1960

1980

2000

Figure 8. Logarithm of GDP per capita by country. Note the different scales.

1960

1980

2000

Stochastic modelling Table 5.

13 495

Parameter estimates, t values and summary statistics for the regression Dt3t ¼ b þ a1 t3t1 þ a2 gt1 . AU

CAN

F

JP

UK

USA

Female b p value (t-statistic) a1 p value (t-statistic) a2 p value (t-statistic) R2 Adj. R2 p value (F-statistic)

3.062 0.000 0.877 0.000 0.367 0.001 0.454 0.433 0.000

1.686 0.000 0.455 0.000 0.205 0.000 0.255 0.227 0.000

1.826 0.001 0.388 0.001 0.208 0.001 0.195 0.164 0.003

0.332 0.053 0.057 0.1941 0.040 0.044 0.076 0.041 0.124

2.894 0.000 0.656 0.000 0.334 0.000 0.335 0.310 0.000

0.664 0.020 0.189 0.015 0.083 0.016 0.109 0.076 0.046

Male b p value (t-statistic) a1 p value(t-statistic) a2 p value (t-statistic) R2 Adj. R2 p value (F-statistic)

1.212 0.001 0.405 0.001 0.136 0.001 0.207 0.177 0.002

0.273 0.091 0.234 0.034 0.037 0.054 0.089 0.055 0.084

1.359 0.003 0.335 0.003 0.156 0.003 0.164 0.132 0.009

0.494 0.029 0.121 0.042 0.057 0.024 0.093 0.059 0.074

1.803 0.001 0.456 0.001 0.195 0.001 0.202 0.172 0.003

0.238 0.143 0.002 0.976 0.025 0.197 0.088 0.054 0.087

US trunc 0.421 0.009 0.341 0.003 0.060 0.004 0.1853 0.1506 0.0081

Note: See Table 1 for description of the test statistics.

The plots of GDP and y2 also somewhat resemble each other, but performing the above analysis performed on y2 did not yield equally good results. 3.4. GDP and financial markets It is generally accepted that GDP is connected with financial markets, see Stock and Watson (2003) for a review. In particular, the term spread, which is broadly defined as the difference between long- and short-term interest rates, seems to be linked with changes in Table 6.

Residual test statistics for the regression Dt3t ¼ b þ a1 t3t1 þ a2 gt1 . AU

CAN

F

JP

UK

USA

Female Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF t-statistic p value KPSS level p value

0.000 0.536 0.330 7.356 B0.01 0.120 0.1

0.002 0.279 0.038 3.921 B0.01 0.312 0.1

0.000 0.009 0.101 1.172 0.1 0.463 0.050

0.000 0.203 0.001 10.599 B0.01 0.084 0.1

0.090 0.880 0.012 6.843 B0.01 0.263 0.1

0.26 0.533 0.734 8.499 B0.01 0.150 0.1

Male Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF t-statistic p value KPSS level p value

0.001 0.284 0.865 9.475 B0.01 0.293 0.1

0.022 0.759 0.674 8.919 B0.01 0.171 0.1

0.000 0.1993 0.007 1.336 0.1 0.446 0.057

0.000 0.075 0.0003 10.512 B0.01 0.118 0.1

0.113 0.220 0.498 3.329 B0.01 0.272 0.1

0.035 0.756 0.235 8.569 B0.01 0.135 0.1

Note: See Table 2 for description of the test statistics.

US trunc 0.059 0.716 0.366 7.607 B0.01 0.176 0.1

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14 496

economic activity [see e.g. Wheelock and Wohar (2009)]. Also, GDP appears to be strongly connected with credit spreads, which reflect how the market perceives default risk [see Duffie and Singleton (2003)]. In addition, there is evidence of a link between stock prices and GDP [see e.g. Bosworth et al. (1975) and Fischer and Merton (1985)]. This section illustrates how such connections can be conveniently analysed in terms of simple linear regressions. Combined with the results of the previous sections, we obtain a link between mortality and financial markets. As an example, we analyse the connections of US GDP to the US government and corporate bond markets. Figure 9 shows the term spreads, the log GDP differences and the credit spreads for the period of 1953
2006. The term spread is the difference between US treasury constant maturity rates of 5 years and 1 year. The credit spread is the difference between Moody’s seasoned corporate bond yields with ratings BAA and AAA. All interest rate data are end-of-year values from Federal Reserve Economic Data (FRED) (2011a, 2011b, 2011c, 2011d). We observe that the log-GDP difference curve often moves in the same direction as term spread, and in opposite direction from credit spread. Combining the findings of for example Wheelock and Wohar (2009) for term spread and (Duffie and Singleton (2003)) for credit spread, we study the connection of US log GDP per capita to both the term spread and the credit spread with the regression: Dgt ¼ b þ a1 sTt1 þ a2 sCt1 ;

(5)

Term spread 6 4 2 0 −2 −4 1955

1960

1965

1970

1975

1980 ∆ log−GDP

1985

1990

1995

2000

2005

1995

2000

2005

1995

2000

2005

0.06 0.04 0.02 0 −0.02 −0.04 1955

1960

1965

1970

1975

1980

1985

1990

Credit spread 2.5 2 1.5 1 0.5 0 1955

1960

1965

1970

1975

1980

Figure 9. Term spread, differenced log-GDP and credit spread.

1985

1990

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15 497

where sT is the difference between the logarithms of long and short interest rates and sC is the logarithm of the difference between the logarithmic yields on BAA- and AAA-rated bonds. The logarithmic transformations are motivated by the positivity of the associated quantities. They will guarantee that all interest rates as well as the credit spread will remain positive in simulations, see Section 5. Regression results are presented in Table 7. Parameter estimates are positive for b and a1 and negative for a2, which is in line with earlier findings. Corresponding t-tests produce very small p values. The regression diagnostics are in line with the requirements of the ordinary least squares regression, apart from some indication of serial correlation. In addition, unit root and stationarity tests suggest that the residuals are stationary. The positivity of a1 means that large term spreads anticipate high GDP growth rates. In other words, a large difference between long- and short-term interest rates predicts strong economic growth, and vice versa. Negativity of a2, on other hand, means that large credit spreads precede small GDP growth rates. In other words, large credit spreads seem to predict slower economic growth, and vice versa. In order to employ the above relations in a simulation model, we need a description of the interest rate spreads as well. To this end, we fit the equations D sTt ¼ b þ asTt1 ;

(6)

asCt1 ;

(7)

D sCt

¼bþ

into US data for term and credit spreads. These are analogous to the mean reverting interest rate model of Vasicek (1977). According to the results in Tables 8 and 9, the regressions fit the data well, although for Equation (6), residual diagnostics do not support the normality hypothesis and for Equation (7), we observe a possibility of serial correlation.

4. Modelling the risk factors Based on the observations in the previous section, we propose to model the mortality risk factors, GDP and the interest rate spreads with the system of equations: Table 7.

Parameter estimates, t values and summary statistics for the regression Dgt ¼ b þ a1 sTt1 þ a2 sCt1 .

b p value (t-statistic) a1 p value (t-statistic) a2 p value (t-statistic)

0.045 0.040 0.028 0.047 0.030 0.005

R2 Adj. R2 p value (F-statistic)

0.17 0.14 0.009

Serial Correlation (BG) Normality (JB) Heteroscedasticity (BP)

ADF t-statistic p value KPSS level p value

Note: See Tables 1 and 2 for descriptions of the test statistics.

0.975 0.632 0.254

0.950 B0.01 0.280 0.1

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Table 8. Parameter estimates, t values and summary statistics for the regression DsTt ¼ b þ asTt1 . b p value (t-statistic) a p value (t-statistic) R2 Adj. R2 p value (F-statistic)

0.049 0.093 0.429 0.000 0.209 0.1931 0.000

Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF t-statistic p value KPSS level p value

0.022 0.000 0.100 0.771 B0.01 0.274 0.1

Note: See Tables 1 and 2 for descriptions of the test statistics. ;1 Dtft ;1 ¼ a11 tft1 þ b1 þ e1t

Dttf ;2 ¼ b2 þ e2t ;3 Dtft ;3 ¼ a33 tft1 þ a37 gt1 þ b3 þ e3t 4 4 Dtm;1 ¼ a44 tm;1 t1 þ b þ et t

Dtm;2 ¼ b5 þ e5t t 67 6 6 ¼ a66 tm;3 Dtm;3 t1 þ a gt1 þ b þ et t

Dgt ¼ a78 sTt1 þ a79 sCt1 þ b7 þ e7t D sTt ¼ a88 sTt1 þ b8 þ e8t D sCt ¼ a99 sCt1 þ b9 þ e9t where eit are random variables describing the unexpected fluctuations in the risk factors. As before, gt denotes the logarithm of GDP per capita, sTt term spread, sCt the credit spread and tft ;i and tm;i t are mortality risk factors for males and females, respectively. The above system can be written as a linear multivariate stochastic difference equation: Dxt ¼ Axt1 þ b þ et ; i m;2 m;3 99 T C where x ¼ tft ;1 ; tft ;2 ; tft ;3 ; tm;1 , b 2 R9 and ot are R9 -valued t ; tt ; tt ; gt ; st ; st , A 2 R random vectors. This compact formulation is straightforward to study, both analytically and numerically. The linear equations above describe a direct link between mortality of ages between 50 and 100 (as y3 depends on these ages by definition) and financial markets. We discover statistical links to the risk factors y1 and y2 through the correlation structure of the vector h

Table 9.

Parameter estimates, t values and summary statistics for the regression DsCt ¼ b þ asCt1 .

b p value (t-statistic) a p value (t-statistic) R2 Adj. R2 p value (F-statistic)

1.188 0.000 0.571 0.000 0.3028 0.2891 0.000

Serial correlation (BG) Normality (JB) Heteroscedasticity (BP) ADF t-statistic p value KPSS level p value

Note: See Tables 1 and 2 for descriptions of the test statistics.

0.646 0.826 0.580 0.972 B0.01 0.138 0.1

Stochastic modelling

17 499

of random variables o below. However, old-age mortality is often the most relevant risk factor when studying mortality-linked cash flows. This is the case for example in pension benefits and longevity bonds, see for example Blake et al. (2006a, 2006b). In order to model the random vectors ot, we examine the residuals of regression equations in Section 3 for US data. Due to the availability of US Treasury Constant Maturity data from 1953 onwards only, we used the residuals for the period 1953
2006 to estimate ot. Except for the interest rate spreads, the residuals appear to follow a Gaussian distribution; see Tables 2, 4 and 6. The correlation matrix R 2 R99 and standard deviation vector r 2 R9 are given below. In matrix R, the corresponding p values for t-tests with null hypothesis of no correlation are given in parenthesis:

2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 R¼6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

1:0000

0:5010 ð0:0001Þ 1:0000

0:4729 0:6820 0:6639 0:2202 0:3123 ð0:0003Þ ð0:0000Þ ð0:0000Þ ð0:1131Þ ð0:0228Þ 0:2475 0:2736 0:7424 0:2299 0:0026 ð0:0740Þ ð0:0474Þ ð0:0000Þ ð0:0977Þ ð0:9854Þ 1:0000 0:0525 0:4452 0:8151 0:2485 ð0:7089Þ ð0:0008Þ ð0:0000Þ ð0:0728Þ 1:0000 0:3301 0:0027 0:1865 ð0:0158Þ ð0:9849Þ ð0:1813Þ 1:0000 0:2089 0:2008 ð0:1333Þ ð0:1493Þ 1:0000 0:2179 ð0:1171Þ 1:0000

s[0.0346 0.0144

0:0518 ð0:7126Þ 0:0710 ð0:6133Þ 0:1282 ð0:3601Þ 0:1509 ð0:2807Þ 0:0023 ð0:9869Þ 0:1855 ð0:1835Þ 0:4351 ð0:0011Þ 1:0000

3 0:0269 ð0:8485Þ 7 7 0:1309 7 7 ð0:3500Þ 7 7 0:0732 7 7 ð0:6024Þ 7 7 0:0103 7 7 ð0:9418Þ 7 7 0:1982 7 7 ð0:1548Þ 7 7 0:1675 7 7 ð0:2305Þ 7 7 0:4724 7 7 ð0:0004Þ 7 7 0:2586 7 7 ð0:0615Þ 5 1:0000

0.0323

0.0493 0.0175

0.0286 0.0197

0.1761 0.2604]T

Correlations between many of the residuals seem to deviate significantly from zero. Thus, in addition to the long-term relations described by the system equations, there seem to be short-term links between the risk factors. In particular, the residuals of equations for Dtft ;1 and Dtft ;3 have negative correlations with the residuals of Dgt. This suggests that mortality is procyclical with economic growth, a phenomenon also observed in Hanewald (2011), Ruhm (2000) and Tapia Granados (2005a, 2005b). Moreover, the residuals for DsTt and DsCt are negatively correlated with those of Dgt, with relatively low p values. Several of the correlation coefficients between residuals for mortality risk factors tit also have small p values. For instance, the residuals for Dtft ;1 are positively correlated with the residuals for all other Dtit s, except for Dtm;3 t . This is a plausible phenomenon, as mortality rates across age groups are likely to depend on some common factors. In order to better capture the uncertainties in mortality from the available data, we will next study the residuals ot on a global level, using the data set from Section 3 for six

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18 500

countries over the years 1950
2006. Due to lack of sufficient financial data for some countries, we study the reduced system: Dtft ;1 ¼ a11 tft ;1 þ b1 þ e1t Dtft ;2 ¼ b2 þ e2t Dtft ;3 ¼ a33 tft ;3 þ a37 gt þ b3 þ e3t þ b4 þ e4t ¼ a44 tm;1 Dtm;1 t t ¼ b5 þ e5t Dtm;2 t þ a67 gt þ b6 þ e6t ¼ a66 tm;3 Dtm;3 t t Dgt ¼ b7 þ e7t ; where GDP is modelled as a geometric Brownian motion, as for example in Kruse et al. (2005). The parameters of the first six rows were estimated from the data in Section 3 for each country. Parameter estimates of the equation for GDP are given in Table 10. 3 2 1:0000 0:0829 0:3369 0:4875 0:2678 0:2368 0:1070 6 ð0:1295Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0499Þ 7 7 6 6 1:0000 0:0042 0:1226 0:6640 0:1023 0:0723 7 7 6 6 ð0:9392Þ ð0:0246Þ ð0:0000Þ ð0:0611Þ ð0:1863Þ 7 7 6 6 1:0000 0:1228 0:2148 0:8343 0:0610 7 7 6 6 ð0:0243Þ ð0:0001Þ ð0:0000Þ ð0:2650Þ 7 7 6 R¼6 1:0000 0:0876 0:1219 0:1803 7 7 6 6 ð0:1089Þ ð0:0255Þ ð0:0009Þ 7 7 6 6 1:0000 0:1156 0:1182 7 7 6 6 ð0:0342Þ ð0:0303Þ 7 7 6 6 1:0000 0:0703 7 7 6 4 ð0:1983Þ 5 1:0000 s [0.0522 0.0254

0.0568

0.0526 0.0261

0.0567 0.0268]T

The results are in general in line with those for the USA. However, in contrast with the f ;2 US data, the residuals of Dtft ;1 are now correlated with Dtm;3 t , but not with Dtt . The f ;3 m;1 same applies to Dtt and Dtt . Also, on a global level, the residuals for male and female Table 10.

Parameter estimates, p values and residual test statistics for the regression Dgt ¼ b7

b4 p values Serial correlation (BG) Normality (JB) Heteroscedasticity (GQ) ADF t-statistic p value KPSS level p value

AU

CAN

F

JAP

UK

USA

0.021 0.000 0.138 0.329 0.559 6.249 B0.01 0.065 0.1

0.021 0.000 0.432 0.022 0.540 5.895 B0.01 0.166 0.1

0.026 0.000 0.002 0.777 0.956 3.188 B0.01 1.455 0.1

0.044 0.000 0.000 0.470 0.998 2.859 B0.01 1.876 0.1

0.021 0.000 0.037 0.389 0.536 6.41 B0.01 0.034 0.1

0.021 0.000 0.68 0.373 0.965 5.457 B0.01 0.050 0.1

Note: See Table 2 for description of the test statistics.

Stochastic modelling

19 501

Dy1 and male Dy2 are negatively correlated with the residuals of Dgt, with low p values, whereas this no longer applies to Dy3. Recall, however, that the equation for Dgt is now different from the one used in the US model where interest rate spreads appeared as regressors.

5. Simulations We study the behaviour of the developed model for the USA by generating 10,000 simulations of 50 years into the future, starting from the observed values of the risk factors at the end of year 2006.

5.1. Historical parameter estimates In the first set of simulations, parameters of the model were estimated from the historical US data of 1953
2006, with the exception of standard deviations of y1, y2 and y3, which were estimated from the global data, in order to get a more comprehensive view on the uncertainty involved in the future development of mortality. Figures 10 and 11 display sample paths and confidence intervals, respectively, for all the nine risk factors. Figure 12 plots the median and the 95% confidence interval for yearly survival probabilities in the final year of simulation as a function of age. Figure 13

υ1 females 8.4

1.5

6.5

8.2 8

υ3 females

υ2 females

6

1

7.8 5.5

7.6

0.5 1960 1980 2000 2020 2040

1960 1980 2000 2020 2040

υ1 males 7.2

6

7

1960 1980 2000 2020 2040

υ2 males

υ3 males 2.5 2

5.5

1.5

6.8 5

1

6.6

0.5 1960 1980 2000 2020 2040

1960 1980 2000 2020 2040

11.5 11 10.5 10 9.5

1960 1980 2000 2020 2040 sC

sT

g 0.8 0.6 0.4 0.2

−1.5

0

−2.5

−2

−0.2 1960 1980 2000 2020 2040

1960 1980 2000 2020 2040

Figure 10. Two sample paths of the risk factors.

1960 1980 2000 2020 2040

H. Aro & T. Pennanen

20 502 υ1 females

υ2 females

υ3 females

8.4

1.5

8.2

6.5

8

6

1

7.8 5.5

7.6

0.5 1960

1980

2000

2020

2040

1960

1980

υ1 males

2000

2020

2040

1960

1980

υ2 males

2000

2020

2040

υ3 males

7.2 2.5

6 7

2 5.5

6.8

1.5

5

1 0.5

1960

1980

2000

2020

2040

1960

1980

2000

2020

2040

1960

1980

sT

g 11.5

0.8

11

0.6

2000

2020

2040

2020

2040

sC −1.5

0.4

10.5

−2

0.2

10

0 −2.5

−0.2

9.5 1960

1980

2000

2020

2040

1960

1980

2000

2020

2040

1960

1980

2000

Figure 11. Development of the medians and 95% confidence intervals of the risk factors.

displays the development of the median and the 95% confidence interval for the survival index of the cohort aged 50 in the year 2006. Figure 14 plots the level sets of an estimated two-dimensional kernel density of the values of the survival index of a female reference cohort aged 50 in the year 2006, and logGDP at the end of the 50-year simulation period. The shape of the probability density p

, females

p

(x,T)

1

1

0.95

0.95

0.9

0.9

0.85

0.85

0.8

0.8

0.75

20

40

60

, males

(x,T)

80

100

0.75

20

40

60

80

100

Figure 12. Median and 99% confidence interval for yearly survival probability in year 2056 as a function on age.

Stochastic modelling

21 503

S(t), females

S(t), males

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 2000

2010

2020

2030

2040

2050

2060

0 2000

2010

2020

2030

2040

2050

2060

Figure 13. Development of the median and 99% confidence interval for survival index as a function of time for the cohort aged 50 in year 2006.

illustrates the long-range dependency between GDP and mortality in the older age groups. In scenarios where the mortality of the elderly improves rapidly, also GDP has grown faster. In general, the simulation results appear reasonable except perhaps for the fast increase in y3 and the corresponding annual survival probabilities for males. This is reflected in the development of the survival indices in Figure 13 where males would appear to outlive females in the long run, an outcome that may be questioned. Another questionable

0.2 0.18 0.16 0.14 S2056 0.12 0.1 0.08 0.06 0.04 10.4

10.6

10.8

11

11.2

11.4 g 2056

11.6

11.8

12

12.2

Figure 14. Kernel density estimate of the joint distribution of the survival index S2056 of a reference cohort of females aged 50 in year 2006 and log GDP g2056 in the year 2056.

22 504

H. Aro & T. Pennanen

feature of the estimated model is the indefinite growth of y2. The growth is well supported by the historical data but it seems unlikely that the growth rate would remain the same when y2 reaches the levels of y1. 5.2. Subjective modifications Due to its simple structure and the natural interpretation of its parameters, the developed model is easy to modify according to the views of the user. For example, if one believes that the period of rapid growth y3 in the 2000s does not provide correct description of its future development, one may choose to use parameters a66 and a67 estimated from the truncated period of 1950
2000. It is also straightforward to add dependencies between the mortality risk factors. If we believe that the future development of y2 depends on y1, we can model this by including y1 in the equation for y2. In the following simulations, we replace the equations for yf,2and ym,2with ;1 ;2 Dtft ;2 ¼ b2 tanhðcf ðvft1  vft1 ÞÞ;

and m;2 Dtm;2 ¼ b5 tanhðcm ðvm;1 t1  vt1 ÞÞ; t

where b2 and b5 are as before, and c f and c m are positive constants. The sigmoid shape of the hyperbolic tangent has the effect that, as long as the difference t1t1  t2t1 remains large, y2 has roughly a constant drift like in the earlier specification. However, when y2 gets close to y1, the drift goes to zero before y2 overtakes y1. We choose cf ¼ 32 and cm 2, so that when the differences between y1 and y2 are at their current levels (approximately 2 for females and 1.5 for males), the value of the hyperbolic tangent function is approximately one, and y2 has the same drift as in the earlier model. These adjustments are, of course, highly subjective, and only reflect the views of the authors. In the following, we rerun the simulations with aforementioned adjustments. We simulate for a longer period of 100 years, so that the dependency of y2 on y1 is better demonstrated. Results are presented in Figures 15
17. As expected, the increase in y2 begins to slow down as it gets closer to y1. The development of survival probabilities and indices is now better in accordance with the common view that women tend to outlive men. Also, the mortality improvement of males is now more moderate than in the previous model. Whether this is a better description of the future development of mortality is again a matter of subjective opinion.

6. Conclusions This paper proposes simple stochastic models for the joint development of mortality and financial markets. Particular emphasis was placed on the long-term patterns in mortality, and its connections with the economy. Due to its simple structure, the model is easy to

Stochastic modelling υ1 females

23 505

υ2 females

8.4

υ3 females 2

7.5

8.2

7

8

6.5

7.8

6

7.6

5.5

1.5 1 0.5

2000

2050

2100

2000

υ1 males

υ2

2050 males

2100

7.2 7

2050 υ3 males

2100

2050

2100

2050

2100

1.5

6.5

7.1

2000

6

6.9

1

5.5

6.8

5

6.7 2000

2050

0.5

2100

2000

2050

2100

sC

0.8 0.6 0.4 0.2 0 −0.2

12 11 10 2000

2000

sT

g

2050

−1.5 −2 −2.5

2100

2000

2050

2100

2000

Figure 15. Development of the medians and 95% confidence intervals of the risk factors.

calibrate to the historical data and to user’s views concerning the future development of mortality and financial markets. Including additional risk factors, such as other financial variables, would also be straightforward. Based on the statistical analysis of an extensive data set, we found both long- and shortterm dependencies between mortality risk factors and GDP. The long-term dependence p

(x,T)

, females

p

(x,T)

1

1

0.95

0.95

0.9

0.9

0.85

0.85

0.8

0.8

0.75

0.75

0.7

, males

0.7 20

30

40

50

60

70

80

90

100

20

30

40

50

60

70

80

90

100

Figure 16. Median and 99% confidence interval for yearly survival probability in year 2056 as a function on age, after parameter adjustments for y2 and y3.

H. Aro & T. Pennanen

24 506 S(t), females

S(t), males

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 2000

2010

2020

2030

2040

2050

2060

0 2000

2010

2020

2030

2040

2050

2060

Figure 17. Development of the median and 99% confidence interval for survival index as a function of time for the cohort aged 50 in year 2006, after parameter adjustments for y2 and y3.

shows in the regression equations, while the short-term connections appear in the correlation structure of the residuals. We also provide corroboration for the relation of GDP to bond markets. The proposed models incorporate these features in simple systems of linear stochastic difference equations. The models are easy to simulate and to adjust according to user’s views. The development of the models was motivated by their potential use in pricing and hedging of mortality-linked financial instruments such as longevity bonds, forwards and swaps. These new instruments have been found to be valuable risk management tools for example for pension insurers [see Blake et al. (2006b), Cairns (2011) and Coughlan et al. (2011)]. The emerging market for these new products would benefit from the development of hedging strategies that would allow the seller of a mortality-linked instrument to reduce his/her mortality exposure by appropriate trading in assets for which liquid markets already exist. The established connections between mortality and bond yields suggest that this might be achievable to some extent by appropriate trading in fixedincome markets. This is a nontrivial problem and it will be studied in a separate article.

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European Accounting Review Vol. 21, No. 3, 533– 564, September 2012

The Adoption and Design of Enterprise Risk Management Practices: An Empirical Study LEEN PAAPE and ROLAND F. SPEKLE´ Nyenrode Business University, Breukelen, the Netherlands

(Received: November 2010; accepted: January 2012) ABSTRACT We examine (1) the extent of enterprise risk management (ERM) implementation and the factors that are associated with cross-sectional differences in the level of ERM adoption, and (2) specific risk management design choices and their effect on perceived risk management effectiveness. Broadly consistent with previous work in this area, we find that the extent of ERM implementation is influenced by the regulatory environment, internal factors, ownership structure, and firm and industryrelated characteristics. In addition, we find that perceived risk management effectiveness is associated with the frequency of risk assessment and reporting, and with the use of quantitative risk assessment techniques. However, our results raise some concerns as to the COSO (Committee of Sponsoring Organizations) framework. Particularly, we find no evidence that application of the COSO framework improves risk management effectiveness. Neither do we find support for the mechanistic view on risk management that is implied by COSO’s recommendations on risk appetite and tolerance.

1. Introduction Over the last decade, there has been a growing interest in risk management. Stakeholders’ expectations regarding risk management have been rising rapidly, especially since the recent financial crisis. In that crisis, weaknesses in risk management practices became painfully visible, and companies are currently under significant pressure to strengthen their risk management systems and to take appropriate actions to improve stakeholder value protection. This pressure is

Correspondence Address: Roland F. Spekle´, Nyenrode Business University, PO Box 130, 3620 AC, Breukelen, the Netherlands. Tel.:+31346291225; Email: [email protected] Paper accepted by Salvador Carmona. 0963-8180 Print/1468-4497 Online/12/030533–32 # 2012 European Accounting Association http://dx.doi.org/10.1080/09638180.2012.661937 Published by Routledge Journals, Taylor & Francis Ltd on behalf of the EAA.

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intensified by regulators and standard setters promulgating new risk management rules and requirements. In addition, credit rating agencies like Standard & Poor’s have begun to evaluate firms’ risk management systems as part of their credit rating analysis. In the wake of these increasing expectations, the idea of enterprise risk management (ERM) has gained substantial momentum as a potentially effective response to risk management challenges. ERM differs from traditional conceptions of risk management in its enterprise-spanning aspirations and in the adoption of a holistic approach in which strategic, operational, reporting and compliance risks are addressed simultaneously rather than separately. Such an integrated approach should help companies to deal with risks and opportunities more effectively, enhancing the entity’s capacity to create and preserve value for its stakeholders (COSO, 2004a). The emergence and evolution of ERM in practice has begun to attract research attention, and an academic risk management literature is starting to develop. One set of papers in this body of work examines the factors that influence ERM adoption (e.g. Beasley et al., 2005; Kleffner et al., 2003; Liebenberg and Hoyt, 2003). Other studies address the effects of ERM adoption on performance (Beasley et al., 2008; Gordon et al., 2009). Yet another cluster of papers explores the details of risk management practices in specific organisational settings (e.g. Mikes, 2009; Wahlstro¨m, 2009; Woods, 2009). With this paper, we seek to add to this incipient literature. Our study aims to make three main contributions. One, based on survey data from 825 organisations headquartered in the Netherlands, we provide new evidence on the factors that are associated with the extent of ERM implementation. This part of our study connects to previous work of Kleffner et al. (2003), Liebenberg and Hoyt (2003) and especially Beasley et al. (2005), and significantly expands the empirical foundations of this research stream. Prior contributions were based largely on US (Beasley et al., 2005; Liebenberg and Hoyt, 2003) and Canadian (Kleffner et al., 2003) data.1 Our observations add a European perspective, allowing some new insights into the generalisability of the earlier findings across different institutional contexts. This is important, because at least some evidence suggests that ERM adoption might be conditional on the legal and regulatory environment, and perhaps also on cultural factors (Beasley et al., 2005; Liebenberg and Hoyt, 2003). In addition, our sample includes small and medium-sized enterprises, as well as public sector and not-for-profit organisations. Both of these groups were absent in earlier studies. Two, we provide a relatively detailed description of current ERM practices, shedding light on the specific design choices organisations make when configuring and implementing their ERM systems. The existing literature has tended to study ERM at a high level of aggregation. For instance, Liebenberg and Hoyt (2003) and Beasley et al. (2008) rely on data on Chief Risk Officer appointments as their sole indicator for ERM adoption. Beasley et al. (2005) use an ordinal scale ranging from ‘no plans exist to implement ERM’ to ‘complete ERM is in

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place’ to capture the extent of ERM implementation.2 And Gordon et al. (2009) measure ERM indirectly through the extent to which the organisation has been successful in realising a number of generic strategic, operational, reporting and compliance objectives. These studies, however, do not address the particulars of ERM practices, nor the differences in ERM design between firms. Yet, there is considerable variety in ERM across organisations. For instance, Arena et al. (2010) provide case study evidence from three Italian firms that demonstrates that ERM can be very different things to different organisations. In her study of two large banks, Mikes (2009) concludes that systematic variations in ERM practices exist, even within a single industry setting (see also Mikes, 2008). In addition, Woods (2009) reports significant variety at the operational level of the ERM system within a single large public sector organisation. As we still know fairly little about the specific ERM design choices organisations make, the attempt to document these choices in a large sample study is instructive in its own right. Three, we explore the relationship between ERM design choices and perceived risk management effectiveness. A whole industry has emerged to assist firms in improving their risk management systems and practices. In addition, several semi-regulatory bodies have published frameworks to guide these efforts. The most prominent of these is the COSO ERM framework (COSO, 2004a; cf. Power, 2009). These normative frameworks implicitly or explicitly suggest that their standards and recommendations represent so-called ‘best practices’. This suggestion, however, does not have a clear theoretical or empirical foundation. Even though some research papers present evidence to indicate that ERM improves firm performance (Beasley et al., 2008; Gordon et al., 2009), we found no studies that examine the effects of the specific recommendations of COSO-type frameworks on risk management effectiveness. Therefore, the question as to whether these frameworks actually help to advance sound risk management is still largely unanswered, and it might well be the case that the effort to standardise and codify risk management practices is premature (Kaplan, 2011). Given the influence of these frameworks, empirical work addressing the effects of the standards and recommendations on risk management effectiveness seems long overdue. We take a step in that direction by analysing the influence of various ERM design alternatives on the perceived quality of the risk management system. Our results regarding the extent of ERM implementation appear to replicate most of the findings of earlier work in this line of research. Broadly consistent with previous studies, we find that the extent of ERM implementation is influenced by the regulatory environment, internal factors, ownership structure, and firm and industry-related characteristics. These findings indicate that the factors that are associated with ERM adoption are similar across different national contexts. As to risk management effectiveness, we find that the frequency of both risk assessment and risk reporting contribute to the perceived quality of the ERM system. In addition, the use of quantitative methods to

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assess risk appears to improve ERM effectiveness. However, our results also raise some concern as to the COSO ERM framework. Particularly, we find no evidence that application of the COSO framework improves risk management effectiveness. Neither do we find support for the mechanistic view on risk management that is implicit in COSO’s recommendations on risk appetite and tolerance. These findings might be taken to suggest that this framework does not fully live up to its purpose, which is to help organisations establish sound risk management. The remainder of this paper is structured as follows. In Section 2, we provide information on the data-set. Section 3 reports on the examination of the extent of ERM implementation. Section 4 explores ERM design choices and their impact on risk management effectiveness. Finally, Section 5 discusses the findings and limitations of this study. 2.

Data Collection and Sample

The survey data we rely on in this paper have been made available to us by a research team involving representatives from PwC, Royal NIVRA (the Dutch Institute of Chartered Public Accountants), the University of Groningen and Nyenrode Business University.3 The composition of this team was quite diverse, bringing together individuals from different professional backgrounds, including two academics specialising in risk management (one of which also has extensive practical experience in internal auditing), a researcher employed by the professional association of auditors and an experienced risk management consultant. To capitalise on these various backgrounds, the design of the questionnaire was set up as a joint team effort. A pre-test among four risk managers and internal auditors confirmed the relevance, composition and clarity of wording of the survey questions. The aim of the research team was to provide a broad, factual picture of current ERM practices and issues in the Netherlands for an audience of practitioners in the field. Although the research team was familiar with the academic literature in the field and used this literature in the construction of the survey, it had no ex ante intention to connect to this literature, and it paid no explicit attention to scholarly scale construction, validation and measurement considerations. As a consequence, some of the variables on which we rely in this paper are somewhat naive. In addition, because the questionnaire has not been designed with the specific purposes of the current study in mind, the match between the available information and the concepts on which we rely in our analyses is not always perfect. Nevertheless, we believe that the data remain valuable because they provide an informative glimpse into ERM adoption, design choices and effectiveness. Using information from company.info, the research team identified organisations located in the Netherlands with annual revenues of more than EUR 10 million and more than 30 employees. A total of 9579 organisations appeared

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Table 1. The data-set

Revenue (× E million) Number of employees Industry Wholesale and retail Transportation Manufacturing Financial services Business services Telecom and IT Energy and utilities Public sector and not-for-profit Unknown

Mean

Std. dev.

Min

Median

Max

791 1177

5213 4.809

11 31

65 260

85,000 80,000

Number 104 40 149 45 88 33 25 334 7

% 12.6 4.8 18.1 5.5 10.7 4.0 3.0 40.5 0.8

to fit these cumulative criteria. The survey was mailed to the board of these organisations in May 2009, asking them to respond within four weeks. To increase the response rate, the team ran a series of ads in several professional journals, announcing the project and emphasising the importance of respondent cooperation. Two hundred and forty questionnaires were undeliverable. Of the remaining 9339 surveys, 928 were returned, resulting in an overall response rate of 9.9%. Upon closer inspection, 103 responses were found not to match the initial selection criteria after all, leaving a final sample of 825 observations. Respondents were board members or CFOs (56%), controllers (20%) or (risk) managers (24%).4 We have no data to examine representativeness in a formal, quantitative way. The sample, however, comprises a relatively large number of (semi-)public sector and not-for-profit organisations (40.5%). We shall control for the potential influence of this dominance in the various analyses. The private sector part of the sample is varied in terms of organisational size and industry, and contains no obvious biases. Table 1 gives a general description of the data-set.

3. Antecedents of ERM Implementation The idea that ERM is a key component of effective governance has gained widespread acceptance. Nevertheless, organisations vary in the extent to which they have adopted it. Some organisations have invested in sophisticated ERM systems, whereas others rely on rather ad hoc responses to risks as they become manifest. In this section of the paper, we explore a number of factors that may help to explain the level of development of ERM practices across organisations. Building on previous studies, we identify five broad groups of factors that we expect to be associated with the extent of ERM implementation: (1)

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regulatory influences; (2) internal influences; (3) ownership; (4) auditor influence; and (5) firm and industry-related characteristics. 3.1. Factors Affecting Implementation: Expectations Regulatory influences In many countries, regulators are pressing firms to improve risk management and risk reporting (Collier et al., 2006; Kleffner et al., 2003). Examples of such regulatory pressure include the NYSE Corporate Governance Rules and the Sarbanes – Oxley Act in the USA, the Combined Code on Corporate Governance in the UK and the Dutch Corporate Governance Code, also known as the Tabaksblat Code. These codes apply to publicly listed firms, and require firms to maintain a sound risk management system. Corporate governance regulation, however, has not been confined to publicly traded companies. Governance rules are also common in parts of the (semi-) public and not-for-profit sector. In addition, some trade associations demand compliance with a governance code as a membership requirement. Like the formal rules that apply to listed companies, these codes typically call for some form of systematic risk management, and we expect that many ERM initiatives arose because of this regulatory pressure (cf. Collier et al., 2006). Accordingly: H1:

Corporate governance regulation is positively associated with the degree of ERM implementation.

Regulatory pressure differs in intensity. Some governance codes are mandatory (e.g. the stock exchange rules and the Dutch public sector codes), whereas others are being presented as optional ‘best practices’. Moreover, to the extent that the codes are in fact mandatory, the intensity of enforcement varies. In the Netherlands, the enforcement of corporate governance regulation seems to be rather weak, except for firms listed on the Amsterdam Stock Exchange. In the discussion leading to H1, we argued that regulatory pressure is a driver of ERM implementation. However, when corporate governance codes are non-binding or when enforcement is weak, this pressure might be easy to ignore. Listed firms are subjected to a special class of regulation, and for these firms, non-compliance is a less viable option. Therefore: H2:

Listed firms have more fully developed ERM systems than nonlisted organisations.

Internal influences The decision to implement ERM is rather consequential, affecting the entire organisation and implying major organisational change. Such far-reaching decisions require strong support from senior management. To emphasise

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their commitment to ERM, many organisations choose to locate ultimate responsibility for risk management explicitly at the senior executive level by appointing a Chief Risk Officer (CRO). Presumably, senior executive leadership is a powerful catalyst for organisational change, and could significantly speed up the process of ERM implementation (Beasley et al., 2005, 2008): H3:

The presence of a CRO is positively associated with the degree of ERM implementation.

Another internal factor to affect ERM development is the presence of an audit committee. Audit committees play an important role in the oversight of risk management practices. In this monitoring role, they can influence the board to ensure that ERM gets adequate management attention, and that sufficient resources are allocated to further ERM development: H4:

The presence of an audit committee is positively associated with the degree of ERM implementation.

Ownership Liebenberg and Hoyt (2003) argue that pressure from shareholders is an important driving force behind ERM adoption (cf. also Mikes, 2009). Proponents of ERM claim that shareholders benefit from integrated risk management because ERM enables companies to improve risk-adjusted decision-making and increase firm value. If that is true, shareholders are likely to be keen ERM supporters. The effectiveness of shareholder pressure, however, differs across firms. If ownership is dispersed, management might find it relatively easy to ignore shareholders’ preferences – at least temporarily. However, institutional investors are more likely to be heard. This additional influence of institutional investors arises from two related sources (Kane and Velury, 2004): (1) as institutional owners hold large shares in the firm, they control a substantial part of the voting rights that can be employed directly to influence management, and (2) the control of large institutional block holders over the supply of capital is such that they can affect the cost of capital of the firm, which ensures a greater receptiveness on the side of management to their preferences – perhaps even to the extent that they no longer need to voice their wishes explicitly, but can expect management to anticipate these preferences. Consistent with this reasoning, Liebenberg and Hoyt (2003) suggest that a higher degree of institutional ownership increases the effectiveness of shareholder pressure, which in turn is positively associated with the extent of ERM adoption: H5:

Institutional ownership is positively associated with the degree of ERM implementation.

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Arguably, insider owners have even more influence over management than institutional owners, especially if they hold a controlling share in the firm. This is the case in owner-managed firms in which ownership and control coincide. However, owner-managers have less incentive to press for ERM. Because agency problems between owners and management are absent in owner-managed firms, the value of ERM is lower in such firms, ceteris paribus. Further, owner-managers tend to rely less on formal control systems (Lovata and Costigan, 2002), which makes them unlikely sponsors of ERM. Therefore: H6:

Owner-managed firms have less developed ERM systems.

Auditor influence In the auditing literature, it is often proposed that larger auditing firms (i.e. the Big 4) provide higher audit quality (see DeAngelo, 1981 for a classic reference and Francis, 2004 for a recent overview). Such high-quality audit firms may be more persuasive in encouraging clients to improve their ERM systems and practices. In addition, it might be the case that organisations that select high-quality auditors are also more committed to risk management (Beasley et al., 2005), and perhaps to good governance in general. Both lines of reasoning seem to imply that firms that engage a Big 4 audit firm are likely to have more elaborate ERM systems: H7:

Engagement of a Big 4 audit firm is positively associated with the degree of ERM implementation.

Firm and industry-related characteristics For some firms, the value of ERM is larger than for others. Liebenberg and Hoyt (2003) hypothesise that ERM is especially important for firms that experience significant growth (cf. also Beasley et al., 2008; Gordon et al., 2009). Such firms face more uncertainties and require better risk management to control the risks that emerge, but also to include the risk profile of various growth opportunities in organisational decision-making (Collier et al., 2007; Liebenberg and Hoyt, 2003). Thus: H8:

Organisational growth is positively associated with the degree of ERM implementation.

The size of the organisation is also likely to affect the extent of ERM adoption (Beasley et al., 2005; Kleffner et al., 2003). Presumably, there are considerable economies of scale involved in operating an ERM system, and it may well be the case that only larger organisations can afford a fully functional ERM system. In addition, larger firms tend to be more formalised, which may be conducive to ERM adoption:

The Adoption and Design of Enterprise Risk Management Practices H9:

541

Organisational size is positively associated with the extent of ERM implementation.

Several studies in the ERM literature have proposed the existence of industry effects. It is commonly assumed that firms in the financial services industry are especially likely to embrace ERM (Beasley et al., 2005; Kleffner et al., 2003; Liebenberg and Hoyt, 2003). Since the release of Basel II, banks have strong incentives to adopt ERM as that may help to reduce capital requirements (Liebenberg and Hoyt, 2003; Mikes, 2009; Wahlstro¨m, 2009). In addition, ERM facilitates better disclosure of the firm’s risk exposure. This is especially important in the banking industry, in which it is relatively easy for firms to opportunistically change their risk profiles. Improved disclosure provides a means to make a credible commitment against this behaviour, thus lowering the cost of capital (Liebenberg and Hoyt, 2003). Another sector that seems more prone to ERM adoption is the energy industry. Kleffner et al. (2003) report that energy firms are relatively heavy ERM users, which they ascribe to the volatile markets in which these firms operate. Because ERM may reduce earnings volatility (Liebenberg and Hoyt, 2003), firms in such markets may value ERM more than firms that face stable market conditions. Accordingly: H10:

Firms in the financial services industry (H10a) and energy sector (H10b) have more fully developed ERM systems than firms in other sectors of the economy.

Unlike previous studies in this line of research, our sample includes public sector and not-for-profit organisations. The surge of risk management has not been confined to the private sector, but has affected the public sector too. Mirroring the private sector, the public sector now commonly sees risk management as an important dimension of good governance and as an aid in the achievement of organisational objectives (Woods, 2009). The various governance codes that have been implemented in parts of the public sector are an expression of this. There is nevertheless reason to believe that the diffusion of ERM has been slower in the public sector relative to the private sector. Operating in a complex and political environment, public sector organisations may find it particularly hard to operationalise their risk management agenda. In addition, available risk management tools and techniques tend to be highly analytical and data driven (Mikes, 2009), which may not accord very well with the dominant culture and management style in the public sector (cf. Bhimani, 2003; Mikes, 2009). These considerations suggest that public sector organisations can be expected to have less developed ERM systems: H11:

Relative to private sector firms, public sector organisations have less developed ERM systems.

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3.2. Summary of Expectations and Measurement of Variables Table 2 summarises the expectations and describes the measurement of the relevant variables. Almost all variables are factual and are based on straightforward survey questions that do not pose special measurement issues. The exception is the dependent Table 2.

Summary of expectations and measurement of variables

Dependent variable: Extent of ERM implementation (STAGE). Ordinal scale, adapted from Beasley et al. (2005): 1 ¼ risk management is mainly incident-driven; no plans exist to implement ERM 2 ¼ we actively control risk in specific areas (e.g. health & safety, financial risk); we are considering to implement a complete ERM 3 ¼ we identify, assess and control risk in specific areas; we are planning to implement a complete ERM 4 ¼ we identify, assess and control strategic, financial, operational and compliance risks; we are in the process of implementing a complete ERM 5 ¼ we identify, assess and control strategic, financial, operational and compliance risks; ERM is an integral part of the (strategic) planning & control cycle Expectation Regulatory influences: † Governance code (+) † Stock exchange listing (+) Internal influences: † Chief Risk Officer (+) † Audit Committee (+) Ownership: † Institutional ownership (+) † Owner-managed firm (2)

Auditor influence: † Big 4 audit firm (+) Firm/industry characteristics: † Growth opportunities (+) † Size (+) † Industry effects: o Financial services (+) o Energy (+) o Public sector (2)

Measurement GOVERNCODE and STOCKEX are dummy variables that take on a value of 1 if the organisation is listed (STOCKEX) or when governance rules apply (GOVERNCODE). CRO and AUDITCOM are dummy variables that take on a value of 1 if a Chief Risk Officer (CRO) or an audit committee (AUDITCOM) is present. INSTOWNER and OWNERMAN are dummy variables that take on a value of 1 if the majority of shares are owned by institutional investors (INSTOWNER), or if the firm is managed by an owner holding a controlling share (OWNERMAN). BIG4 is a dummy variable that takes on a value of 1 if the auditor is a Big 4 audit firm. GROWTH is an ordinal variable expressing the average yearly growth of revenues over the last three years (1 ¼ less than 10%, 2 ¼ between 10 and 25%, 3 ¼ more than 25%). lnREVENUE is indicative of size and is calculated as the natural log of revenue (or the size of the budget in case of public sector organisations). FINSERV, ENERGY and PUBSEC are industry dummies that take on a value of 1 if the organisation belongs to the financial services industry, the energy sector or the public sector.

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variable, that is, the extent of ERM implementation (STAGE). STAGE is measured using an ordinal scale, based on Beasley et al. (2005), but adapted to suit the research interests of the team that constructed the survey (see Section 2). Specifically, whereas the original scale of Beasley et al. was based on broad statements regarding (intentional) ERM implementation,5 the survey items on which we rely contain additional descriptive detail regarding manifest ERM practices. From the perspective of the current study, these adaptations are potentially problematic because the added description of actual practices in a particular answer category need not coincide with the ERM intentions described in that same category. For instance, it is possible that an organisation actively controls risk in specific areas (which should lead to a score of 2 or 3 on STAGE; see Table 2), yet has no intentions to implement full-scale ERM (which should lead to a score of 1). However, although this problem is quite serious in principle, it does not appear to be so in fact. As none of the respondents checked more than one of the relevant categories (which they could have done given the technical design of the questionnaire), respondents apparently had no major difficulty in scoring their organisations on the scale. This suggests that in the real world as perceived by the respondents, the answer categories are descriptively accurate after all. The questionnaire contains additional information to support the implementation metric on which we rely. Respondents were asked to provide information on the scope of their periodic risk identification and assessment efforts by indicating whether or not they include each of the categories strategic, financial, operational, reporting and compliance risks in the exercise. Because ERM is characterised by its broadness and comprehensive ambitions, the scope of risk assessment is a highly relevant indicator of the extent of ERM implementation,6 and we expect a significant correlation between the scope of risk assessment (measured by the number of risk categories included in periodic risk assessment) and STAGE. In support of our implementation proxy, the data corroborate this expectation (r ¼ 0.397; p ¼ 0.000). In addition, a factor analysis confirms that both variables load on a single factor. 3.3. Analysis and Results Table 3 provides an overview of the descriptive statistics. These statistics show that approximately 11% of the organisations in the sample report having a fully functional ERM system in place. Another 12.5% is currently in the process of implementing such a system. Fourteen per cent do not seem to have a systematic and proactive approach to risk management. Table 4 presents the correlation matrix. The bivariate correlations indicate that the extent of ERM implementation is associated with most of the independent variables discussed in Section 3.1, providing some initial evidence in support of the hypotheses. The matrix also indicates that the correlations between the independent variables are low, and prompt no multicollinearity concerns.

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L. Paape and R.F. Spekle´ Table 3.

Descriptive statistics extent of implementation

Ordinal and ratio variables Mean

Std. dev.

Min

Median

Max

2.68 1.34 4.43

1.190 0.557 1.447

1 1 2.35

2 1 4.17

5 3 11.35

4 102 (12.5%)

5 91 (11.1%)

STAGE GROWTH lnREVENUE

Distribution of ordinal variables over response categories 1 2 3 STAGE 114 (14.0%) 318 (38.9%) 192 (23.5%) GROWTH 571 Nominal variables Yes GOVERNCODE 492 STOCKEX 75 CRO 154 AUDITCOM 399 INSTOWNER 82 OWNERMAN 123 BIG4 625 FINSERV 45 ENERGY 25 PUBSEC 334

(70.6%)

203 (25.1%)

(¼ 1) (60.8%) (9.1%) (18.8%) (51.3%) (10.6%) (16.0%) (77.2%) (5.5%) (3.1%) (40.8%)

No (¼ 0) 317 (39.2%) 747 (90.9%) 666 (81.2%) 420 (48.7%) 688 (89.4%) 647 (84.0%) 185 (22.8%) 773 (94.5%) 793 (96.9%) 484 (59.2%)

35 (4.3%)

–

–

Because the dependent variable (extent of ERM implementation; STAGE) is measured on an ordinal scale, and because most of the independent variables are categorical, we test the hypotheses using ordinal logistic regression. Table 5, panel A reports the results of this analysis. The model appears to fit the data quite well (x2 ¼ 152.767, p ¼ 0.000; Cox and Snell pseudo R2 ¼ 0.193). Further, the test of parallel lines (not reported) indicates that the slope coefficients are the same across the various levels of the dependent variable and that, consequently, ordinal logistic estimation is appropriate. Due to missing values on some variables for various cases, we lose a number of observations in the regression, and the sample size drops from 825 to 714. The results offer support for several of our expectations. We find that publicly traded firms do in fact have more mature ERM systems (H2), whereas ownermanaged firms appear less inclined to invest in ERM development (H6). In addition, we find the presence of both a CRO (H3) and an audit committee (H4) to contribute to the degree of ERM implementation. Finally, we observe that larger organisations (H9) and firms in the financial sector (H10a) tend to have more sophisticated ERM systems. These results are generally consistent with the findings of previous studies. Contrary to hypothesis H5, we do not find an effect for institutional ownership. A potential explanation could be that in the Netherlands, institutional block holders are traditionally rather reluctant to interfere with management.

1 1: STAGE 1 2: GOVERNCODE 0.198∗∗∗ 3: STOCKEX 0.216∗∗∗ 4: CRO 0.245∗∗∗ 5: AUDITCOM 0.283∗∗∗ 6: INSTOWNER 0.143∗∗∗ 7: OWNERMAN 20.167∗∗∗ 8: BIG4 0.117∗∗∗ 9: GROWTH 0.010 10: lnREVENUE 0.261∗∗∗ 11: FINSERV 0.150∗∗∗ 12: ENERGY 0.064∗ 13: PUBSEC 20.045 ∗

2

3

4

5

6

7

8

9

10

11

12

1 0.194∗∗∗ 0.108∗∗∗ 0.183∗∗∗ 0.286∗∗∗ 20.088∗∗ 0.101∗∗∗ 0.104∗∗∗ 0.192∗∗∗ 20.019 20.006 20.226∗∗∗

1 0.105∗∗ ∗ 0.409∗∗ ∗ 20.005 20.291∗∗ ∗ 0.214∗∗ ∗ 20.063∗ 0.163∗∗ ∗ 0.030 20.032 0.289∗∗ ∗

1 0.208∗∗∗ 0.078∗∗ 20.051 0.016 0.028 0.021 0.077∗∗ 0.024 20.057

1 0.100∗∗∗ 20.304∗∗∗ 0.236∗∗∗ 20.106∗∗∗ 0.222∗∗∗ 0.115∗∗∗ 0.054 0.254∗∗∗

1 20.151∗∗∗ 0.094∗ 0.162∗∗∗ 0.115∗∗∗ 0.222∗∗∗ 0.038 20.239∗∗∗

1 20.252∗∗∗ 0.149∗∗∗ 20.175∗∗∗ 0.004 20.014 20.336∗∗∗

1 20.106∗ ∗∗ 0.230∗ ∗∗ 0.080∗ ∗ 20.006 0.189∗ ∗∗

1 20.042 0.024 0.074∗ 20.282∗∗∗

1 0.119∗∗∗ 0.023 20.032

1 20.043 20.200∗∗∗

1 20.147∗∗∗

p , 0.10; ∗∗ p , 0.05; ∗∗∗ p , 0.01 (2-tailed).

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Table 4. Spearman correlation matrix (extent of implementation)

545

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L. Paape and R.F. Spekle´ Table 5.

Ordinal logistic regression results

Estimatea Panel A: full sample results Dependent variable: STAGE Sample: 825; included observations: 714 GOVERNCODE 0.180 STOCKEX 0.674 CRO 0.882 AUDITCOM 0.577 INSTOWNER 0.142 OWNERMAN 20.419 BIG4 0.168 GROWTH (¼ 2) 0.017 GROWTH (¼ 3) 0.065 lnREVENUE 0.226 FINSERV 0.857 ENERGY 0.275 PUBSEC 20.202 x2 ¼ 152.767; p ¼ 0.000 Cox and Snell pseudo R2 ¼ 0.193

Std. error

Wald

p (2-tailed)

0.166 0.273 0.180 0.166 0.249 0.221 0.178 0.364 0.354 0.054 0.343 0.397 0.185

1.184 6.072 23.890 12.048 0.326 3.596 0.884 0.002 0.034 17.495 6.247 0.481 1.196

0.276 0.014 0.000 0.001 0.568 0.058 0.347 0.963 0.854 0.000 0.012 0.488 0.274

Panel B: sample excluding (semi-) public sector and not-for-profit organisations Dependent variable: STAGE Sample: 484; included observations: 424 GOVERNCODE 0.119 0.214 0.309 0.578 STOCKEX 0.787 0.289 7.394 0.007 CRO 0.876 0.229 14.614 0.000 AUDITCOM 0.593 0.237 6.242 0.012 INSTOWNER 0.142 0.262 0.293 0.588 OWNERMAN 20.409 0.226 3.287 0.070 BIG4 0.153 0.210 0.526 0.486 GROWTH (¼ 2) 0.188 0.389 0.232 0.630 GROWTH (¼ 3) 0.265 0.379 0.489 0.484 lnREVENUE 0.194 0.063 9.359 0.002 FINSERV 0.854 0.347 6.049 0.014 ENERGY 0.274 0.398 0.475 0.491

x2 ¼ 119.035; p ¼ 0.000 Cox and Snell pseudo R2 ¼ 0.245 a

To facilitate interpretation, the estimates we report for dummy variables express the effect when the variable of interest takes on a value of 1. For example, the positive estimate we report for the variable CRO in panel A (0.882, p ¼ 0.000) indicates that firms that have a CRO also have more fully developed ERM systems (i.e. higher scores on STAGE), ceteris paribus. This is different from the output generated by most statistical software packages (including SPSS) that typically return estimates expressing the effect when the variable of interest is zero.

However, Liebenberg and Hoyt (2003) did not find an effect in their US-based sample either. This could mean that investors do not value ERM adoption after all. However, it can also mean that if ERM contributes to shareholder value

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creation, companies will invest in ERM regardless of explicit shareholder pressure. Our data do not allow us to differentiate between these rival explanations. The analysis does not lend support for the supposed influence of governance codes (H1), suggesting that governance regulation and the associated pressure to invest in risk management do not affect ERM development. However, we do find that listed firms have more fully developed ERM systems (as per H2). In conjunction, these findings might be taken as evidence that strong code enforcement (which is typically absent, except for stock exchange regulation) is required to affect the risk management behaviour of organisations. However, these findings are also consistent with an alternative explanation that holds that the absence of significant differences between organisations that are subject to governance regulation and those to which no governance codes apply is caused by widespread voluntary adoption of corporate governance regulation by the latter group (cf. Deumes and Knechel, 2008), and that sustains that the effect of stock exchange listing has nothing to do with stricter enforcement, but merely indicates that listed firms are more sophisticated in general. The data do not allow a further examination of these alternative explanations. Prior studies provide mixed evidence on the effect of growth (H8). Gordon et al. (2009) report a positive effect of growth opportunities on the extent of ERM implementation. The results of Beasley et al. (2008) and Liebenberg and Hoyt (2003), however, do not support this finding. The results of our study are also negative in this respect. However, it should be emphasised that our sample includes only 35 firms reporting high growth. Additional analysis (details not reported here) shows that many of these high-growth firms are owner-managed (42%), and that almost all of them are small. Given these numbers, our null finding may not be very instructive. It is interesting to note that in our analysis, auditor quality (H7) has no effect on ERM development. This is at odds with Beasley et al. (2005), who did find a significant and positive auditor effect. An explanation of this difference can perhaps be found in the high quality of the Dutch audit profession. At least the Dutch professional association of certified accountants believes that the quality of the Dutch CPA is truly world class, even in smaller firms. If this is actually true, there is no reason to expect that auditor identity matters in this Dutch sample after all. Finally, the industry effects (H10 and H11) seem to be limited to the financial services sector. We observe no effects for the energy sector. However, the number of energy firms in our sample is very small (25, or only about 3% of the total sample size). Neither do we observe a public sector effect, suggesting that ERM implementation is not hindered by the inherently larger complexity of the political environment in which public sector organisations operate.

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3.4. Additional Analysis Despite the fact that we do not observe a public sector effect in the previous analysis, we cannot rule out the possibility that the results of this analysis are affected by the dominance of (semi-) public sector and not-for-profit organisations in our sample (as reported in Table 1, 40.5% of the observations come from such organisations). For this reason – and also to facilitate comparison with previous studies in this line of research (e.g. Beasley et al., 2005; Kleffner et al., 2003) – we rerun the analysis with a sample that excludes these organisations. The results of this additional analysis are in Table 5, panel B. The findings are qualitatively similar to the full sample results, and reinforce our original inferences.

4.

ERM Design and Perceived Effectiveness

In configuring their ERM systems, organisations need to face numerous design choices. Over the years, a large industry has emerged to assist firms in making these choices. In addition, several semi-regulatory bodies have published frameworks to guide organisations in their ERM design and implementation efforts. The best-known example is the COSO ERM framework (COSO, 2004a). This framework, however, only provides very broad guidance, suggesting key principles and concepts but leaving the details to the adopting organisations themselves. That is to say, COSO does actually attend to the everyday details of risk management practice, but only in an annex to the framework (COSO, 2004b). This annex is explicitly not part of the framework itself, but intends to provide practical illustrations that might be useful to those seeking to apply ERM techniques. COSO emphasises that these illustrations should not be interpreted as preferred methods or best practices. COSO’s cautious and unassuming position as to the practical side of ERM is quite sensible, given the paucity of evidence-based knowledge of effective ERM system design. Comprehensive ERM theories do not exist, and as far as we know, there are no empirical studies that systematically document specific ERM practices and their contribution to ERM effectiveness.7 Therefore, we must assume that the application techniques described in the annex are based on anecdotal evidence at best, and COSO is right not to present these illustrations as actual prescriptions. However, as a consequence, ERM-adopting organisations face very open-ended design problems, with little concrete guidance at the operational and instrumental level. The data-set that has been made available to us includes information on specific ERM design choices, particularly in the areas of risk identification and assessment, and risk reporting and monitoring. We also have information on risk tolerance definition. These data allow us to describe current practices, tools and techniques in these areas, providing a broad overview of their incidence and prevalence. In addition, the data-set contains information on perceived risk

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management effectiveness, providing the opportunity empirically to examine possible relationships between ERM practices and effectiveness. This analysis will be distinctly exploratory in nature. We have no clear theory to build on. Neither is there any prior empirical research to inform a systematic development of hypotheses. Therefore, we shall structure the analysis around a number of research questions, and instead of trying to test theory, we shall focus the analysis on finding answers to these questions. We organise the research questions in three broad themes, corresponding with the areas identified above, that is, risk tolerance, risk identification and assessment, and risk reporting and monitoring. However, we begin with a preliminary question: does application of the COSO framework help organisations to implement an effective ERM system? 4.1. ERM Design Choices The 2004 COSO report is generally viewed as the most authoritative ERM framework. Given this reputation, one would expect to observe widespread application of the framework in practice. Furthermore, if this framework deserves its reputation, one would expect that its application improves risk management effectiveness. Whether this is empirically true, however, is still an open question. Hence: Q1:

Does application of the COSO ERM framework contribute to risk management effectiveness?

Risk tolerance The entity’s risk appetite is a key concept in the COSO ERM framework. Risk appetite refers to ‘the amount of risk, on a broad level, an entity is willing to accept in pursuit of value’ (COSO, 2004a, p. 19). Risk appetite, thus, expresses the organisation’s risk attitude at the level of the organisation as a whole. Risk appetite is the starting point of COSO-type ERM, and according to COSO, organisations need to consider and define their risk appetite, essentially as a precondition for successful risk management. Risk appetite may be expressed in qualitative or quantitative terms, and COSO declares to be indifferent between these two options. However, the COSO framework also proposes that in addition to the expression of the entity’s high-level risk attitude, organisations need to define their risk attitudes at a lower level of aggregation, that is, at the level of specific objectives. At this lower level, COSO refers to the notion of risk tolerance, which is ‘the acceptable level of variation relative to achievement of a specific objective’ (COSO, 2004a, p. 20). These risk tolerances are a further specification of the entity’s high-level risk appetite, and they should help the organisation to remain within the boundaries of its stated risk appetite. At this lower level, COSO conveys a clear preference for quantification: ‘risk tolerances can be measured, and often are best measured in the same units as the related objectives’ (COSO, 2004a, p. 40).

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In these recommendations on risk appetite and tolerance, COSO espouses a highly mechanistic view on risk management. It is, however, uncertain whether such a view is realistic and practicable. For instance, Collier et al. (2007) find that subjective, heuristic methods of risk management are much more common than the systems-based approaches advocated in much professional training and in the professional literature. Power (2009) argues that the idea of organisation-wide risk appetite and risk tolerance assumes that organisations are unitary and intentional actors, which he regards as reductive, simplistic and potentially misleading. In addition, COSO works from a very traditional perspective on human decision-making in which agents are fully rational and risk attitudes are explicable and stable. Such a perspective is hard to maintain in the face of years of behavioural studies documenting systematic biases and situational and path dependencies in risky choice problems (e.g. Thaler and Johnson, 1990; Tversky and Kahneman, 1992; see also Power, 2009).8 These divergent positions feed into the second research question: Q2:

Does explication and quantification of risk tolerance improve risk management effectiveness?

Risk identification and assessment Organisations need to address the question as to the frequency of risk identification and analysis. How often should the organisation go over the risks to ascertain the availability of sufficient up-to-date information to act upon? Risk exposure is not static, and it is plausible to assume that the frequency of risk assessment should keep pace with changes in the environment. Additionally, some minimum level of frequency may be required to ensure that risk management becomes ingrained sufficiently deeply in the functioning of the organisation, and to prevent it from becoming a merely ceremonial compliance exercise (cf. Arena et al., 2010). The COSO framework is silent on these issues, but they seem important nonetheless: Q3:

Is the frequency of risk assessment associated with risk management effectiveness?

Another choice variable in the area of risk identification and assessment is the number of management levels to include in the risk appraisal exercise. Is it sufficient to localise risk assessment at the senior management level? Alternatively, is it better to involve middle management as well? And if so, how far down does one need to go? Although COSO does not provide a clear answer to these questions, it does indicate that even though the CEO has ultimate responsibility for ERM, ERM is ‘the responsibility of everyone in an entity and therefore should be an explicit or implicit part of everyone’s job description’ (COSO, 2004a, p. 88). This seems altogether reasonable, not just from a shared responsibility

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perspective, but also from an information asymmetry point of view: if middle managers enjoy an information advantage as to the specifics of their business units or departments, it makes sense to engage them in risk identification and assessment: Q4:

Does engagement of lower levels of management in risk assessment contribute to risk management effectiveness?

Formally, COSO requires organisations to apply a combination of qualitative and quantitative risk assessment techniques. However, a closer reading of the discussion in the COSO report (2004a, p. 52) strongly suggests that this requirement is merely a diplomatic way to express COSO’s preference for quantification, and that the provision actually means that organisations should not rely on qualitative methods alone, but should apply quantitative techniques if at all possible. Accordingly: Q5:

Is the use of quantitative risk assessment techniques positively associated with risk management effectiveness?

Risk reporting and monitoring The process of risk management – from initial risk identification via risk response selection to monitoring and evaluation – requires relevant, timely and reliable information, and organisations that implement ERM need to invest in information systems to support the risk management function. The COSO ERM framework acknowledges this need, but its guidance as to the actual setup of these information systems is highly generic. COSO does, however, argue that monitoring should ideally proceed on an ongoing basis, as continuous monitoring is more effective than separate evaluations (cf. COSO, 2004a, pp. 75 –76): separate evaluations take place after the fact and, consequently, are less able to assure a timely response to problems as and when they occur. This suggests that high-frequency risk reporting is valuable, and may enhance the quality of risk management: Q6:

Does the frequency of risk reporting positively affect risk management effectiveness?

In discussing the contents of risk reporting, COSO emphasises the need to report all indentified ERM deficiencies (COSO, 2004a, p. 80). Internal risk reporting, however, will typically be broader. The process of risk management can meaningfully be conceptualised in terms of Demski’s decision-performance control framework. This framework emphasises both the need for feedforward information to calibrate and feed the decision model to arrive at an ‘optimal’ decision (e.g. the organisation’s risk response), as well as feedback information to monitor

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the implementation of the risk response policy and to adapt the model and its implementation to environmental disturbances (cf. Demksi, 1969). To support monitoring and model calibration, internal risk reporting should include retrospective diagnostic data on current risk profiles and ongoing risk management processes. For feedforward purposes, organisations might also demand prospective information as part of their regular risk reporting practices, for example, information on important internal or external changes that may affect their risk exposure and that should be included in future decision-making. It seems plausible to assume that risk management effectiveness is affected by the richness of both this retrospective and prospective information: Q7:

Does the richness of retrospective (Q7a) and prospective (Q7b) risk reporting enhance risk management effectiveness?

Control variables Our sample contains data from organisations in varying stages of ERM implementation. Some have fully fledged ERM systems in place, whereas others have adopted ERM, but are still in the process of implementation. Stage of implementation is likely to have an effect on perceived risk management effectiveness. In addition, it may be the case that the extent of implementation is correlated with various specific design choices. For these reasons, we control for differences in ERM implementation in the analysis. We also control for size and potential industry effects. Consistent with previous research, we found that firms in the financial services industry tend to have more fully developed ERM systems in place (see Section 3). However, because the value of these systems is typically higher in the financial services industry than in other sectors, it might well be that the aspiration level as to the quality of the risk management system is higher here as well. This would imply that perceived effectiveness is lower, ceteris paribus. We also argued in Section 3 that the extent of ERM implementation is likely to be lower in the public sector, because the standard ERM approach does not seem to fit very well with the political environment in which public sector organisations operate, nor with the dominant culture and management style of these organisations. The data, however, did not support these contentions, and we observed no differences in ERM implementation between the public and the private sector. However, it is still possible that there is a public sector effect in the analysis of risk management effectiveness, and we explore this possibility by including a public sector dummy in the model. 4.2. Summary of Research Questions and Measurement of Variables In Table 6, we summarise the research questions and describe the operationalisation of the variables. Most variables are rather factual in nature, and are based on relatively uncomplicated survey questions that do not require much interpretation

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Table 6. Research questions and measurement of variables Dependent variable: Effectiveness (EFFECTIVENESS) is measured by asking respondents to score the quality of their risk management on a 10-point scale (1 ¼ deeply insufficient, 6 ¼ adequate, 10 ¼ excellent) Research question Application of COSO

Measurement COSO is a dummy variable that takes on a value of 1 if the organisation reports application of COSO.

Risk tolerance: † Explication/quantification of risk tolerance (TOLERANCE)

TOLERANCE is measured using an ordinal scale: 1 ¼ no explication of risk tolerance 2 ¼ risk tolerance is explicated in qualitative terms 3 ¼ risk tolerance is quantified

Risk identification and assessment: † Frequency of risk assessment (ASSESSFREQ) † Engagement of lower management levels (LEVEL) † Quantitative risk assessment (QUANTMETHODS)

ASSESSFREQ expresses the frequency of the entity-wide risk identification/ assessment exercise (1 ¼ never, 2 ¼ yearly, 3 ¼ quarterly, 4 ¼ monthly, 5 ¼ weekly). LEVEL counts the number of management levels involved in risk identification/ assessment. A score of 1 means that only the board is involved, 2 means that the exercise includes the board and the management level just below the board, etc. QUANTMETHODS is a dummy that takes on a value of 1 if the organisations use one or more of the following four techniques: scenario analysis, sensitivity analysis, simulation and stress testing.

Risk reporting and monitoring: † Reporting frequency (REPORTFREQ) † Richness of reporting: retrospective (RETROSPECT) and prospective (PROSPECT) information

REPORTFREQ indicates how often the organisation reports on risk to internal constituencies (1 ¼ never, 2 ¼ ad hoc, 3 ¼ yearly, 4 ¼ quarterly, 5 ¼ monthly, 6 ¼ weekly). Both RETROSPECT and PROSPECT count the number of items from a 4-item list that the organisation includes in its standard risk reporting format. The list for RETROSPECT includes general information on risks, the status of risk control activities, critical risk indicators and incidents. The PROSPECT list comprises developments in risk profile, significant internal changes, significant external changes and risk control improvements.

Control variables:

See Table 2.

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or judgement on the side of the respondent. This is, however, different for the dependent variable, that is, risk management effectiveness. The scores on this variable are based on the following survey question: ‘how would you rate your organisation’s risk management system on a scale of 1 to 10?’ This question is broad and open, and appears to be designed to capture respondents’ subjective assessment of the contribution of the risk management system to the attainment of the organisation’s (implicit or explicit) risk management objectives. The question does not specify what is meant by a risk management system,9 nor the dimensions that should be included in the quality assessment. For our purposes, this is unfortunate, because it forces us to measure an inherently complex and multifaceted construct with a single survey item. However, the survey contains additional information to support this metric. In another part of the questionnaire, respondents were asked to indicate whether they believe that their risk management system has helped them to cope with the effects of the financial crisis. The scores on this question (measured on a 5-point Likert scale) seem a somewhat narrow but relevant additional indicator of the quality of the risk management system. This alternative proxy correlates significantly with the risk management effectiveness measure on which we rely (r ¼ 0.197; p ¼ 0.000), suggesting that our measure is at least reasonable. 4.3. Sample Selection and Descriptive Statistics on Current ERM Practices Consistent with the aims of this study, we base the analysis of ERM practices and effectiveness on the questionnaire responses from the organisations that have actually adopted ERM. We define ERM adopters as organisations that report to be in stage 4 or 5 on the scale we use to measure the extent of ERM implementation (STAGE; see Section 3.2 and Table 2). These organisations have implemented broad and inclusive risk management systems that encompass strategic, financial, operational and compliance objectives, and have integrated ERM in their (strategic) planning and control cycle – or are in the process of doing so. The other organisations in the sample (i.e. those that are in stages 1 – 3) apply more traditional, silo-based approaches to risk management, and although they may consider ERM as an interesting alternative, they have not (yet) adopted ERM. The number of ERM adopters in the sample is 193. This number includes 55 (semi-) public sector organisations (28.5%) and 25 firms from the financial services sector (13.0%). All the following analyses are based on this subsample of ERM adopters. Table 7 presents descriptive statistics on perceived risk management effectiveness and the various design choices. The data indicate that the average organisation believes that the effectiveness of its risk management system is quite good (7.33 on a 10-point scale). Only about 4% of the respondents report their system to be less than satisfactory (score ≤ 5; details not tabulated), whereas approximately 16% consider their system to be outstanding (score ≥ 9).10 Despite its acclaimed authority, application of the COSO ERM framework is not

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Table 7. Descriptive statistics design and effectiveness variables Ordinal and ratio variables

EFFECTIVENESS TOLERANCE ASSESSFREQ LEVEL REPORTFREQ RETROSPECT PROSPECT STAGE lnREVENUE

Mean

Std. dev.

Scale

Min

Median

Max

7.33 1.84 2.41 2.67 4.15 2.20 1.79 4.47 5.09

1.083 0.906 0.801 1.154 1.035 1.044 1.282 0.500 1.964

1–10 1–3 1–5 1–5 1–6 0–4 0–4 4–5 –

3 1 1 1 1 0 0 4 2.40

7 1.5 2 3 4 2 2 4 4.55

10 3 5 5 6 4 4 5 11.35

Yes (¼ 1) 83 (43%) 114 (60.6%)

No (¼ 0) 110 (57%) 74 (39.4%)

Nominal variables COSO QUANTMETHODS

widespread. Only 43% of the organisations in the sample of ERM adopters apply the COSO framework.11 However, this figure might underestimate COSO’s true influence. Casual observation suggests that many organisations have hired consulting firms to help design and implement the ERM system, and it seems plausible that the solution packages offered by these consultants are in fact heavily influenced by COSO. If this is actually the case, the (indirect) impact of COSO on current ERM practices is much larger than the reported application rate suggests. Another interesting fact is that the mean score on TOLERANCE is 1.84, which seems to imply that a quantification of risk tolerance is not very common. This is indeed the case. A further analysis of this variable (not tabulated) reveals that only 31% of the respondents express risk tolerance in quantitative terms. Almost half of the organisations convey that they do not explicate risk tolerance; not even in qualitative terms. 4.4. The Relationship between ERM Design and Effectiveness: Analysis and Results Table 8 displays the correlation matrix. This matrix provides initial evidence that perceived risk management effectiveness is in fact dependent on most of the design choices identified in Section 4.1, at least in a bivariate analysis. This table also shows that the design choices are to some extent interrelated, as many correlations between the independent variables are significant. The correlations are, however, low enough not to signal multicollinearity issues.12 We explore the impact of the various design choices on risk management effectiveness by estimating a multivariate OLS model that includes all design variables on which we have information, and a number of control variables to

556

1 1: EFFECTIVENESS 1 2: COSO 0.032 3: TOLERANCE 0.134∗ 4: ASSESSFREQ 0.221∗∗∗ 5: LEVEL 0.143∗ 6: QUANTMETHODS 0.240∗∗∗ 7: REPORTFREQ 0.189∗∗∗ 8: RETROSPECT 0.011 9: PROSPECT 0.051 10: STAGE 0.318∗∗∗ 11: lnREVENUE 0.055 12: FINSERV 0.017 13: PUBSEC 20.173∗∗ ∗

2

3

Spearman correlation matrix (design and effectiveness) 4

5

6

7

1 0.019 1 20.131∗ 0.130∗ 1 0.193∗∗∗ 0.019 20.041 1 20.016 0.290∗∗∗ 0.212∗∗∗ 20.087 1 ∗∗∗ 20.119 0.103 0.340 0.047 0.050 1 ∗ 0.062 0.026 0.046 0.139 0.028 0.169∗∗ ∗ ∗ ∗∗ 0.124 0.048 0.147 20.061 20.092 20.129 20.045 0.165∗∗ 0.177∗∗ 0.206∗∗∗ 0.066 0.001 0.346∗∗∗ 0.098 20.055 0.170∗∗ 20.009 20.041 0.098 0.075 0.042 0.056 0.119 0.244∗∗∗ 20.091 20.078 20.039 20.035 20.197∗∗∗ 20.267∗∗∗

p , 0.10; ∗∗ p , 0.05; ∗∗∗ p , 0.01 (2-tailed).

8

9

10

11

12

1 0.255∗∗∗ 1 0.130∗ 0.065 1 0.216∗∗∗ 20.033 20.008 1 0.324∗∗∗ 0.147∗∗ 20.055 0.044 1 0.023 0.049 0.095 20.098 20.247∗∗∗

L. Paape and R.F. Spekle´

Table 8.

The Adoption and Design of Enterprise Risk Management Practices Table 9.

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OLS regression results

Dependent variable: EFFECTIVENESS Sample: 193; included observations: 156

Intercept COSO TOLERANCE ASSESSFREQ LEVEL QUANTMETHODS REPORTFREQ RETROSPECT PROSPECT STAGE lnREVENUE FINSERV PUBSEC

Coefficient

Std. error

t

p (2-tailed)

4.191 0.129 20.010 0.196 20.009 0.321 0.133 0.021 0.018 0.431 0.007 20.248 20.284

0.690 0.142 0.079 0.090 0.061 0.150 0.075 0.073 0.055 0.141 0.035 0.208 0.158

6.072 0.909 20.121 2.185 20.138 2.137 1.778 0.292 0.327 3.064 0.201 21.195 21.796

0.000 0.365 0.904 0.031 0.890 0.034 0.078 0.771 0.744 0.003 0.841 0.234 0.075

F ¼ 3.534; p ¼ 0.000 R2 ¼ 0.229; adjusted R2 ¼ 0.164

capture possible size and industry effects. Due to missing values, we lose 37 observations in this analysis, and we run the model using data from 156 organisations. Table 9 reports the results. The model explains 22.9% of the variance in the dependent variable (adjusted R2 ¼ 0.164). The regression results show that perceived risk management effectiveness is affected by the extent of implementation: unsurprisingly, organisations that have adopted ERM but are still in the process of implementation are less positive about the effectiveness of their systems than those that have already finalised implementation. Furthermore, the results suggest that application of the COSO ERM framework as such does not help to improve risk management quality (Q1), and that quantification of risk tolerances does not contribute to perceived risk management effectiveness (Q2). These results raise some rather profound questions as to the generally acclaimed authority of COSO as the leading framework in ERM. We defer the discussion of these questions to Section 5. Several research questions in the areas of risk identification/assessment and risk reporting/monitoring can be answered affirmatively. Thus, we find that the frequency of risk assessment (Q3), the use of quantitative risk assessment techniques (Q5) and the frequency of risk reporting (Q6) contribute to perceived risk management effectiveness. However, we do not find a significant effect for the engagement of lower levels of management (Q4): apparently, devolvement of risk assessment does not generally improve risk management effectiveness. Neither do we find that the richness of retrospective and prospective risk reporting (Q7a and Q7b) helps to advance the quality of risk management.

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Organisational size does not seem to matter. There are, however, industry effects. Although firms in the financial services industry do not appear to differ from the average firm in the sample, we do observe a significant negative effect for organisations in the (semi-) public sector. This finding will also be discussed more fully in Section 5. 4.5. Additional Analysis The sample of ERM adopters contains a relatively large number of (semi-) public sector and not-for-profit organisations (55 organisations, or 28.5% of the sample). Although we included a dummy variable in the analysis to control for sector effects, this does not fully exclude the possibility that the results reported in Table 9 are affected by the relatively large proportion of (semi-) public sector and not-for-profit organisations in the sample. Therefore, we run an additional analysis with a sample that only includes observations from private sector firms. This (untabulated) analysis returns similar results, confirming the original findings. The questionnaire asked respondents to score the quality of their risk management system on a 10-point scale. This being a rather fine-grained scale, respondents might have had some difficulties in scoring their systems. Although we do expect respondents generally to know whether their risk management is poor, sufficient or excellent, they may have trouble judging whether their system is worth a 7 or a 6. To mitigate this potential problem, we transform the original effectiveness scores into three broader levels of effectiveness,13 that is, poor (≤ 5), sufficient (6 – 7) and excellent (≥ 8), and rerun the analysis, now using logistic regression. The results of this analysis (not tabulated for sake of brevity) are very similar to the original findings, except for the effect of the use of quantitative risk assessment methods (QUANTMETHODS), which is no longer significant. All other findings remain unaffected, reinforcing our earlier inferences. 5.

Discussion

In this paper, we examined two main themes relating to ERM. First, we studied the extent of ERM implementation and the factors that may help to explain crosssectional differences in the level of adoption. In the second part of the study, we explored specific ERM design choices and their association with perceived risk management effectiveness. The first part of the paper builds on the findings of previous research into the extent of implementation (e.g. Beasley et al., 2005; Kleffner et al., 2003; Liebenberg and Hoyt, 2003). Using data from 825 organisations, our study considerably broadens the empirical basis underlying this stream of research. Specifically, whereas prior studies were based mainly on US and Canadian data, we work with data from organisations headquartered in the Netherlands, allowing some insights into the generalisability of the earlier findings in a different institutional context. In addition, unlike earlier studies,

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we include small and medium-sized enterprises, as well as public sector organisations. The findings largely corroborate the results of prior work, suggesting that the factors that are associated with ERM implementation are similar across different national contexts. Particularly, we found that publicly traded firms and organisations with a CRO and audit committee have more mature ERM systems, whereas the applicability of corporate governance regulation does not appear to influence ERM adoption. In addition, we found that larger organisations and firms in the financial sector tend to have more sophisticated ERM systems. These results are all consistent with prior empirical work. There is no evidence of an effect of institutional ownership, which is also consistent with previous findings. We do, however, observe that owner-managed firms are less prone to invest in ERM. Earlier studies did not include this factor. Finally, we found no auditor-related influences, suggesting that in the Netherlands, Big 4 and nonBig 4 audit firms are equally effective in promoting high-quality ERM among their clients. The second part of the paper addresses specific ERM design choices and their relation with risk management effectiveness. As far as we know, our paper is the first larger scale empirical study to examine this relation. In the analysis, we found that the frequency of risk assessment, the use of quantitative risk assessment techniques and the frequency of risk reporting contribute to perceived risk management effectiveness. In addition, we observe that on average, public sector organisations report lower risk management effectiveness than private sector organisations. In the part of the paper focusing on ERM adoption, we argued that public sector organisations may experience unique problems in ERM implementation due to the complex political environment in which they operate and their dominant culture and management style. The data did not corroborate this expectation. However, in conjunction with the negative public sector effect in the analysis of risk management effectiveness, the analysis suggests the following interpretation: it may be the case that in the public sector, organisations seek to conform to general expectations by implementing ERM systems that are relatively sophisticated from a technical point of view, even though the generic ERM concepts, tools and techniques are less effective in a public sector context. This suggests that there may be considerable value in developing an ERM approach that is more tailored to the specific needs and circumstances of the public sector. This seems an important challenge for regulators and standard-setting agencies, and also an interesting research opportunity. Research in this area is remarkably scarce, and further theoretical and empirical work would certainly help to gain a deeper understanding of the functioning of risk management in a public sector context. Several factors included in the analysis do not appear to affect risk management effectiveness. Some of these null results are perhaps even more interesting than the positive findings. For instance, even though the data might underestimate the true influence of COSO ERM, we found that only 43% of ERM adopters actually use that framework. In addition, we found that application of the COSO ERM

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framework does not contribute to risk management effectiveness. These findings raise concern as to the assumed authoritative status of this framework. If the framework is actually good, why do so many firms choose not to use it? Moreover, why are firms that do use it not more successful than those that don’t? Another interesting observation is that the majority (66%) of organisations do not quantify risk tolerances. In fact, approximately half of them do not explicate risk tolerances at all. This practice is contrary to COSO, which claims that explication of risk appetite and, subsequently, quantification of risk tolerances is essentially a conditio sine qua non for reliable risk management. The regression results, however, indicate that explication and quantification of risk tolerances do not contribute to perceived risk management effectiveness. Taken together, these negative findings challenge the validity of some of COSO’s key assumptions. The COSO framework is predicated on the idea that sound risk management should be highly structured, starting with an explicit definition of the organisation’s overall risk appetite, followed by a deductive process to decompose the risk appetite into quantified risk tolerances at the level of specific objectives, which subsequently need to be matched with appropriate risk responses and control activities to ensure that the organisation remains within the pre-set boundaries. This mechanistic and technocratic perspective has been criticised for being built on a reductionist notion of organisations as unitary and intentional actors, and for assuming hyperrational human agency (Power, 2009). Our empirical evidence also suggests that this perspective might be misguided. That is to say, the majority of organisations do not seem to embrace COSO’s systematic, ‘calculative’ (Mikes, 2009) approach, and apparently without loss of effectiveness. This may be taken to imply that a less structured, more heuristic approach to risk management is feasible (cf. also Collier et al., 2007; Mikes, 2009). A practical implication of this inference would be that standard setters may wish to reconsider their frameworks to accommodate more subjective and exploratory risk management styles. Overall, the findings of our study reflect that ERM is still in a developing stage, and that important knowledge gaps remain – both in practice and in academe. Seeking to conform to rapidly increasing expectations from stakeholders and regulators, organisations struggle with the (re-)design of their risk management systems, looking for effective approaches that suit their needs. Although standard setters present ERM as an effective response to risk management challenges, only a minority of respondents (11%) report to have fully functional ERM systems in place, while another 13% is in the process of implementing such a system. A significantly larger group is still considering ERM implementation (39%). These observations indicate that the practice of risk management has not yet matured, and that new and innovative approaches might emerge over time. Remarkably, standard setters appear quite eager to codify ‘best practices’, apparently ignoring that such practices may be hard to identify at a point in time at which risk management practices are still being tried and tested by organisations. In the meantime, academic research offers precious little guidance to inform the design of

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effective risk management systems, and many questions remain unanswered – including some very basic ones. For instance, we have only got a rudimentary understanding of how decision-makers respond to information on risk, and how these responses can be influenced to ensure a proper weighting of risk in the decision. In addition, we know virtually nothing about how organisations integrate risk management in their management control structures to guide the behaviour of lower level managers in decentralised organisations. This list could easily be expanded. Given the relevance of risk management in contemporary organisations, questions such as these should be high on the agenda of the accounting research community (cf. Kaplan, 2011). This study has several limitations that should be recognised when interpreting the evidence. Although we cherish the permission to use the data, the fact that we rely on secondary data forced us to focus the analysis on the factors on which we have information, rather than on the factors that are most interesting from the point of view of this paper. Fortunately, the two largely coincide. Nevertheless, it would have been interesting to include, for instance, the way in which organisations have integrated risk management in a strategy setting, or to study the impact of the internal environment on risk management effectiveness – to name but a few of the central themes of the COSO ERM framework. For now, we must leave the exploration of these factors to future research. Another consequence of our reliance on this particular data-set is that some of our measures are rather naive. Most variables are single item metrics. Although this is adequate for the more factual variables (actually, most of the independent variables qualify as such), several other constructs are so complex that measuring them with only one indicator is clearly not ideal. Especially, the measurement of perceived risk management effectiveness could be improved, and we expect future studies to make significant progress in that area. It should also be emphasised that the effectiveness measure is based on perceptions rather than on ‘hard’ data. Therefore, the scores on this measure may be biased, and they may be an inaccurate reflection of the actual contribution of ERM to the functioning of the organisation (cf. Ittner and Larcker, 2001). On the other hand, the organisational position of the respondents seems to ensure that they are knowledgeable about the functioning of the risk management systems and able to make meaningful evaluative statements about these systems. Therefore, we believe that the analysis is at least informative of current risk management practices and of the contribution of elementary design choices to overall risk management quality. As a first step towards a more rigorous, evidence-based understanding of successful risk management practices, this seems well worth the effort. Acknowledgements The data on which this paper relies have been collected by a research team involving PwC, Royal NIVRA, the University of Groningen and Nyenrode Business

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University. We thank the research team for generously sharing its data with us. This paper has benefited from insightful comments and suggestions from two anonymous reviewers and the (past) editor Salvador Carmona. We also acknowledge valuable feedback on earlier versions from Max Brecher, Ivo De Loo, Jacques de Swart, Anne-Marie Kruis, Hans Strikwerda, Frank Verbeeten, Sally Widener, and conference participants at the 2010 Conference of the Management Control Association, the 2011 European Conference on Corporate Governance & Internal Auditing and the Annual Congress 2011 of the European Accounting Association. Notes 1

In addition to these US-based and Canadian studies, Collier et al. (2007) surveyed risk management practices in the UK. However, because their examination of the drivers of ERM implementation is limited to an analysis of bivariate correlations, it is difficult to relate their findings to the other studies in this line of research. 2 In our analysis of the factors associated with the extent of ERM implementation, we adopt a similar metric. 3 One of the authors of the current paper was involved in this team. 4 The fact that the respondents come from different functional groups and hierarchical levels does not affect our analyses. We included dummy variables in all regressions to control for possible effects related to the organisational position of the respondents, but found none. The dummies were insignificant in the regressions, and their inclusion did not alter the substantive findings of the analyses. The regression results we report in Sections 3 and 4 exclude the respondent dummies. 5 The Beasley scale is as follows: (1) no plans exist to implement ERM; (2) investigating ERM, but no decision made yet; (3) planning to implement ERM; (4) partial ERM is in place; and (5) complete ERM is in place (Beasley et al., 2005). 6 We considered using this information as an alternative to STAGE in the analyses. This, however, is not an attractive option for two reasons. First, using the alternative metric would complicate comparison with the results from previous studies (particularly Beasley et al., 2005). Second, using the alternative metric would cost us well over 200 observations due to missing values (the survey instrument instructed respondents only to complete the relevant questions in specific circumstances). 7 To the best of our knowledge, the only study that explores the relationship between ERM design and effectiveness is Collier et al. (2007). This study, however, examines risk management practices at a high level of aggregation, using broad categories of practices as independent variables, rather than specific instruments and techniques. 8 Remarkably, COSO appears to be well aware of this behavioural literature (see, for instance, COSO, 2004a, pp. 51– 52). It is unclear why COSO has chosen to ignore the implications of this work. 9 We mitigate this specific problem by restricting the empirical analysis of ERM effectiveness to organisations that have adopted ERM (see Section 4.3). This restriction ensures that all included respondents subscribe to the notion of ERM, and we can be reasonably assured that their point of reference in scoring their risk management system is sufficiently similar to allow a meaningful comparison. We thank an anonymous reviewer for suggesting this approach. 10 In the total sample (i.e. including organisations in stages 1 –3), the average grade is 6.44, with approximately 20% of respondents indicating that their system is not sufficient. 11 In the total sample (including firms that have not (yet) adopted ERM), 21.5% of the respondents report to apply (elements of) COSO.

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12

A diagnosis of the variance inflation factors (VIF) confirms this; the highest VIF in the analysis is only 1.382. 13 We thank an anonymous reviewer for this suggestion.

References Arena, M., Arnaboldi, M. and Azzone, G. (2010) The organizational dynamics of enterprise risk management, Accounting, Organizations and Society, 35(7), pp. 659–675. Beasley, M. S., Clune, R. and Hermanson, D. R. (2005) Enterprise risk management: an empirical analysis of factors associated with the extent of implementation, Journal of Accounting and Public Policy, 24, pp. 521–531. Beasley, M., Pagach, D. and Warr, R. (2008) Information conveyed in hiring announcements of senior executives overseeing enterprise-wide risk management processes, Journal of Accounting, Auditing & Finance, 23, pp. 311–332. Bhimani, A. (2003) A study of the emergence of management accounting system ethos and its influence on perceived system success, Accounting, Organizations and Society, 28, pp. 523–548. Collier, P. M., Berry, A. J. and Burke, G. T. (2006) Risk and management accounting: best practice guidelines for enterprise-wide internal control procedures, CIMA Research Executive Summary Series, 2(11). Collier, P. M., Berry, A. J. and Burke, G. T. (2007) Risk and Management Accounting: Best Practice Guidelines for Enterprise-Wide Internal Control Procedures (Oxford: CIMA/Elsevier). COSO (2004a) Enterprise Risk Management – Integrated Framework. Executive Summary & Framework (Committee of Sponsoring Organizations of the Treadway Commission). COSO (2004b) Enterprise Risk Management – Integrated Framework: Application Techniques. Executive Summary & Framework (Committee of Sponsoring Organizations of the Treadway Commission). DeAngelo, L. E. (1981) Auditor size and audit quality, Journal of Accounting and Economics, 3, pp. 183–199. Demski, J. S. (1969) Decision-performance control, The Accounting Review, XLIV, pp. 669–679. Deumes, R. and Knechel, W. R. (2008) Economic incentives for voluntary reporting on internal risk management and control systems, Auditing: A Journal of Practice & Theory, 27, pp. 35–66. Francis, J. R. (2004) What do we know about audit quality?, British Accounting Review, 36, pp. 345–368. Gordon, L. A., Loeb, M. P. and Tseng, C. (2009) Enterprise risk management and firm performance: a contingency perspective, Journal of Accounting and Public Policy, 28, pp. 301–327. Ittner, C. D. and Larcker, D. F. (2001) Assessing empirical research in management accounting: a value-based perspective, Journal of Accounting and Economics, 32, pp. 349–410. Kane, G. D. and Velury, U. (2004) The role of institutional ownership in the market for auditing services: an empirical investigation, Journal of Business Research, 57, pp. 976–983. Kaplan, R. S. (2011) Accounting scholarship that advances professional knowledge and practice, The Accounting Review, 86, pp. 367–383. Kleffner, A. E., Lee, R. B. and McGannon, B. (2003) The effect of corporate governance on the use of enterprise risk management: evidence from Canada, Risk Management and Insurance Review, 6, pp. 53–73. Liebenberg, A. P. and Hoyt, R. E. (2003) The determinants of enterprise risk management: evidence from the appointment of chief risk officers, Risk Management and Insurance Review, 6, pp. 37–52. Lovata, L. M. and Costigan, M. L. (2002) Empirical analysis of adopters of economic value added, Management Accounting Research, 13, pp. 215–228. Mikes, A. (2008) Chief risk officers at crunch time: compliance champions of business partners?, Journal of Risk Management in Financial Institutions, 2, pp. 7–25.

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Quantitative Finance Letters, 2013 Vol. 1, 47–54, http://dx.doi.org/10.1080/21649502.2013.803757

The case for convex risk measures and scenario-dependent correlation matrices to replace VaR, C-VaR and covariance simulations for safer risk control of portfolios WILLIAM T. ZIEMBA∗ University of British Columbia, Canada Sabanci University, Turkey (Received 19 March 2013; in final form 6 May 2013 ) Value at risk (VaR) is the most popular risk measure and is enshrined in various regulations. It postulates that portfolio losses are less than some prescribed amount most of the time. Therefore a loss of $10 million is the same as a loss of $5 billion. C-VaR tries to correct this by linearly penalizing the loss so a loss of $20 million is twice as damaging as that of $10 million with the same probability. This is an improvement but is not enough of a penalty to force investment portfolios to be structured to avoid these losses. The author has used convex risk measures since 1974 in various asset–liability management (ALM) models such as the Russell Yasuda Kasai and the Vienna InnoALM. They penalize losses at a much greater rate than linear rate so that double or triple losses are more than two or three times as undesirable. Also scenario-dependent correlation matrices are very important in model applications because ordinary average correlations tend to work when you do not need them and fail by giving misleading results when you need them. For example, in stock market crash situations, bonds and stocks are no longer positively correlated. Adding these two features to stochastic asset–liability planning models is a big step towards improving risk control and performance. Keywords: Convex risk measures; Value at risk; Scenarios; Optimization; Multi-period asset–liability planning models; Correlations; Crash modelling

1. Introduction The greatest investor in terms of wealth achieved is Warren Buffett. He has two rules of investing: Rule 1 Do not lose money. Rule 2 Do not forget Rule 1. My experience with great investors from my papers, books, lectures, consulting, and money management and various courses where I have taught is that the great investors focus first on not losing and the ones who blow up focus mainly on winning. Ziemba and Ziemba (2007, 2013) discuss great investors and several hedge fund and bank trading blowups. The latter includes TLCM in 1998, Niederhoffer in 1997 and in other years, and Amaranth in 2006. There are numerous other blowups including the January 2008 Societé General equity index loss that affected many other investors, the February 2008 MF Global wheat losses and the 2013 Banca Monte dei Paschi di Siena and J. P. Morgan Whale losses. All of these losses were in the billions and involve rogue traders so they grab our attention. But, there are countless other losses of a smaller but significant level. I argue that the recipe for disaster has two elements: • the overall position is overbet relative to its existing and readily available capital and • the position is not diversified in all scenarios.

∗ Emails:

Once a trader or firm enters the danger zone by violating these two risk control elements, profits at a high level can continue to be made for a substantial period and large fees can be earned. But if a bad scenario occurs, then there can be great losses. In most cases, once the portfolio is in deep trouble, it can only be saved with more cash or marginable securities. And there bailout resources are frequently unavailable. Therefore a prudent plan is to simply avoid this possible situation by not overbetting and diversifying across all plausible scenarios. This sounds simple but many many traders and portfolio managers cannot or will not follow these protective measures. A major reason is the incentives of the trader. There are large fees to be made by making profits and taking a percentage. Therefore a trader can rack up large fees and then if a blowout occurs, the penalties are not great enough. The trader is fired, discredited a bit, possibly sued a bit but more than likely survives with a large wealth. Then the trader can get a new job or fund from others who value the skills in trading and excuse the blowout by saying it was a one in 9 or 10 sigma event as Jorion (2007) and Till (2006) discuss in their analyses of the demise of Amaranth. Ziemba and Ziemba (2007, 2013) curred largely because of a $7 to $5 price drop. Since the price range in previous years was $2–11, 9 or 10 sigma seems way over the mark. It is more like 1 in 50 or 100. Such overbetting in basically one asset sets up a big loss in a plausible but unlikely event. An update on the gas market by Rachel Ziemba is

[email protected]; [email protected]

© 2013 William T. Ziemba. Published by Taylor & Francis. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The moral rights of the named author(s) have been asserted.

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in Ziemba and Ziemba (2013). A valuable book on risk management using Value at risk (VaR) is Jorion (2000b) and Jorion (2007). See also Alexander (2008) and Jorion (2000a). But VaR is very dangerous especially if you are levered since it encourages overbetting. Long-term capital management, armed with many Massachusetts Institute of Technology (MIT) PhDs, simulations, stress tests and past data, managed to lose 95% of their capital in 1998. I argue in Section 5 that multiple scenario-dependent correlation matrixes are crucial. Among other things they can pick up negative stock–bond correlations in stock market crashes. Taleb (2001, 2007, 2012) calls these rare bad events Black Swans and he discusses some ways to minimize the bad effects or even profit from such events. Our earlier stochastic programming asset–liability literature uses the term bad scenarios because the models use optimization over possible scenarios; see, for example, Kallberg et al. (1982), Kusy and Ziemba (1986), Cariño and Ziemba (1998) and Ziemba (2003, 2007, 2012). In the stochastic programming ALM models, a wide variety of scenarios is considered including some possible bad ones that cover the range of future possibilities. It is their impact not necessarily the actual event that is modelled. Typically, the possibility of such bad events is not even considered in most models. One place where they are considered besides the stochastic programming ALM literature is in the pricing of put options. These options may include the possibility of bad outcomes even if they do not occur. See Tompkins et al. (2008) who study all the S&P500 options from 1985 to 2002. A great trader in these puts is Warren Buffett of Berkshire Hathaway. While I trade them over about a three month period in exchanges with careful risk control, Buffett with enormous resources makes private over the counter contracts over 15–18 years with various pension funds that want to guarantee that they will not lose money. Besides having about $4.6 billion to invest now, Buffett most likely will owe nothing when these puts expire many years from now; see Ziemba and Ziemba (2013) for an analysis. Bad events occur in other areas such as weather, see for example, Kunreuther and Michel-Kerjan (2009) and the activities of their Wharton risk centre project on managing and financing extreme events such as earthquakes, hurricanes and other catastrophes. As I observed in Ziemba (2003), the number of rare events, that is, events outside the range of all previous event realizations is growing. For example, 20 of the 30 most expensive insured catastrophes worldwide during 1970 still are post 2001 and fully 13 were in the USA. The hurricanes’ Katrina, Rita and Wilma cost more than $180 billion and hurricane Sandy in 2012 costs $65 billion. Rouge traders, such as Nick Leeson and Brian Hunter, also rack up large fees by forcing the market prices at option expiry time to expand their mark-to-the-market paper profits. Of course, with their large positions, they cannot actually trade at these prices. Therefore there is a regulatory issue here. I just discuss here how one might do better risk control. But it is clear that even if the rogue trader knew how to do a better risk control, it likely pays to disregard it and shoot for the fences and rack up large fees and then

mop up the mess should the bad scenario occur. Lleo and Ziemba (2013) show that even if a big trader understands risk control, overbetting is actually optimal in an expected utility model since the penalties for losses are small and the payoff for winning is large. Section 2 shows how a modification of Markowitz mean-variance analysis, where the weighted sum of target deviations over various targets and time periods, replaces variance where the target violations are penalized by a convex risk measure. Then one has a concave maximization problem where long-run expected discounted wealth at the horizon is traded off for a risk measure by convex target violations with an Arrow Pratt risk-aversion parameter. In Sections 3–5, I show how this procedure plus scenario-dependent correlation matrices were used in the pension fund model InnoALM for Siemens Austria fund. Section 6 concludes. 2. Modifying mean-variance for multi-period ALM planning models Markowitz mean-variance analysis is widely used in portfolio applications and has been the standard approach of many investment management systems. But its use is questionable in multiple periods and with liabilities. This section presents a modified mean-variance like tradeoff for multi-period asset–liability planning models. In static portfolio theory suppose there are n assets, i = 1, . . . , n, with random returns ξ1 , . . . , ξn . The return on asset i, namely ξi , is the capital appreciation plus dividends in the next investment period such as monthly, quarterly, or yearly or some other time period. The n assets have the distribution F(ξ1 , . . . , ξn ) with known mean vector ξ¯ = (ξ¯1 , . . . , ξ¯n ) and known nxn variance–covariance matrix with typical covariance σij for i = j and variance σi2 for i = j. A basic assumption is that the return distributions are independent of the asset weight choices, so F = ϕ(x). A mean-variance frontier is generated by the problem φ(δ) = Maximize ξ¯ ′ x s.t. x′ x  δ e′ x = w0 x ∈ K, where e is a vector of ones, x = (x1 , . . . , xn ) are the asset weights, K represents other constraints on the x’s and w0 is the investor’s initial wealth. When variance is parameterized with δ > 0, it yields a concave curve, as in figure 1(a). This is a Markowitz (1952, 1987), and Markowitz and van Dijk (2006) mean-variance efficient frontier and optimally trades off mean which is desirable with variance which is undesirable. Tobin (1958) extended the Markowitz model to include a risk free asset with mean ξ0 and no variance. Then the efficient frontier concave curve becomes the straight line as shown in figure 1(b). The standard deviation here is plotted rather than the variance to make the line straight. An investor will pick an optimal portfolio in the Markowitz model using a utility

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Convex risk measures (a) Expected return

(b)

Optimal portfolio A for given utility function

Expected return Optimal portfolio B for given utility function

rf

Efficient frontier

Efficient frontier Market index

Variance

Standard deviation

Figure 1. Two efficient frontiers. (a) Markowitz mean-variance efficient frontier. (b) Tobin risk free asset and separation theorem.

function that trades off mean for variance or, equivalently, standard deviation as shown in figure 1(a) to yield portfolio A. For the Tobin model, one does a simpler calculation to find the optimal portfolio which will be on the straight line in figure 1(b) between the risk free asset and the market index M . Here the investor picks portfolio B that is two-thirds cash (risk free asset) and one-third market index. The market index may be proxied by the S&P500 or Wilshire 5000 value weighted indices. Since all investors choose between cash and the market index, this separation of the investor’s problem into finding the market index independent of the investor’s utility function and then where to be on the line for a given utility function is called Tobin’s separation theorem. Ziemba et al. (1974) discuss this and show how to compute the market index and optimal weights of cash and the market index for various utility functions and constraints. In the Markowitz model the variance is the measure of risk. If ρ(x) = risk measure, then ρ1 (x) = x′ x and the variance constraint controls risk. An equivalent formulation of the mean-variance problem moves the risk to the objective with the function μ(x) − λρ1 (x), where μ(x) is the mean. This objective emphasizes the risk-return tradeoff and includes the risk-aversion penalty parameter λ. Risk measures should emphasize the downside. Suppose x is defined as the loss relative to a benchmark. Then popular risk measures include ρ2 (x) = VaR α (x) = −F −1 (1 − α) and  1 −1 ρ3 (x) = C-VaR α (x) = F −1 (t) dt. 1 − α 1−α VaR treats all losses above the (1 − α)th percentile the same, whereas C-VaR weights losses in the tail linearly by size. Greater control over risk is achieved with a convex risk measure ρ4 (x) = CVα (x) =  1−α (1/(1 − α)) 0 c(F −1 (t)) dt, where c(x) is a positive, decreasing convex function with c(0) = 0. Then μ(x) − λρ4 (x) is a concave utility function. Starting with the papers of Kallberg et al. (1982) and Kusy and Ziemba (1986) as devised in my 1974 University of British Columbia PhD class on stochastic programming and asset–liability management (ALM), I have proposed and used as an objective function the concave utility function maxx {μ(x) − λρ4 (x)}.

Such concave objective functions with convex risk measures were used in the Russell Yasuda Kasai model (Cariño et al. 1994, Cariño and Ziemba 1998, Cariño et al. 1998), and are justified in an axiomatic sense in Rockafellar and Ziemba (2000) which is reprinted in MacLean and Ziemba (2013). Non-technical decision-makers find the increasing penalty for target violations a good approach to lower risk and try to avoid shortfalls and it is easy for them to understand. Of course, the exact form of the convex function is important to the control of risk. In the next section, a successful application illustrates the approach.

3. InnoALM: the innovest Austrian pension fund financial planning model Siemens Oesterreich, part of the global Siemens Corporation, is the largest privately owned industrial company in Austria. Its businesses, with revenues of ≈2.4 billion in 1999, include information and communication networks, information and communication products, business services, energy and travelling technology, and medical equipment. Its pension fund is the largest corporate pension plan in Austria and is a DCP. More than 15 000 employees and 5000 pensioners are members of the pension plan, which had ≈510 million in assets under management as of December 1999. Innovest Finanzdienstleistungs, founded in 1998, is the investment manager for Siemens Oesterreich, the Siemens pension plan and other institutional investors in Austria. The motivation to build InnoALM, as described in Geyer and Ziemba (2008), was based on the desire to have superior performance and good decision aids to help achieve this ranking. Various uncertain aspects, possible future economic scenarios, stocks, bonds, and other investments, transactioncosts, liquidity, currency aspects, liability commitments over time, Austrian pension fund law, and company policy suggested that a good way to approach this asset–liability problem was via a multi-period stochastic linear programming model. This model has innovative features, such as state-dependent correlation matrixes, fat-tailed assetreturn distributions, convex penalties on target violations, a concave objective function, simple computational schemes and easy to interpret output.

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W. T. Ziemba

InnoALM was produced in six months in 2000, with Geyer and Ziemba serving as consultants and with assistance from Herold and Kontriner who were Innovest employees. InnoALM demonstrates that a small team of researchers with a limited budget can quickly produce a valuable modelling system that can easily be operated by nonstochastic programming specialists on a single personal computer. The IBM OSL stochastic programming software provides a good solver. The model can be used to explore possible European, Austrian and Innovest policy alternatives. The liability side of the Siemens pension plan consists of employees for whom Siemens is contributing DCP payments and retired employees who receive pension payments. Contributions are based on a fixed fraction of salaries. The annual pension payments are based on a discount rate of 6% and the remaining life expectancy at the time of retirement. These annuities grow by 1.5% annually to compensate for inflation. Hence, the wealth of the pension fund must grow by 7.5% a year to match liability commitments. The model determines the optimal purchases and sales for each of N assets in each of T planning periods. Typical asset classes used at Innovest are US, Pacific, European, and emerging market equities and US, UK, Japanese, and European bonds. The objective is to maximize the concave risk-averse utility function ‘expected terminal wealth’ less convex penalty costs subject to various linear constraints. Austria’s limits of 40% maximum in equities, 45% maximum in foreign securities and 40% minimum in Eurobonds. The convex risk measure is approximated by a piecewise linear function, so the model is a multi-period stochastic linear programme. Typical targets that the model tries to achieve (and is penalized for if it does not) are for a growth of 7.5% a year in wealth (the fund’s assets), a deterministic target, and for portfolio performance returns to exceed benchmarks, a stochastic target. Excess wealth is placed into surplus reserves, and a portion of the excess is paid out in succeeding years. 4. Formulation of InnoALM as a multistage stochastic linear programming model The non-negative decision variables are wealth (after transactions) Wit , purchases Pit and sales Sit for each asset (i = 1, . . . , N ). Purchases and sales are in periods t = 0, . . . , T − 1. Purchases and sales are scenariodependent except for t = 0.† Wealth accumulates over time for a T period model according to Wi0 = Wiinit + Pi0 − Si0 ,

˜ i1 = R˜ i1 Wi0 + P˜ i1 − S˜ i1 , W

N  i=1

Pi0 (1 + tcpi ) =

t = 1, t = 2, . . . , T − 1,

and ˜ iT = R˜ it Wi,T −1 , W This section is based on Geyer and Ziemba (2008).

N 

Si0 (1 − tcsi ) + C0 ,

i=1

t = 0,

and N  i=1

P˜ it (1 − tcp)i =

N  i=1

S˜ it (1 − tcs)i + Ct ,

t = 1, . . . , T − 1, where tcpi and tcsi denote asset-specific linear transactioncosts for purchases and sales, and Ci is the fixed (nonrandom) net cashflow (inflow if positive). Portfolio weights can be constrained over linear combinations (subsets) of assets or individual assets via  i∈U

˜ it − θU W

N  i=1

˜ it  0 W

and −

 i∈L

˜ it + θL W

N  i=1

˜ it  0, W

t = 1, . . . , T − 1,

where θU is the maximum percentage and θL is the minimum percentage of the subsets U and L of assets i = 1, . . . , N included in the restrictions. The θU ’s, θL’s, U ’s and L’s may be time dependent. Risk is measured as a weighted discounted convex function of target violation shortfalls of various types in various periods. In a typical application, the deterministic wealth target is assumed to grow by 7.5% in each year. The wealth targets are modelled via N  ˜ it − P˜ it + S˜ it ) + M ˜ tW  W ¯ t, (W i=1

t = 1, . . . , T ,

˜ tW are wealth target shortfall variables. where M The benchmark-target B˜ t is scenario dependent. It is based on stochastic asset returns and fixed asset weights αi defining the stochastic benchmark portfolio

t = 0,

˜ it = R˜ it Wii,t−1 + P˜ it − S˜ it , W

†

Wiinit is the prespecified initial value of asset i. There is no uncertainty in the initialization period t = 0. Tildas denote scenario-dependent random parameters or decision variables. Returns are associated with time intervals. R˜ it (t = 1, . . . , T ) are the (random) gross returns for asset i between t − 1 and t. The scenario generation and statistical properties of returns are discussed below. The budget constraints are

B˜ t = W0

t  N  i=1 j=1

αi R˜ ij ,

with shortfall constraints N  ˜ it − P˜ it + S˜ it ) + M ˜ tβ  B˜ t , (W i=1

t = 1, . . . , T ,

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Convex risk measures ˜ tB is the benchmark-target shortfall. These shortwhere M falls are also penalized with a piecewise linear convex risk measure. If total wealth exceeds the target, a fraction γ = 10% of the exceeding amount is allocated to a reserve account and invested in the same way as other available funds. However, the wealth targets at future stages are adjusted. Additional ˜ t are introduced and the non-negative decision variables D wealth target constraints become N  i=1

˜ it − P˜ it + S˜ it − D ˜ t) + M ˜ tW = W ¯t + (W

t−1  j=1

˜ t−j , γD

t = 1, . . . , T − 1, ˜ 1 = 0. where D The objective function is ⎛ ⎞⎤ ⎡ N T    ˜ tk )⎠⎦ . ˜ iT − λ d t wt ⎝ vk ck (M W Max EST ⎣dT t=1

i=1

k∈{W ,B}

This maximizes the expected discounted value of terminal wealth in period T net of the expected discounted penalty costs over the horizon from the convex risk measures ck (·) for various desired targets, respectively. The expectation is over T period scenarios ST . The discount factors dt are related to the interest rate r by dt = (1 + r)−t . Usually r is taken to be the three or six month treasury-bill rate. The λ is 1 the Arrow Pratt risk-aversion index and tradeoff risk and 2 return. The vk are weights for the wealth and benchmark shortfalls and the wt are weights for the weighted sum of shortfalls at each stage. The weights are normalized via 

k∈{W ,B}

vk = 1 and

T  t=1

wt = T .

Since pension payments are based on wealth levels, increasing these levels increases pension payments. The reserves provide security for the pension plan’s increase of pension payments at each future stage.

5. Results Geyer and Ziemba (2008) presented results for a problem sample model with four asset classes (European stocks, US stocks, European bonds and US bonds) with five periods (six stages). The periods are twice 1 year, twice 2 years and 4 years (10 years in total). They assumed discrete compounding. Using a 100-5-5-2-2 node structure, they generated 10 000 scenarios. Initial wealth equals 100 units, and the wealth target is assumed to grow at an annual rate of 7.5%. They do not consider a benchmark target or cash in- and outflows in this sample application. The risk-aversion index was RA = 4, and the discount factor 5%, which corresponds roughly with a simple static mean-variance model to a standard 60/40 stock/bond pension fund mix (see Kallberg and Ziemba 1983). Assumptions about the statistical properties of returns measured in nominal euros are based on monthly data from January 1970 for stocks and 1986 for bonds to September 2000. These include means, standard deviations and correlations for the applications of InnoALM and appear in table 1. The monthly stock returns were nonnormal, negatively skewed and fat-tailed, whereas monthly bond returns were close to normal (the critical value of the Jarque–Bera test for a = 0.01 is 9.2). For long-term planning models such as InnoALM, with its one-year review period, however, the statistical properties of monthly returns are less relevant. Although average returns and volatilities remained about the same, one year of data is lost when annual returns are computed and the distributional properties changed dramatically. There was negative skewness, but no evidence existed for fat tails in

Table 1. Mean, standard deviation and correlation assumptions. Stocks Europe Normal periods (70% of the time)

High volatility (20% of the time)

Extreme periods (10% of the time)

Average period

All periods Source: Geyer and Ziemba (2008).

Stocks US Bonds Europe Bonds US Standard deviation Stocks US Bonds Europe Bonds US Standard deviation Stocks US Bonds Europe Bonds US Standard deviation Stocks US Bonds Europe Bonds US Standard deviation Mean

0.755 0.334 0.514 14.6 0.786 0.171 0.435 19.2 0.832 −0.075 0.315 21.7 0.769 0.261 0.478 16.4 10.6

Bonds US

Europe

US

0.286 0.780 17.3

0.333 3.3

10.9

0.100 0.715 21.1

0.159 4.1

12.4

−0.182 0.618 27.1

−0.104 4.4

12.9

0.202 0.751 19.3 10.7

0.255 3.6 6.5

11.4 7.2

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W. T. Ziemba

annual returns, except for European stocks (1970–2000) and US bonds. The mean returns from this sample are comparable to the 1900–2011 112-year mean returns estimated by Dimson et al. (2012). The correlation matrixes in table 1 for the three different regimes are based on the regression approach of Solnik, Boucrelle and Le Fur (1996). Moving average estimates of correlations among all assets are functions of standard deviations of US equity returns. The estimated regression equations are then used to predict the correlations in the different regimes. Three regimes are considered, and the assumption is that 10% of the time, equity markets are extremely volatile; 20% of the time, markets are characterized by high volatility; and 70% of the time, markets are normal. The 35% quantile of US equity return volatility defines ‘normal’ periods. ‘Highly volatile’ periods are based on the 80% volatility quantile, and ‘extreme’ periods, on the 95% quartile. The associated correlations reflect the return relationships that typically prevailed during those market conditions. The correlations in table 1 show a distinct pattern across the three regimes. Correlations among stocks tend to increase as stock return volatility rises, whereas the correlations between stocks and bonds tend to decrease. European bonds may serve as a hedge for equities during extremely volatile periods because bond and stock returns, which are usually positively correlated, are then negatively correlated. The latter is a major reason using scenariodependent correlation matrixes is a major advance over the sensitivity of one correlation matrix. Optimal portfolios were calculated for seven cases with and without mixing of correlations and with normal, t- and historical distributions; see table 2. The ‘mixing’ cases NM, TM and HM use mixing correlations. Case NM assumes normal distributions for all assets. Case HM uses the historical distributions of each asset. Case TM

assumes t-distributions with five degrees of freedom for stock returns, whereas bond returns are assumed to have normal distributions. The ‘average’ cases NA, HA, and TA use the same distribution assumptions, with no mixing of correlation matrixes. Instead, the correlations and standard deviations used in these cases correspond to an ‘average’ period in which 10%, 20% and 70% weights are used to compute the averages of correlations and standard deviations in the three different regimes. Comparisons of the average (A) cases and mixing (M) cases are mainly intended to investigate the effect of mixing correlations. TMC maintains all assumptions of case TM but uses Austria’s constraints on asset weights (see table 2). Eurobonds must be at least 40% and equity at most 40%, and these constraints are binding. Some test results. Table 2 shows the optimal initial asset weights at Stage 1 for the various cases. Table 3 shows results for the final stage (expected weights, expected terminal wealth, expected reserves and shortfall probabilities). These tables exhibit a distinct pattern: the mixingcorrelation cases initially assign a much lower weight to European bonds than the average-period cases. Singleperiod, mean-variance optimization and average-period cases (NA, HA, and TA) suggest an approximate 45%/55% stock/bond mix. The mixing-correlation cases (NM, HM and TM) imply a 65%/35% stock/bond mix. Investing in US bonds is not optimal at Stage 1 in any of the cases, an apparent result of the relatively high volatility of US bonds relative to European bonds. Table 3 shows that the distinction between A and M cases becomes less pronounced over time. European equities, however, still have a consistently higher weight in the mixing cases than in the no-mixing cases. This higher weight is mainly at the expense of Eurobonds. In general, the proportion of equities at the final stage is much higher

Table 2. Optimal initial asset weights at Stage 1 (in %). Stocks Europe

Stocks US

Bonds Europe

Bonds US

34.8 27.2 40.0 44.2 47.0 37.9 53.4 35.1

9.6 10.5 4.1 1.1 27.6 25.2 11.1 4.9

55.6 62.3 55.9 54.7 25.4 36.8 35.5 60.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Single-period, mean-variance optimal weights (average periods) Case NA: no mixing (average periods), normal distributions Case HA: no mixing (average periods), historical distributions Case TA: no mixing (average periods), t-distributions for stocks Case NM: mixing correlations, normal distributions Case HM: mixing correlations, historical distributions Case TM: mixing correlations, t-distributions for stocks Case TMC: mixing correlations, historical distributions constraints on asset weights

Table 3. Expected terminal wealth, and expected reserves and probabilities of shortfalls with a target wealth, Wt = 206.1.

NA HA TA NM HM TM TMC

Stocks Europe

Stocks US

Bonds Europe

Bonds US

Expected terminal wealth

Expected reserves at Stage 6

Probability of target shortfall

Probability shortfall >10%

34.3 33.5 35.5 38.0 39.3 38.1 20.4

49.6 48.1 50.2 49.7 46.9 51.5 20.8

11.7 13.6 11.4 8.3 10.1 7.4 46.3

4.4 4.8 2.9 4.0 3.7 2.9 12.4

328.9 328.9 327.9 349.8 349.1 342.8 253.1

202.8 205.2 202.2 240.1 235.2 226.6 86.9

11.2 13.7 10.9 9.3 10.0 8.3 16.1

2.7 3.7 2.8 2.2 2.0 1.9 2.9

Convex risk measures than in the first stage. This result may be explained by the fact that the expected portfolio wealth at later stages is far above the target wealth level (206.1 at Stage 6), and the higher risk associated with stocks is less important. The constraints in case TMC lead to lower expected portfolio wealth throughout the horizon and to a higher shortfall probability than in any other case. Calculations show that initial wealth would have to be 35% higher to compensate for the loss in terminal expected wealth stemming from those constraints. In all cases, the optimal weight of equities is much higher than the historical 4.1% in Austria. The expected terminal wealth levels and the shortfall probabilities at the final stage shown in table 3 make the difference between mixing and no-mixing cases even clearer. This table shows the performance in terms of risk-return. The mixing-correlation cases yield higher levels of terminal wealth and lower shortfall probabilities. If the level of portfolio wealth exceeds the target, the surplus is allocated to a reserve account. The reserves in t are computed from and are shown in table 2 for the final stage. These values are in monetary units given an initial wealth level of 100. They can be compared with the wealth target 206.1 at Stage 6. Expected reserves exceed the target level at the final stage by up to 16%. Depending on the scenario, the reserves can be as high as 1800. Their standard deviation (across scenarios) ranges from 5 at the first stage to 200 at the final stage. The constraints in case TMC lead to a much lower level of reserves compared with the other cases, which implies, in fact, less security against future increases of pension payments. Optimal allocations, expected wealth and shortfall probabilities are mainly affected by considering mixing correlations but the type of distribution chosen has a smaller impact. This distinction is primarily the result of the higher proportion allocated to equities, if different market conditions are taken into account by mixing correlations. The

53

results of any asset-allocation strategy depend crucially on the mean returns. Geyer and Ziemba investigated the effect by parameterizing the forecasted future means of equity returns. Assume that an econometric model forecasts that the future mean return for US equities is some value between 5% and 15%. The mean of European equities is adjusted accordingly so that the ratio of equity means and the mean bond returns shown in table 1 are maintained. Geyer and Ziemba retain all other assumptions of case NM (normal distribution and mixing correlations). The results are sensitive to the choice of the mean return; see Chopra and Ziemba (1993) and Kallberg and Ziemba (1981, 1984). If the mean return for US stocks is assumed to equal the long-run mean of close to 12%, as estimated by Dimson et al. (2012), the model yields an optimal initial weight for equities of 100. A mean return for US stocks of 9%, however, implies an optimal weight of less than 30% for equities. The sensitivity of the mean carries into multi-period models. There the effect is strongest in period 1 then less and less in future periods, see Geyer and Ziemba (2008). Whereas in later periods, this sensitivity is less and, by period 5, it is almost non-existent. 6. Model tests To emphasize the difference between the cases TM and TA, figure 2 compares the cumulated monthly returns obtained from the rebalancing strategy for the two cases with a buyand-hold strategy that assumes that the portfolio weights on January 1992 were fixed at the optimal TM weights throughout the test period. In comparison to the buy-andhold strategy or the performance using TA results, for which rebalancing does not account for different correlation and volatility regimes, rebalancing on the basis of the optimal TM scenario tree provided a substantial gain. The model, once developed in 2000, proved to be very useful for Innovest. In 2006, Konrad Kontriner (Member of

Figure 2. Cumulative monthly returns for different strategies, 1992–2002. Source: Geyer and Ziemba (2008).

54

W. T. Ziemba

the Board) and Wolfgang Herold (Senior Risk Strategist) of Innovest stated that: The InnoALM model has been in use by Innovest, an Austrian Siemens subsidiary, since its first draft versions in 2000. Meanwhile it has become the only consistently implemented and fully integrated proprietary tool for assessing pension allocation issues within Siemens AG worldwide. Apart from this, consulting projects for various European corporations and pensions funds outside of Siemens have been performed on the basis of the concepts of InnoALM. The key elements that make InnoALM superior to other consulting models are the flexibility to adopt individual constraints and target functions in combination with the broad and deep array of results, which allows to investigate individual, path dependent behaviour of assets and liabilities as well as scenario-based and Monte-Carlo like risk assessment of both sides. In light of recent changes in Austrian pension regulation the latter even gained additional importance, as the rather rigid asset-based limits were relaxed for institutions that could prove sufficient risk management expertise for both assets and liabilities of the plan. Thus, the implementation of a scenario-based asset-allocation model will lead to more flexible allocation restraints that will allow for more risk tolerance and will ultimately result in better long-term investment performance. Furthermore, some results of the model have been used by the Austrian regulatory authorities to assess the potential risk stemming from less constraint pension plans.

References Alexander, C., Market Risk Analysis, 2008 (Wiley: Chichester). Cariño, D. and Ziemba, W.T., Formulation of the Russell-Yasuda Kasai financial planning model. Oper. Res., 1998, 46(4), 433–449. Cariño, D., Myers, D. and Ziemba, W.T., Concepts, technical issues and uses of the Russell-Yasuda Kasai financial planning model. Oper. Res., 1998, 46(4), 450–462. Cariño, D. R., Kent, T., Myers, D.H., Stacey, C., Sylvanus, M., Turner, A.L., Watanabe, K. and Ziemba, W.T., The Russell-Yasuda Kasai model: An asset/liability model for a Japanese insurance company using multistage stochastic programming. Interfaces, 1994, 24(1), 29–49. Chopra, V.K. and Ziemba, W.T., The effect of errors in mean, variance and co-variance estimates on optimal portfolio choice. J. Portfolio Manage., 1993, 19, 6–11. Dimson, E., Marsh, P. and Staunton, M., Investment Returns Yearbook, 2012 (Credit Suisse Global: London). Geyer, A. and Ziemba, W.T., The innovest Austrian pension fund financial planning model InnoALM. Oper. Res., 2008, 56(4), 797–810. Jorion, P., Risk management lessons from long-term capital management. Eur. Financ. Manage., 2000a, 6, 277–300. Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, 2000b (McGraw Hill: New York). Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed., 2007 (McGraw Hill: New York). Kallberg, J.G. and Ziemba, W.T., Remarks on optimal portfolio selection. Methods of Operations Research, Oelgeschlager, edited by G. Bamberg and O. Optiz, Vol. 44, pp. 507–520, 1981 (Gunn and Hain: Boston, MA). Kallberg, J.G., White, R. and Ziemba, W.T., Short term financial planning under uncertainty. Manage. Sci., 1982, XXVIII, 670–682.

Kallberg, J.G. and Ziemba, W.T., Comparison of alternative utility functions in portfolio selection problems. Management Sci., 1983, 29, 1257–1276. Kallberg, J.G. and Ziemba, W.T., Mis-specifications in portfolio selection problems. In Risk and Capital, edited by G. Bamberg and K. Spremann, pp. 74–87, 1984 (Springer Verlag: New York). Kunreuther, H. and Michel-Kerjan, E.O., At War with the Weather: Managing Large Scale Risks in a New Era of Catastrophes, 2009 (MIT Press: Cambridge, MA). Kusy, M.I. and Ziemba, W.T., A bank asset and liability management model. Oper. Res., 1986, 34, 356–376. Lleo, S. and Ziemba, W.T., Optimality of Rouge trading. Working paper, 2013, Reims Management School. MacLean, L.C. and Ziemba, W.T. (Eds.), Handbook of the Fundamentals of Financial Decision Making, 2013 (World Scientific: Singapore). Markowitz, H.M., Portfolio selection. J. Finance, 1952, 7(1), 77–91. Markowitz, H.M., Mean-Variance Analysis in Portfolio Choice and Capital Markets, 1987 (Basil Blackwell: Cambridge, MA). Markowitz, H.M. and van Dijk, E., Risk-return analysis. In Handbook of Asset and Liability Management, edited by S.A. Zenios and W.T. Ziemba, Vol. 1, pp. 139–197. Handbooks in Finance, 2006 (North Holland: Amsterdam). Rockafellar, R.T. and Ziemba, W.T., Modified Risk Measures and Acceptance Sets. Working paper, 2000. Solnik, B., Boucrelle, C. and Le Fur, Y., International market correlation and volatility. Financial Analysts J., 1996, 52, 17–34. Taleb, N.N., Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets, 2001 (Random House: New York). Taleb, N.N., The Black Swan: The Impact of the Highly Improbable, 2007 (Random House: New York). Taleb, N.N., Antifragile: Things That Gain from Disorder, 2012 (Random House: New York). Till, H., EDHEC comments on the Amaranth case: early lessons from the debacle. Technical report, EDHEC, 2006. Tobin, J., Liquidity preference as behavior towards risk. Review of Economic Studies, 1958, 25(2), 65–86. Tompkins, R., Ziemba, W.T. and Hodges, S., The favorite-longshot bias in S&P500 futures options: the return to bets and the cost of insurance. In Handbook of Sports and Lottery Markets, edited by D.B. Hausch and W.T. Ziemba, pp. 161–180. Handbooks in Finance, 2008 (North Holland: Amsterdam). Ziemba, W.T., The Stochastic Programming Approach to Asset Liability and Wealth Management, 2003 (AIMR: Charlottesville, VA). Ziemba, W.T., The Russell-Yasuda Kasai InnoALM and related models for pensions, insurance companies and high net worth individuals. In Handbook of Asset and Liability Management, edited by S.A. Zenios and W.T. Ziemba, Vol. 2, pp. 861–962. Handbooks in Finance, 2007 (North Holland: Amsterdam). Ziemba, W.T., Calendar Anomalies and Arbitrage, 2012 (World Scientific: Singapore). Ziemba, R.E.S. and Ziemba, W.T., Scenarios for Risk Management and Global Investment Strategies, 2007 (Wiley: Chichester). Ziemba, R.E.S. and Ziemba, W.T., Investing in the Modern Age, 2013 (World Scientific: Singapore). Ziemba, W.T., Parkan, C. and Brooks-Hill, F.J., Calculation of investment portfolios with risk free borrowing and lending. Manage. Sci., 1974, 21, 209–222.

THE EFFECTIVENESS OF USING A BASIS HEDGING STRATEGY TO MITIGATE THE FINANCIAL CONSEQUENCES OF WEATHER-RELATED RISKS Linda L. Golden,* Charles C. Yang,† and Hong Zou‡

ABSTRACT This paper examines the effectiveness of using a hedging strategy involving a basis derivative instrument to reduce the negative financial consequences of weather-related risks. We examine the effectiveness of using this basis derivative strategy for both summer and winter seasons, using both linear and nonlinear hedging instruments and the impacts of default risk and perception errors on weather hedging efficiency. We also compare the hedging effectiveness obtained using weather indices produced by both the Chicago Mercantile Exchange (CME) and Risk Management Solutions, Inc. (RMS). The results indicate that basis hedging is significantly more effective for the winter season than for the summer season, whether using the CME or RMS weather indices, and whether using linear or nonlinear derivative instruments. It is also found that the RMS regional weather indices are more effective than the CME weather indices, and the effectiveness of using either linear or nonlinear hedging instruments for weather risk management can vary significantly depending on the region of the country. In addition, the results indicate that default risk has some impact on nonlinear basis hedging efficiency but no impact on linear basis hedging efficiency, and reasonable perception errors on default risk have no impact on either linear or nonlinear basis hedging efficiency.

1. INTRODUCTION With the deregulation of the energy and power industries, energy companies face increased financial exposure due to weather changes. In a regulated, monopolistic setting the financial consequences of these risks could be absorbed in allowed pricing; with deregulation, such risks are now not necessarily compensated in this manner. To address this problem, weather-related financial instruments have been created.1 Essentially, certain market participants see they can address the adverse consequences of weatherrelated events in a manner similar to the securitization process used to mitigate the adverse consequences of commodity price risk in the commodity market. By measuring and indexing various weather recordings across the United States, one can create financial instruments whose pay off values are dependent upon the level of the recorded underlying weather index. Initially, only over-the-counter (OTC) trades on such derivative instruments could be made; however, in 1999 exchange-traded weather futures, and options on futures, began to be offered at the Chicago * Linda L. Golden, PhD, is the Marlene and Morton Myerson Centennial Professor in Business at the Red McCombs School of Business, University of Texas at Austin, Austin, TX 78712, [email protected]. † Charles C. Yang, PhD, is Assistant Professor in Insurance and Risk Management in the Department of Finance, College of Business, Florida Atlantic University, Boca Raton, FL 33431, [email protected]. ‡ Hong Zou, PhD, is Associate Professor in Finance in the Department of Economics and Finance, City University of Hong Kong, Hong Kong, China, [email protected]. 1 The first transaction was a weather derivative contract based on Heating Degree Days in Milwaukee between Koch Industries and Enron Corporation (see Climetrix at www.climetrix.com).

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Mercantile Exchange (CME). Although OTC transactions, being private transactions, carry default risk (the potential of counterparty default on the contract), CME transactions are contracts traded on the open market, are marked to the market, are guaranteed by the exchange, and do not carry default risk. On the other hand, by being constructed as standardized in form and based on the location of the weather station reading the underlying weather index used for the CME contract, the weather derivative contracts traded at CME possess a different risk: basis risk. Basis risk for a market participant using a particular weather-indexed derivative involves the possibility that the location of the source of weather data used to specify the weather index underlying the derivative contract may not precisely match the location of the enterprise’s exposure to weather-related losses. Managing weather risks by trading weather derivatives is a rapidly growing business. In a weather derivative transaction, variables such as temperature, precipitation, wind, or snow are measured and indexed covering a specified amount of time at a specified location. A threshold limit regarding the actionable level of the measured variable is agreed upon by the buyer and seller. The most common weather derivatives are contracts based on indices that involve Heating Degree Days (HDD) for the winter season and Cooling Degree Days (CDD) for the summer season. Using 65⬚F as the baseline, HDD and CDD values are determined by subtracting the day’s average temperature from 65⬚F for HDD and subtracting 65⬚F from the day’s average temperature for CDD values. If the temperature exceeds 65⬚F in the winter, the HDD is 0; if the temperature is lower than 65⬚F in the summer, the CDD is 0 (because generally one does not need to heat in the winter if the temperature is above 65⬚F, and one does not need to cool in the summer if the temperature is below 65⬚F). CME creates HDD and CDD derivative instruments based upon temperature index values centered in 15 U.S. cities (see Appendix A). Another company, Risk Management Solutions, Inc. (RMS),2 also supplies CDD and HDD indices for the management of weather risks. RMS produces 10 regional indices; each is created by averaging the temperature index values centered in 10 chosen cities within the region. Derivative instruments can be created based on the RMS indices as well. Because the financial impact of short bursts of cold or hot weather can be absorbed by most firms, most weather derivatives accumulate the HDDs or CDDs over a specified contract period, such as one week, one month, or a winter/summer season. To calculate the degree days over a multiday period, one aggregates the daily degree measure for each day in that period (cf. Golden et al. [2007] for the requisite formulas and for further discussion). Initially, energy companies were the main enterprises hedging weather risks by trading weather derivatives. Today, diverse enterprises such as resorts, hotels, restaurants, universities, governments, airlines, and farms are using weather derivatives to manage weather risks. As the market for managing weather risks continues to grow, the enterprises that hedge weather risk using exchange-traded contracts want to minimize their basis risk, and, conversely, those that hedge using OTC contracts want to minimize their counterparty default risk. In the literature Briys et al. (1993), Poitras (1993), Moschini and Lapan (1995), Vukina et al. (1996), Li and Vukina (1998), and Coble et al. (2000) have conducted studies related to the issue of optimal hedging of basis risk. Other studies have analyzed optimal hedging strategies when the hedging enterprise faces default risk (Hentschel and Smith 1997). Additional research has investigated the demand for insurance by policyholders who purchase insurance from credit-risky insurers but without consideration of basis risk (see, e.g., Tapiero et al. 1986; Doherty and Schlesinger 1990). An OTC derivative instrument called a basis derivative mitigates the basis risk inherent in exchangetraded weather derivatives and when combined with them lessens the hedger’s default risk exposure compared to just using an OTC weather derivative (MacMinn 1999; Considine 2000). This OTC basis derivative combines a local temperature index and an exchange-traded temperature index to moderate

2

Risk Management Solutions, Inc., is the world’s leading provider of products and services for the quantification and management of natural hazard risks.

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the impact of the temperature difference between the location of the exchange-traded derivative and the local weather index. Using a basis derivative in conjunction with an exchange-traded derivative can improve the basis risk exposure faced by the hedging enterprise. Although the hedging enterprise again faces default risk (basis derivatives are traded OTC), the basis derivative instrument is expected to be less volatile, being the difference between the local index and the exchange-traded index. This stability should lessen the enterprise’s default risk exposure compared to an OTC weather contract written directly on the local index. By setting up mathematical models and varying the model parameters in a simulation study, Golden et al. (2007) analyze the joint impact of basis risk and the tradeoff between basis risk and default risk regarding the effectiveness of the linear and nonlinear hedging models. The actual indices used for trade, however, may behave differently from the model-based indices, so it remains for us to examine the results for the real series and to ascertain if either of the two indices (CME or RMS) is superior regarding region, index, or season. Accordingly, this paper empirically analyzes the effectiveness of using linear and nonlinear 3 basis hedging instruments developed using either the CME or RMS weather indices. Effectiveness is examined by region of the country, source of the weather index used (CME or RMS), and season of the year (summer or winter), as well as by examining any potential interactive effects, such as region by index, season by index, region by season, or region by index by season. Both the hedger and the issuer are aware that the probability of the issuer’s default is nonzero. However, because such an event is uncertain, the economic agents may have perceptions different from the issuer’s real default risk. Another advancement in the paper is that it takes into consideration the agents’ subjective perception errors on the issuers’ default risk and analyzes the impacts of this perception error on basis hedging efficiency. Finally, in contrast to Golden et al. (2007), which fixes the strike levels of options at the expected values of the underlying weather indexes, the current paper allows the strike levels of options to be decision variables, leading to different results for the impact of default risk on nonlinear basis hedging efficiency.

2. WEATHER HEDGING MODELS We follow the hedging models from Golden et al. (2007) with some modifications to incorporate perception errors on default risk. We assume the enterprise is attempting to hedge the negative effects that weather-related events may have on the quantity demand of their products or services (e.g., energy producers need to hedge demand risk due to variations in the quantity of energy demand as a function of the weather, or a ski resort has an interest in hedging against too many warm weather days). We assume that the risk in the quantity demand (q) is mainly affected by a weather event, and we assume that the hedging enterprise wishes to choose weather derivatives to maximize or minimize an objective function of the end-of-period wealth. To hedge against the risk of a weather-related loss, an enterprise will choose one of the following strategies: A. Use an OTC derivative based on the weather index of the site where the hedger’s business is located. This carries counterparty default risk but no basis risk. B. Use an exchange-traded derivative based on a regional weather index or one of CME’s 15 city weather indices. This strategy carries basis risk but no default risk.

3 By linear hedging instruments we mean derivative contracts such as futures and forwards whose payoff is linear in structure. By nonlinear hedging instruments we mean derivative contracts such as options whose payoff is nonlinear (e.g., zero below the exercise threshold and linear thereafter).

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C. Employ a combination strategy and buy an exchange-traded derivative to mitigate the majority of the weather-related risk at minimal default risk, and then supplement this with an OTC basis hedging derivative designed to decrease the enterprise’s basis risk.4 This third strategy (C) is intended to reduce the risk of weather-related loss while simultaneously improving the basis risk by creating a hedge that corrects for the difference between CME indices and OTC local weather indices. Analyzing the effectiveness of this third strategy is the focus of this paper. To analyze the impact of default risk, we need to model the likelihood of counterparty default. To this end, let ␪ denote the proportion of the required payoff on a weather derivative that will actually be paid to the hedging enterprise. Here we take ␪ as either 1 or 0 (i.e., the counterparty either performs entirely or fails to perform entirely). The probability distribution of ␪ is defined as Prob(␪ ⫽ 1) ⫽ p and Prob(␪ ⫽ 0) ⫽ 1 ⫺ p. The default risk is represented by Prob(␪ ⫽ 0) ⫽ 1 ⫺ p. The hedging enterprise and the weather derivative issuer may have different perceptions about the issuer’s default risk, but through the process of resolving a price and entering into a weather derivative contract, they essentially agree to a subjective probability distribution for the issuer’s default risk (possibly only implied). This subjective or market probability distribution is defined as Prob(␪ ⫽ 1) ⫽ ps and for default risk Prob(␪ ⫽ 0) ⫽ 1 ⫺ ps. The subjectively agreed upon common nondefault parameter ps may or may not be the same as the ‘‘true’’ or real likelihood of nondefault p. Our analysis employs the risk/variance minimization objective function. The variance of the hedger’s end-of-period wealth (W) is denoted by v(W) ⫽ ␴2(W). To compare the effectiveness of using a basis derivative, we first consider the baseline situation, A above, wherein a linear OTC weather derivative contract drawn on a local weather index is used. Let Tl be the strike price of the contract, and let tl denote the local weather index used to write the contract. Let r pl ⫽ max(Tl ⫺ tl, 0) denote the payoff of a put option on the weather index and r cl ⫽ max(tl ⫺ Tl, 0) denote the payoff of a call option on the weather index. Let hl⫺ll denote the hedge ratio used for the local weather forward derivative contract. Without consideration of default risk, the final wealth resulting from hedging using a forward contract with a price of ␲l⫺ll is simply W ⫽ q ⫺ hl⫺ll␲l⫺ll ⫹ hl⫺ll(Tl ⫺ tl ), where q represents the (uncertain and weather-related) demand quantity.5 However, taking into consideration the possibility of default risk, this final wealth is ␪ W ⫽ q ⫺ hl⫺ll␲l⫺ll ⫹ hl⫺ll(␪r lp ⫺ r lc),6

(1)

because default on the contract occurs only if the counterparty is forced to pay (when Tl ⬎ tl ). Here ␲␪l⫺ll denotes the price of the weather forward with consideration of default risk. Suppose now that the weather-related quantity of goods or services demanded of the enterprise employing the hedging strategy can be decomposed such that there is a portion that is systematically related to the local weather index and a portion that constitutes nonsystematic or idiosyncratic nonweather-dependent individual variation in demand (see, e.g., Davis 2001). This may occur, for example, because of the particular sensitivities of the hedging enterprise (e.g., better insulation, better snow

4

The hedger could also hedge a significant portion of its credit risk, for example, by using a credit default swap written on the weather derivative counterparty. This research assumes that the credit risk is not hedged and analyzes its impact on the effectiveness of weather hedging. The incorporation of credit derivatives will possibly be affected by incoming potential regulations. This legislation, as currently understood, will not necessarily affect the use or effectiveness of basis weather derivatives. 5 Because weather derivatives are designed to hedge weather risks/quantity risks, this paper focuses on quantity risk and analyzes the effectiveness of weather derivatives in reducing this risk. Furthermore, if a hedger considers using weather derivatives to hedge its quantity risk, most probably, no significant correlation exists between the price and quantity risk, and price derivatives are not effective to hedge the quantity risk. Therefore, it is reasonable to consider only quantity risk in weather hedging. However, if one finds significant correlation between the price and quantity risk, both the price and quantity risk (and both weather derivatives and price derivatives) should be included in the hedging model. The tick sizes of all the weather derivatives in this paper are assumed to be 1. 6 Because of the symmetrical structure of the weather forward/future, which consists of a call option and a put option, similar analysis can be conducted on the impact of the hedger’s default probability (and similar results should be obtained). This paper considers only the counterparty’s default probability to avoid repetitive analysis (and for a clear presentation).

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plows). We express this relationship as q ⫽ ␣ ⫹ ␤tl ⫹ ε, where ε ⬃ N(0, ␴2ε ) is the nonsystematic quantity risk, which is assumed to be independent of the weather indices. The optimal hedging problem is MINhl⫺ll␴2(W), subject to Prob(␪ ⫽ 1) ⫽ ps. Performing the maximization using (1), we derive the optimal hedge ratio as7 h*l⫺ll ⫽

⫺ps␤␴tl ,r pl ⫹ ␤␴tl ,r cl 2p s rl

2 r lc

p ␴ ⫹ ␴ ⫹ (ps ⫺ p2s )␮2r lp ⫺ 2ps␴r lp,r lc

.

(2)

The minimum variance of the hedger’s final wealth when using the OTC contract and the linear local hedging strategy with default risk is Vmin⫺ll ⫽ Var[W(h*l⫺ll)], subject to Prob(␪ ⫽ 1) ⫽ p; that is, 2 2p 2 2 2 2 2p 2 p p Vmin⫺ll ⫽ ␤2␴t2l ⫹ ␴ε2 ⫹ h* l⫺ll p␴r l ⫹ h* l⫺ll ␴r lc ⫹ h* l⫺ll(p ⫺ p )␮r l ⫺ 2ph* l⫺ll␴r l ,r lc ⫹ 2h* l⫺ll␤p␴tl ,r l ⫺ 2h* l⫺ll␤␴tl ,t lc.

(3) We turn now to the exchange-traded weather hedging alternative involving the use of an exchangetraded linear contract. This contract involves basis risk, so we also add an adjustment compensating for basis risk via a basis derivative risk hedge that ‘‘corrects’’ for the difference tl ⫺ te between the local weather index tl and the index te used by the exchange in creating a traded derivative. This ‘‘correction’’ is done by creating an OTC contract with strike price Td based on the difference in indices td(td ⫽ tl ⫺ te). The resulting hedging model (for strategy C) is W ⫽ q ⫺ he⫺lb␲e⫺lb ⫺ hd⫺lb␲␪d⫺lb ⫹ he⫺lb(Te ⫺ te) ⫹ hd⫺lb(␪r pd ⫺ r cd),

(4)

where r pd ⫽ max(Td ⫺ td, 0) and r dc ⫽ max(td ⫺ Td, 0) with the put and call option designation being symbolized by superscripts. Here he⫺lb and hd⫺lb are the hedge ratios of the exchange-traded weather futures contract and the basis weather forward, and ␲e⫺lb and ␲␪d⫺lb are the price of these weather contracts, respectively. The optimal hedging problem in strategy C involves optimally selecting a hedge ratio for both the exchange-traded derivative and the basis derivative simultaneously; that is, MINhe⫺lb,hd⫺lb␴2(W), subject to Prob(␪ ⫽ 1) ⫽ ps. Upon minimization, the optimal linear hedge ratios are determined to be h*d⫺lb ⫽

␤␴tl ,te(ps␴te,r pd ⫺ ␴te,r cd) ⫹ ␤␴2te(␴tl ,r cd ⫺ ps␴tl ,r pd )

(5)

␴2te[ps␴2r dp ⫹ (ps ⫺ p2s )␮2r dp ⫹ ␴2r dc ⫺ 2ps␴r dp ,r dc ] ⫺ (ps␴te,r dp ⫺ ␴te,r dc )2

and h*e⫺lb ⫽

␤(ps␴te,r dp ⫺ ␴te,r cd)(␴tl ,r cd ⫺ p␴tl ,r dp) ⫹ ␤␴tl ,te[ps␴2r pd ⫹ ␴2r cd ⫹ (ps ⫺ p2s )␮2r pd ⫺ 2ps␴r pd ,r cd] ␴t2e[ps␴r2pd ⫹ (ps ⫺ ps2)␮r2pd ⫹ ␴r2cd ⫺ 2ps␴r pd,r cd] ⫺ (ps␴te,r pd ⫺ ␴te,r cd)2

.

(6)

Similarly, the minimum variance of the final wealth when using linear basis hedging is Vmin⫺lb ⫽ Var[W(h*e⫺lb, h*d⫺lb)], subject to Prob(␪ ⫽ 1) ⫽ p; that is, 2 2 2 2 2 2 2 2 2 2 Vmin⫺lb ⫽ ␤2␴t2l ⫹ ␴2ε ⫹ h* e⫺lb␴te ⫹ h* d⫺lb p␴r pd ⫹ h* d⫺lb␴r cd ⫹ h* d⫺lb(p ⫺ p )␮r pd ⫺ 2ph* d⫺lb␴r pd ,r cd p ⫺ 2h*e⫺lb␤␴tl ,te ⫹ 2h*d⫺lb␤p␴tl ,r dp ⫺ 2h* d⫺lb␤␴tl ,r dc ⫺ 2h* e⫺lb h* d⫺lb p␴te,r d ⫹ 2h* e⫺lb h* d⫺lb␴td ,r dc .

(7)

Turning now to the nonlinear local and basis hedging models (i.e., those involving instruments with nonlinear payoff functions such as put and call options),8 we write 7

Details of this and other optimizations not provided in the paper can be obtained from the authors. This research assumes that a hedger uses ‘‘the most effective weather contract’’ instead of a mix of weather contracts based on different locations or with different strike prices to hedge its weather risk. However, it would be interesting to test if a portfolio of different weather contracts is more effective than the most effective weather contract. 8

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W ⫽ q ⫺ hl⫺nl␲␪l⫺nl ⫹ ␪hl⫺nl r pl

(8)

W ⫽ q ⫺ hd⫺nb␲␪d⫺nb ⫺ he⫺nb␲e⫺nb ⫹ he⫺nbr ep ⫹ ␪hd⫺nbr dp,

(9)

and

where nl and nb denote the nonlinear local hedging and the nonlinear basis hedging, respectively. In a manner similar to the derivations in Golden et al. (2007) and to the derivation of equations (2)–(3) and (5)–(7), we can obtain the optimal nonlinear hedge ratios and the nonlinear local and basis minimum final wealth variances: h*l⫺nl ⫽ h*d⫺nb ⫽ h*e⫺nb ⫽

⫺␤␴tl ,r p* l ␴2r p* ⫹ (1 ⫺ ps)␮r2p* l l

,

p* ⫺ ␤␴ ␤␴tl ,r p* ␴r p* ␴r2p* tl ,r p* e e ,r d d e

␴2r p* ␴2r dp*(1 ⫺ ps␳2r ep*,r p* ) ⫹ (1 ⫺ ps)␴2r p* ␮2r dp* e d e

(10) ,

2 p* ⫺ ␤␴ (ps ⫺ 1)␤␮2r dp*␴tl ,r p* ⫹ ps␤␴tl ,r dp*␴r p* tl ,r ep*␴r dp* e e,r d 2 2 p*) ⫹ (1 ⫺ p )␴ p*␮ p* ␴2r ␴r2p* (1 ⫺ ps␳r2p* s re rd d e,r d p* e

(11) ,

2p* 2 2 2p* p* Vmin⫺nl ⫽ ␤2␴t2l ⫹ ␴ε2 ⫹ h* l⫺nl p␴r l ⫹ h* l⫺nl(p ⫺ p )␮r l ⫹ 2h* l⫺nl␤p␴tl ,r l ,

(12) (13)

and 2 2 2 2 2 2 Vmin⫺nb ⫽ ␤2␴t2l ⫹ ␴2ε ⫹ h*e⫺nb ␴2r ep* ⫹ h* d⫺nbp␴r dp* ⫹ h* d⫺nb(p ⫺ p )␮r dp* ⫹ 2h* e⫺nb␤␴tl ,r ep* p*, ⫹ 2h*d⫺nb␤p␴tl ,r p* ⫹ 2h* e⫺nbh* d⫺nbp␴r p* d e,r d

(14)

p* p* 9 where r p* l , r e , and r d are the payoffs of the put options with optimal strike prices. The hedging effectiveness of the linear and nonlinear basis hedging is measured by

Vmin⫺ll ⫺ Vmin⫺lb Vmin⫺ll

(15)

Vmin⫺nl ⫺ Vmin⫺nb , Vmin⫺nl

(16)

and

respectively. These ratios express the extent to which the use of the basis derivative in conjunction with an exchange-traded derivative (strategy C) has reduced the uncertainty in final wealth compared to the simpler hedging strategy A that uses a local weather index with associated default risk. Although the formulas to this point are general, in the numerical empirical results that follow we shall assume a strong dependence of demand on weather (␴2ε ⫽ 0) and, by rescaling if necessary, that ␤ ⫽ 1.

3. WEATHER DATA USED

FOR

EMPIRICAL ANALYSES

CME offers futures and options on futures for 15 selected cities throughout the United States (see Appendix A).10 CME selected these cities based on population, variability in their seasonal temperatures, and their extensive use in OTC derivative trading. The RMS framework comprises 10 indices made up of data gathered from 10 regions with 10 cities in each region. The RMS selected these regions because they represent significant meteorological patterns, predominant weather risk trading places, population, and correlations with other weather stations throughout the regions. To construct a

9

The optimal strike prices were determined by a global search over all possible strike values in the process of risk minimization. Data for degree days from 1978 to 2004 are provided by Earth Satellite Corporation, the weather data provider to CME and RMS.

10

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weather index for a particular region, RMS collects HDD and CDD data from 10 city weather stations within the region and averages them.11 One hundred separate city weather records are used in the RMS data set. Of these, 13 are locations that are also used by the CME derivative products (and hence would have no basis risk when compared to the CME index). Accordingly, we have used the 87 remaining city locations for the comparative assessment of basis risk hedging effectiveness. The most common contract terms in the weather derivative market are seasonal (summer season and winter season), monthly, and weekly.12 In this paper, we consider only seasonal contracts.13 The CME index is either an HDD or CDD index for the particular city being traded. We compare the RMS indices with the CME indices to determine which index provides better basis hedging results.

4. BASIS HEDGING EFFICIENCY The analysis presented in this paper concerns the effectiveness of basis hedging under the assumption that the subjective perceived counterparty nondefault probability ps (which was incorporated into the weather derivative contract’s price) is the same as the actual likelihood of nondefault p, that is, when the market implied default risk is the true default risk. Three default risk levels are considered: Prob(␪ ⫽ 0) ⫽ 1 ⫺ p ⫽ 0.10, 0.05, and 0.01. Appendix B presents the computed effectiveness measure for the 87 cities in the summer and winter seasons using both the CME and the RMS weather indices and utilizing both linear and nonlinear basis hedging at the 0.05 default risk level.14 With regard to the computation of the basis hedging effectiveness for any of the 87 local cities, the basis hedging effectiveness can be compared against 15 different CME-based city indices (with derivative instruments). CME table entries in Appendix B are the maximum comparison measurements obtained using the 15 CME city indices. This is done because the closest of the 15 cities geographically may not be the best city index to use for basis risk hedging. Similarly, for any given city, 10 measurements of basis hedging effectiveness are obtained by using the 10 RMS regional weather indices, and the RMS basis hedging effectiveness is defined as the maximum of these 10 measurements.15 Generally, default risk has no impact on linear basis hedging efficiency. For example, the hedging efficiency for city 15 (Madison, WI) is 0.94 in the winter season and 0.77 in the summer season for all three default risk levels. For some cities, a minor difference is seen in the hedging efficiency for the summer season. For example, the hedging efficiency for city 6 (Richmond, VA) is 0.49, 0.48, and 0.47 for the three default risk levels, respectively. Compared to linear basis hedging efficiency, nonlinear basis hedging efficiency is affected more significantly by default risk. For some cities, nonlinear basis hedging efficiency increases with an increase in default risk. For example, the nonlinear basis hedging efficiency is 0.80, 0.85, and 0.87 for the winter season for city 14 (Louisville, KY), and 0.56, 0.62, and 0.67 for the summer season for city 12 (Grand Rapids, MI), for the three default risk levels of 0.01, 0.05, and 0.10, respectively. For other

11

The RMS regional indices are not currently exchange listed; however, OTC contracts can be created using them. See Climetrix at www.climetrix.com. 13 Winter season is November 1 through March 31, and summer season is May 1 through September 30. April and October are often referred to as the ‘‘shoulder months’’ and are not considered in this paper. 14 The results at the other two default risk levels are not presented because of space limitations but are available from the authors. These do not affect the relevance and results of this research as to whether the default risk is hedged or not. The basis hedging strategy is always more effective for most of the cities at all three default risk levels considered (0.10, 0.05, and 0.01) and should also be more effective if there is no default risk. Reasonable perception errors on default risk have no impact. Other analyses are conducted for one specific default risk level (0.05), which can also be 0.01 or ‘‘no default risk.’’ 15 When the sample is a random sample, there might be a significant positive bias. However, our sample is not a random sample. The group of cities are selected to represent all the different regional weather patterns (weather risks) in the United States. For any of the representative weather patterns/risks, the most effective weather contract should be selected for hedging. Therefore the hedging effectiveness is defined as the maximum of the RMS hedging effectiveness measurements (when the hedger uses the contracts based on the RMS regional indexes) or the maximum of the CME effectiveness measurements (when the hedger uses the contracts based on the CME city indexes). 12

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Figure 1 Difference in the Nonlinear Basis Hedging Efficiency for the Winter Season between Default Risk Levels of 0.10 and 0.01 (NHE 0.10 ⴚ NHE 0.01) 0.30 0.25

NHE0.10 - NHE0.01

0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 1

11

21

31

41

51

61

71

81

Cities Note: Miami, FL, with a value of 2.99 is not shown in this figure for a better illustration.

cities, nonlinear basis hedging efficiency decreases with an increase in default risk. For example, the nonlinear basis hedging efficiency is 0.74, 0.70, and 0.68 for the winter season for city 20 (Cheyenne, WY), and 0.90, 0.88, and 0.85 for the summer season for city 40 (Fargo, ND), for the three default risk levels of 0.01, 0.05, and 0.10, respectively. The difference in nonlinear basis hedging efficiency between two of the three default risk levels (0.10 and 0.01) is displayed in Figures 1 and 2. The difference ranges from ⫺0.10 to 0.05 for the winter season, and ⫺0.10 to 0.30 for the summer season,

Figure 2 Difference in the Nonlinear Basis Hedging Efficiency for the Summer Season between Default Risk Levels of 0.10 and 0.01 (NHE 0.10 ⴚ NHE 0.01) 0.6 0.5

NHE 0.10 - NHE 0.01

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 1

11

21

31

41

51

61

71

81

Cities Note: Olympia, WA, with a value of 0.93 and Seattle, WA, with a value of 1.29 are not shown in this figure for a better illustration.

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for most of the cities. This indicates that, for the nonlinear basis hedging in the winter season, the impact of default risk is more significant for those cities whose nonlinear basis hedging efficiency decreases with an increase in default risk, whereas for the nonlinear basis hedging in the summer season, the impact of default risk is more significant for those cities whose nonlinear basis hedging efficiency increases with an increase in default risk. A three-way analysis of variance (ANOVA)16 ascertained if region of the country, season of the year, or weather index used (CME or RMS) had a statistically significant impact on the effectiveness of using basis risk derivatives with respect to hedging the uncertainty in final wealth. Separate ANOVAs were carried out for linear hedging (eq. 15) and nonlinear hedging (eq. 16) effectiveness. Also considered was the possibility of interaction effects (e.g., it might be that RMS is more effective in the Northeast during the winter), so interactions were also assessed statistically. The outcomes of these analyses are presented in Table 1. As can be observed, the models incorporating region, season, and index explain a highly significant amount of the variance in the effectiveness of hedging uncertainty in both the linear hedging and nonlinear hedging contexts (p ⬍ 0.0001). Moreover, the main effects of region, index used, and season were all statistically significant for both linear and nonlinear hedging methods. This implies that the levels of basis risk hedging effectiveness differ significantly across regions, seasons, and indices used. For example, the average effectiveness measure for the CME index in the winter season in the Northeast region is much higher than the average effectiveness for the CME index in the winter season in the Southeast region. Figure 3 presents the average effectiveness measure for linear basis risk hedging across seasons and across regions, and Figure 4 presents the same information for nonlinear basis risk hedging.

Table 1 ANOVA of the Effectiveness of Basis Risk Hedging According to Region, Index, and Season for Linear Hedging Instruments (Forwards and Futures) and Nonlinear Hedging Instruments (Options) Source

DF

Sum of Squares

F-Value

p⬍

2.48 5.86 9.81 11.01 0.45 1.68 2.12 0.21

0.0001 0.0001 0.0019 0.0010 0.9082 0.0921 0.1464 0.9928

6.10 13.57 21.24 64.69 1.09 1.12 6.21 0.41

0.0001 0.0001 0.0001 0.0001 0.3681 0.3492 0.0132 0.9271

a

Modelb Region Index Season Region ⫻ index Region ⫻ season Index ⫻ season Three-way interaction Error

39 9 1 1 9 9 1 9 308

Linear Hedging Effectiveness 27.00 14.71 2.74 3.07 1.12 4.23 0.59 0.53 85.99

Nonlinear Hedging Effectivenessc d

Model Region Index Season Region ⫻ index Region ⫻ season Index ⫻ season Three-way interaction Error

39 9 1 1 9 9 1 9 308

14.28 7.33 1.28 3.89 0.59 0.60 0.37 0.22 18.50

a

The dependent variable is the effectiveness measure (15). R2 ⫽ 0.24. c The dependent variable is the effectiveness measure (16). d 2 R ⫽ 0.44. b

16

Only the results for the default risk level 1 ⫺ p ⫽ 0.05 are presented for the ANOVA analysis.

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Figure 3 Average Effectiveness of Linear Basis Hedging 1.2

1

0.8

Effectiveness

0.6 Winter CME Winter RMS

0.4

Summer CME Summer RMS

0.2

0

-0.2

-0.4 Region

Figure 4 Average Effectiveness of Nonlinear Basis Hedging 1

0.9

0.8

Effectiveness

0.7

0.6 Winter CME Winter RMS

0.5

Summer CME Summer RMS

0.4

0.3

0.2

0.1

0 MidAtlantic

Midwest

Mountain States

Northeast

Northern Plains

Pacific Northwest

Region

Plains

South Central

Southeast

Southwest

THE EFFECTIVENESS

OF

USING

A

BASIS HEDGING STRATEGY

167

Figure 5 Difference in CME Linear Basis Risk Hedging Effectiveness between Winter and Summer Seasons (LHE winter ⴚ LHEsummer) 1.6 1.4

LHEwinter - LHEsummer

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 1

11

21

31

41

51

61

71

81

Cities Note: Miami, FL, with a value of ⫺6.13 is not shown in this figure.

Figure 6 Difference in CME Nonlinear Basis Risk Hedging Effectiveness between Winter and Summer Seasons (NHE winter ⴚ NHEsummer) 1.4 1.2

NHE winter - NHE summer

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 1

11

21

31

41 Cities

Note: Miami, FL, with a value of ⫺1.27 is not shown in this figure.

51

61

71

81

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From the ANOVA table we observe that there is also a marginally significant interaction between region and season for linear hedging instruments and a significant interaction between index used and season for nonlinear hedging instruments. All other interaction effects were not statistically significant. To illustrate the seasonal effect on hedging effectiveness using linear derivatives, Figure 5 presents the difference in linear basis hedging effectiveness for the winter and summer seasons. We observe that the linear basis hedging for the winter season is generally more effective than for the summer season. Similarly, Figure 6 shows the difference between the effectiveness for the CME index using nonlinear hedging instruments, and it again shows that basis risk hedging is more effective during the winter than during the summer. The ANOVA results show that these differences are significant.

5. EFFECTIVENESS HEDGING

OF

CME INDICES COMPARED

TO

RMS INDICES

FOR

BASIS

The difference in the linear basis hedging effectiveness for winter and summer seasons using the CME and the RMS indices is displayed in Figures 7 and 8. Consistent with the significant main effect of index in the ANOVA, we observe that, in general, linear basis hedging is more effective using weather derivatives written on the RMS indices than the CME indices. For the winter season, linear basis hedging is more effective using weather derivatives written on the RMS indices for 67 of the 87 cities, and there are nearly 30 cities whose linear basis hedging effectiveness is at least 0.10 higher using the RMS indices than using the CME indices. The difference is only 0.05 or less for 13 of the 20 cities whose linear basis hedging effectiveness is higher using the CME indices. The difference in linear basis hedging effectiveness between the CME and the RMS indices is more pronounced for the summer season. The linear basis hedging model is more effective using weather derivatives written on the RMS indices for 80 of the 87 cities for the summer season, and there are more than 60 cities whose linear basis hedging effectiveness is at least 0.10 higher using the RMS indices. Similar results apply to the nonlinear basis hedging model. The difference in the nonlinear basis hedging effectiveness between the CME and the RMS indices for winter and summer seasons is displayed in Figures 9 and 10. Generally, the nonlinear basis hedging is also more effective using weather deriv-

LHERMS - LHECME

Figure 7 Difference in Linear Basis Hedging Effectiveness between the CME and the RMS Winter Season Indices (LHE RMS ⴚ LHE CME) 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 1

11

21

31

41 Cities

51

61

71

81

THE EFFECTIVENESS

OF

USING

A

BASIS HEDGING STRATEGY

169

LHERMS - LHECME

Figure 8 Difference in the Linear Basis Hedging Effectiveness between the CME and the RMS Summer Season Indices (LHE RMS ⴚ LHE CME) 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 1

11

21

31

41

51

61

71

81

Cities

atives written on the RMS indices than the CME indices. In addition, the difference exhibits a pattern similar to the linear basis hedging model effectiveness. The nonlinear basis hedging model is more effective using weather derivatives written on the RMS indices for 64 of the 87 cities for the winter season, and 75 of the 87 cities for the summer season.

Figure 9 Difference in Nonlinear Basis Hedging Effectiveness between the RMS and the CME Winter Season Indices (NHE RMS ⴚ NHE CME) 1.2 1.0

NHERMS - NHECME

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 1

11

21

31

41 Cities

51

61

71

81

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Figure 10 Difference in Nonlinear Basis Hedging Effectiveness between the RMS and the CME Summer Season Indices (NHE RMS ⴚ NHE CME) 1.0

NHERMS - NHE CME

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 1

11

21

31

41

51

61

71

81

Cities

6. IMPACTS

OF

PERCEPTION ERRORS

This section examines the impact of perception errors on basis hedging efficiency using the CME weather indexes. Four different levels of perception errors are analyzed: ⫺0.04, ⫺0.02, 0.02, and 0.04 with the real default risk assumed to be 0.05. A sampling of the results is presented in Table 2. These empirical results indicate that reasonable perception errors have no impact on either linear or nonlinear basis hedging efficiency. For example, the linear basis hedging efficiency is 0.93 for the winter season and 0.51 for the summer season for all the four levels of perception errors for Providence, RI (city 35); and the nonlinear basis hedging efficiency is 0.89 for the winter season and 0.78 for the summer season for all four levels of perception errors for Sioux Falls, SD (city 45). It is interesting to note that there is no difference in the basis hedging efficiency between a positive perception error (say, 0.04) and a negative perception error (say, ⫺0.04) for any given city. One extreme case of perception errors is that the perceived probability (ps) is always 1; that is, the investors believe that the real default risk is 0 even though it is not. This extreme case is defined as ‘‘unanticipated’’ default risk in this paper. Because the default risk is unanticipated, it is not incorporated into the premium of weather contracts. We now present some empirical results concerning the impact of unanticipated default risk on basis hedging efficiency. The unanticipated default risk levels considered are 0.10, 0.05, and 0.01. The results indicate that unanticipated default risk has no impact on linear basis hedging efficiency, as is the same as the impact of the anticipated default risk. For example, the linear hedging efficiency is 0.91 for the winter season and 0.45 for the summer season for Albany, NY (city 28) for all the three levels of the unanticipated default risk. It is also interesting to note that linear hedging efficiency is almost the same for any given city no matter whether the default risk is anticipated or not. For example, the linear basis hedging efficiency for Rapid City, SD (city 43) is always 0.79 for the winter season and 0.63 for the summer season when there is no perception error or when default risk is unanticipated. For a given default risk, nonlinear basis hedging efficiency is also almost the same for a given city no matter whether the default risk is anticipated or not. For example, at the 0.10 default risk level, the nonlinear basis hedging efficiency for Charleston, WV (city 3) is 0.83 for the winter season and 0.77 for the summer season when there is no perception error, and 0.82 for the winter season and 0.76 for the summer season when the default risk is unanticipated. Some difference is seen in the

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OF

USING

A

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171

Table 2 Impact of Perception Errors on the Basis Hedging Efficiency: 0.05 Real Default Risk Level Linear Basis Hedging

Nonlinear Basis Hedging

City

Perception Error

Winter

Summer

Winter

Summer

5

⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04 ⫺0.04 ⫺0.02 0 0.02 0.04

0.92 0.92 0.92 0.92 0.92 0.94 0.94 0.94 0.94 0.94 0.58 0.58 0.58 0.58 0.58 0.93 0.93 0.93 0.93 0.93 0.88 0.88 0.88 0.88 0.88 0.86 0.87 0.87 0.87 0.86 0.86 0.86 0.86 0.86 0.86 0.64 0.64 0.64 0.64 0.64 0.15 0.15 0.15 0.15 0.15

0.65 0.66 0.66 0.66 0.65 0.77 0.77 0.77 0.77 0.77 0.21 0.21 0.21 0.21 0.21 0.51 0.51 0.51 0.51 0.51 0.76 0.76 0.76 0.76 0.76 0.65 0.66 0.66 0.66 0.65 0.51 0.51 0.51 0.51 0.51 0.37 0.38 0.38 0.38 0.37 ⫺0.03 ⫺0.03 ⫺0.03 ⫺0.03 ⫺0.03

0.91 0.91 0.91 0.91 0.91 0.94 0.94 0.94 0.94 0.94 0.77 0.77 0.77 0.77 0.77 0.88 0.88 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.89 0.68 0.68 0.68 0.68 0.68 0.65 0.65 0.65 0.65 0.65

0.82 0.82 0.83 0.82 0.82 0.88 0.88 0.88 0.88 0.88 0.34 0.34 0.34 0.34 0.34 0.87 0.87 0.87 0.87 0.87 0.78 0.78 0.78 0.78 0.78 0.85 0.85 0.85 0.85 0.85 0.64 0.64 0.64 0.64 0.64 0.72 0.72 0.72 0.72 0.73 0.20 0.20 0.20 0.20 0.20

15

25

35

45

55

65

75

85

nonlinear basis hedging efficiency among different unanticipated default risk levels, and the difference is of a similar pattern to that among different default risk levels with no perception errors. For some cities, nonlinear basis hedging efficiency increases with an increase in unanticipated default risk. For example, the nonlinear basis hedging efficiency is 0.25, 0.33, and 0.38 for the winter season for Winslow, AZ (city 87), and 0.27, 0.43, and 0.48 for the summer season for Richmond, VA (city 6), for the three unanticipated default risk levels of 0.01, 0.05, and 0.10, respectively. For some other cities, the nonlinear basis hedging efficiency decreases with an increase in unanticipated default risk. For example, the nonlinear basis hedging efficiency is 0.74, 0.66, and 0.59 for the winter season for Los Angeles, CA (city 85), and 0.75, 0.69, and 0.64 for the summer season for Tulsa, OK (city 71), for the three unanticipated default risk levels of 0.01, 0.05, and 0.10, respectively.

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7. CONCLUSION This paper has explored the effectiveness of using a basis derivative hedging strategy (involving either linear or nonlinear derivative instruments) in an attempt to mitigate both default and basis risks inherent in OTC and exchange-traded weather derivatives. We find, using actual weather data, that linear and nonlinear basis hedging are both much more effective for the winter season than for the summer season, a finding that should be of use to potential hedgers deciding whether or not to use weather derivatives and how to implement a weather derivative strategy. This article shows that the effectiveness of hedging using a basis derivative varies significantly across regions of the country. Moreover, when it comes to the difference in effectiveness between the two most popular types of standardized weather indices, generally the RMS regional indices are more effective than the CME city indices for implementing a basis hedging strategy. These results have important implications for determining which type of indices to use to create a hedging strategy to control weather-related financial consequences for enterprises contemplating weather risk management in different parts of the United States and in different seasons. The results also indicate that default risk has some impact on nonlinear basis hedging efficiency but no impact on linear basis hedging efficiency, and reasonable perception errors on default risk have no impact on both linear and nonlinear basis hedging efficiency. It is very interesting to note that linear basis hedging efficiency as well as nonlinear basis hedging efficiency are almost the same for a weather risk in most cities no matter whether default risk is anticipated or unanticipated.

APPENDIX A THE CME TRADING CITIES

AND

WEATHER STATIONS

Station Name

Symbol

Region

State

City

Atlanta Hartsfield International Airport Boston Logan International Airport Chicago O’Hare International Airport Cincinnati Northern Kentucky Airport Dallas–Fort Worth International Airport Des Moines International Airport Houston Bush Intercontinental Airport Kansas City International Airport Las Vegas McCarran International Airport Minneapolis–St. Paul International Airport New York La Guardia Airport Philadelphia International Airport Portland International Airport Sacramento Executive Airport Tucson International Airport

ATL BOS ORD CVG DFW DSM IAH MCI LAS MSP LGA PHL PDX SAC TUS

Southeast Northeast Midwest Midwest South Central Plains South Central Plains Southwest Northern Plains Northeast Mid-Atlantic Pacific Northwest Southwest Southwest

GA MA IL KY TX IA TX MO NV MN NY PA OR CA AZ

Atlanta Boston Chicago Covington Dallas Des Moines Houston Kansas City, MO Las Vegas Minneapolis New York Philadelphia Portland Sacramento Tucson

APPENDIX B THE EFFECTIVENESS

OF

BASIS HEDGING

FOR

87 CITIES

Linear Basis Hedging Winter Region

State

Mid-Atlantic Mid-Atlantic Mid-Atlantic Mid-Atlantic Mid-Atlantic Mid-Atlantic

NJ MD WV NJ PA VA

City Atlantic City Baltimore Charleston Newark Pittsburgh Richmond

CME 0.88 0.88 0.68 0.91 0.92 0.8

Nonlinear Basis Hedging

Summer

RMS 0.94 0.9 0.85 0.95 0.93 0.91

CME ⫺0.1 0.77 0.81 0.57 0.66 0.48

RMS 0.16 0.89 0.76 0.74 0.83 0.77

Winter CME 0.9 0.88 0.82 0.91 0.91 0.8

Summer

RMS 0.89 0.89 0.86 0.93 0.88 0.93

CME 0.03 0.78 0.77 0.64 0.83 0.44

RMS 0.28 0.8 0.8 0.7 0.86 0.77

THE EFFECTIVENESS

OF

USING

A

BASIS HEDGING STRATEGY

173

APPENDIX B (CONTINUED) THE EFFECTIVENESS

OF

BASIS HEDGING

FOR

87 CITIES

Linear Basis Hedging Winter Region

State

City

Mid-Atlantic Mid-Atlantic Mid-Atlantic Midwest Midwest Midwest Midwest Midwest Midwest Midwest Midwest Mountain States Mountain States Mountain States Mountain States Mountain States Mountain States Mountain States Mountain States Mountain States Mountain States Northeast Northeast Northeast Northeast Northeast Northeast Northeast Northeast Northeast Northern Plains Northern Plains Northern Plains Northern Plains Northern Plains Northern Plains Northern Plains Northern Plains Northern Plains Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Pacific Northwest Plains Plains Plains Plains Plains Plains Plains Plains Plains South Central South Central South Central South Central South Central

VA VA PA OH MI MI IN KY WI WI IL ID WY WY CO CO ID CO NV UT NV NY VT NH CT NY NY ME RI NY MT ND MN ND MT MT SD MN SD OR ID OR WA OR OR WA WA WA MO KS IA NE NE IL MO KS KS LA AR TX LA OK

Roanoke Dulles Williamsport Columbus Detroit Grand Rapids Indianapolis Louisville Madison Milwaukee Peoria Boise Casper Cheyenne Colorado Springs Grand Junction Pocatello Pueblo Reno Salt Lake City Winnemucca Albany Burlington Concord Hartford New York City New York City Portland Providence Rochester Billings Bismarck Duluth Fargo Helena Missoula Rapid City Rochester Sioux Falls Eugene Lewiston Medford Olympia Pendleton Salem Seattle Spokane Yakima Columbia Dodge City Dubuque North Platte Omaha Springfield St. Louis Topeka Wichita Baton Rouge Little Rock Lubbock New Orleans Oklahoma City

CME 0.8 0.89 0.88 0.94 0.93 0.76 0.95 0.88 0.94 0.93 0.91 0.52 0.51 0.51 0.54 0.24 0.47 0.36 0.58 0.52 0.39 0.91 0.9 0.89 0.94 0.95 0.91 0.95 0.93 0.86 0.66 0.79 0.86 0.88 0.44 0.58 0.79 0.97 0.88 0.71 0.54 0.38 0.8 0.7 0.85 0.63 0.74 0.69 0.87 0.59 0.93 0.71 0.91 0.9 0.92 0.96 0.84 0.69 0.86 0.82 0.73 0.86

Nonlinear Basis Hedging

Summer

RMS 0.89 0.93 0.93 0.91 0.96 0.86 0.97 0.86 0.9 0.95 0.94 0.68 0.8 0.84 0.68 0.42 0.73 0.61 0.44 0.77 0.54 0.97 0.95 0.96 0.97 0.96 0.9 0.95 0.96 0.95 0.86 0.91 0.8 0.89 0.7 0.72 0.93 0.89 0.86 0.73 0.68 0.44 0.78 0.79 0.8 0.6 0.82 0.79 0.89 0.65 0.84 0.77 0.89 0.91 0.92 0.97 0.88 0.83 0.89 0.75 0.72 0.83

CME

RMS

0.53 0.59 0.55 0.69 0.68 0.64 0.83 0.73 0.77 0.68 0.71 0.43 ⫺0.2 ⫺0.8 ⫺0.6 ⫺0.3 0.22 ⫺0.1 0.21 0.18 0.41 0.46 0.13 ⫺0.6 0.56 0.63 0.66 ⫺0.1 0.51 0.52 0.09 0.6 ⫺0.5 0.67 0.02 0.21 0.63 0.66 0.76 0.58 0.42 0.66 ⫺0.7 0.65 0.71 0.59 0.49 0.55 0.66 0.53 0.77 0.47 0.74 0.65 0.72 0.75 0.48 0.41 0.51 ⫺0.4 0.64 0.57

0.8 0.81 0.69 0.84 0.87 0.8 0.94 0.79 0.77 0.73 0.8 0.55 0.66 0.52 0.41 0.66 0.74 0.38 0.42 0.72 0.56 0.7 0.73 0.52 0.92 0.76 0.74 0.66 0.83 0.83 0.65 0.83 0.29 0.73 0.65 0.44 0.81 0.67 0.66 0.55 0.67 0.73 ⫺1.1 0.86 0.78 0.35 0.81 0.82 0.87 0.66 0.68 0.67 0.77 0.7 0.82 0.87 0.6 0.52 0.79 0.25 0.56 0.67

Winter CME 0.72 0.85 0.85 0.93 0.84 0.81 0.94 0.85 0.94 0.94 0.92 0.42 0.66 0.7 0.76 0.61 0.46 0.45 0.77 0.36 0.68 0.91 0.79 0.88 0.94 0.87 0.85 0.84 0.88 0.85 0.66 0.83 0.88 0.89 0.62 0.79 0.75 0.94 0.89 0.7 0.86 0.59 0.88 0.81 0.86 0.8 0.83 0.79 0.89 0.65 0.95 0.74 0.91 0.92 0.93 0.94 0.86 0.63 0.88 0.78 0.7 0.86

Summer

RMS 0.78 0.93 0.91 0.92 0.93 0.9 0.97 0.87 0.87 0.92 0.89 0.58 0.83 0.78 0.82 0.65 0.74 0.64 0.66 0.57 0.71 0.95 0.86 0.94 0.95 0.94 0.84 0.91 0.93 0.94 0.83 0.88 0.74 0.91 0.78 0.84 0.83 0.9 0.89 0.77 0.89 0.63 0.84 0.84 0.89 0.78 0.89 0.81 0.89 0.73 0.86 0.8 0.89 0.93 0.94 0.97 0.86 0.72 0.93 0.72 0.71 0.86

CME

RMS

0.63 0.59 0.71 0.74 0.77 0.62 0.91 0.81 0.88 0.71 0.87 0.31 ⫺0.1 ⫺0 0.2 0.02 0.02 ⫺0 0.34 0.37 ⫺0.1 0.2 0.4 0.12 0.59 0.63 0.73 0.45 0.87 0.57 0.36 0.57 0.34 0.88 0.27 0.26 0.68 0.72 0.78 0.31 0.52 0.56 ⫺0.3 0.68 0.8 ⫺0.3 0.54 0.57 0.85 0.58 0.88 0.44 0.73 0.68 0.78 0.8 0.76 0.6 0.64 0.14 0.48 0.85

0.72 0.84 0.74 0.85 0.92 0.68 0.91 0.81 0.83 0.78 0.81 0.63 0.55 0.56 0.24 0.67 0.81 0.15 0.45 0.74 0.4 0.79 0.77 0.58 0.87 0.73 0.8 0.75 0.82 0.79 0.74 0.74 0.72 0.87 0.77 0.54 0.83 0.69 0.78 0.55 0.69 0.76 ⫺0.6 0.88 0.73 0.05 0.79 0.84 0.87 0.67 0.81 0.69 0.74 0.7 0.84 0.72 0.78 0.69 0.83 0.33 0.7 0.87

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APPENDIX B (CONTINUED) THE EFFECTIVENESS

OF

BASIS HEDGING

FOR

87 CITIES

Linear Basis Hedging Winter Region South Central South Central South Central Southeast Southeast Southeast Southeast Southeast Southeast Southeast Southeast Southeast Southwest Southwest Southwest Southwest Southwest Southwest Southwest

State TX LA OK AL SC NC MS FL AL TN NC FL NM CA CA CA CA AZ AZ

City San Antonio Shreveport Tulsa Birmingham Charleston Charlotte Jackson Miami Mobile Nashville Raleigh Tampa Albuquerque Bakersfield Fresno Long Beach Los Angeles Phoenix Winslow

Nonlinear Basis Hedging

Summer

CME

RMS

CME

0.87 0.83 0.77 0.81 0.63 0.61 0.64 ⫺6.2 0.64 0.77 0.67 ⫺0.1 0.6 0 0.52 ⫺0.4 0.15 0.53 0.42

0.81 0.91 0.75 0.91 0.88 0.82 0.84 ⫺4.9 0.82 0.76 0.88 0.58 0.46 0.21 0.67 0.12 0.37 0.61 0.39

0.05 0.64 0.46 0.55 ⫺0.1 0.27 0.38 ⫺0 0.61 0.75 0.34 ⫺0.1 0.03 ⫺0 0.29 ⫺0.1 ⫺0 ⫺0 0

RMS 0.42 0.79 0.64 0.81 0.68 0.49 0.71 0.04 0.56 0.79 0.81 0 0.33 0.41 0.57 0.28 0.13 0.29 0.34

Winter

Summer

CME

RMS

CME

0.89 0.84 0.76 0.85 0.72 0.79 0.68 ⫺1.7 0.6 0.77 0.75 0.32 0.55 0.5 0.11 0.72 0.65 0.79 0.34

0.79 0.95 0.78 0.92 0.83 0.87 0.83 ⫺0.6 0.8 0.84 0.89 0.77 0.43 0.17 0.21 0.75 0.64 0.79 0.34

⫺0.1 0.84 0.69 0.71 0.39 0.37 0.72 ⫺0.5 0.47 0.52 0.34 ⫺0.2 ⫺0.1 0.03 0.53 ⫺0 0.2 0.34 0.17

RMS 0.2 0.92 0.7 0.77 0.79 0.66 0.89 0.04 0.66 0.6 0.72 0.01 0.47 0.07 0.6 0.26 0.37 0.44 0.55

8. ACKNOLWEDGMENTS The authors gratefully acknowledge the valuable and insightful comments of the anonymous referee and data support from Earth Satellite Corporation (www.earthsat.com). REFERENCES BRIYS, E., M. CROUHY, AND H. SCHLESINGER. 1993. Optimal Hedging in a Futures Market with Background Noise and Basis Risk. European Economic Review 37: 949–960. COBLE, K. H., R. G. HEIFNER, AND M. ZUNIGA. 2000. Implications of Crop Yield and Revenue Insurance for Producer Hedging. Journal of Agricultural and Applied Economics 25(2): 432–452. CONSIDINE, G. 2000. Introduction to Weather Derivatives. Weather Derivatives Group, Aquila Energy. Available at: www.cme.com/ weather/introweather.pdf. DAVIS, M. 2001. Pricing Weather Derivatives by Marginal Value. Quantitative Finance 1: 1–4 DOHERTY, N., AND H. SCHLESINGER. 1990. Rational Insurance Purchasing: Consideration of Contract Nonperformance. Quarterly Journal of Economics 105: 243–253. GOLDEN, L. L., M. WANG, AND C. YANG. 2007. Handling Weather Related Risks through the Financial Markets: Considerations of Credit Risk, Basis Risk, and Hedging. Journal of Risk and Insurance 74: 319–346. HENTSCHEL, L., AND C. W. SMITH, JR. 1997. Derivatives Regulation: Implications for Central Banks. Journal of Monetary Economics 40: 305–346. LI, D., AND T. VUKINA, T. 1998. Effectiveness of Dual Hedging with Price and Yield Futures. Journal of Futures Markets 18(5): 541– 561. MACMINN, R. D. 1999. Risk and Choice: A Perspective on the Integration of Finance and Insurance. Risk Management and Insurance Review 3: 69–79. MOSCHINI, G., AND H. LAPAN. 1995. The Hedging Role of Options and Futures under Joint Price, Basis and Production Risk. International Economic Review 36: 1025–1049. POITRAS, G. 1993. Hedging and Crop Insurance. Journal of Futures Markets 13(4): 373–388. TAPIERO, C., Y. KAHANE, AND L. JACQUE. 1986. Insurance Premiums and Default Risk in Mutual Insurance. Scandinavian Actuarial Journal (Dec.): 82–97.

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VUKINA, T., D. LI, AND D. HOLTHAUSEN. 1996. Hedging with Crop Yield Futures: A Mean-Variance Analysis. American Journal of Agricultural Economics 78: 1015–1025.

Discussions on this paper can be submitted until October 1, 2010. The authors reserve the right to reply to any discussion. Please see the Submission Guidelines for Authors on the inside back cover for instructions on the submission of discussions.

Journal of Risk Research Vol. 15, No. 2, February 2012, 169–186

The first five years of the EU Impact Assessment system: a risk economics perspective on gaps between rationale and practice Jacopo Torritia* and Ragnar Löfstedtb a

London School of Economics and Political Science, London, UK; bKing’s Centre for Risk Management, King’s College London, London, UK (Received 10 November 2010; final version received 12 June 2011) In 2003, the European Commission (EC) started using Impact Assessment (IA) as the main empirical basis for its major policy proposals. The aim was to systematically assess ex ante the economic, social and environmental impacts of European Union (EU) policy proposals. In parallel, research proliferated in search for theoretical grounds for IAs and in an attempt to evaluate empirically the performance of the first sets of IAs produced by the EC. This paper combines conceptual and evaluative studies carried out in the first five years of EU IAs. It concludes that the great discrepancy between rationale and practice calls for a different theoretical focus and a higher emphasis on evaluating empirically crucial risk economics aspects of IAs, such as the value of statistical life, price of carbon, the integration of macroeconomic modelling and scenario analysis. Keywords: better regulation; European Union; Impact Assessment; risk economics

1. Introduction Since 2003, the European Commission (EC) has employed an integrated Impact Assessment (IA) system for estimating ex ante the impacts of its policy and regulatory proposals in economic, social and environmental terms. In principle, the EC established IAs with the aim of providing a ground work for evidence-based policymaking, assessing the impact of the proposals in terms of costs, benefits and risks, opening the spectrum of policy alternatives and systematically including stakeholder opinions in the decision-making process. IAs consist of the most important aid to policy-making within the EU ‘better regulation’ agenda. In the first five years of existence, more than 200 IAs have been carried out by the EC. An increase in the quantity of IAs did not necessarily correspond to an increase in the quality. These five years of EU IAs did not pass unnoticed by the research community. In fact, two types of literatures have been examining the IA phenomenon: studies that made conceptual assumptions about the rationale of IAs and empirical evaluations that measured their performance. On the one hand, the rationale for the IA system has been charged of different expectations. In a paper focusing on risk regulation, Löfstedt (2004) envisaged IA as the instrument that would enable the shift of the regulatory pendulum from *Corresponding author. Email: [email protected] ISSN 1366-9877 print/ISSN 1466-4461 online Ó 2011 Taylor & Francis http://dx.doi.org/10.1080/13669877.2011.634512 http://www.tandfonline.com

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precautionary principle to evidence-based risk regulation. Did the pendulum actually swing? And which direction is it going to take for the future? Other commentators (Pelkmans, Labory, and Majone 2000; Radaelli 2003) focused on the idea that IAs would be instrumental in improving the quality of regulatory policies at the Community level. Some argued that IAs would prove capable of enhancing stakeholder participation in EU decision-making (Cavatorto 2001). Several of these contributions did not in fact differ substantially from the institutional literature, which generically referred to IA as an instrument for ‘better regulation’. On the other hand, empirical studies which evaluated IA performances looked mainly at procedural correctness of the reports (Kirkpatrick and Lee 2004); interaction with the ‘better regulation’ agenda (Evaluation Partnership [EP] 2007); inclusion of sustainable development (Adelle, Hertin, and Jordan 2006; IEEP 2004; Kirkpatrick and Franz 2007); level of quantification of the impacts (Vibert 2004); use of risk analysis techniques (Torriti 2007a) and consideration of policy alternatives (Renda 2006). Most of the empirical studies associated with IAs carried out in the first five years of the IA system used the EC guidelines as benchmarks for evaluating the content of a defined sample of IAs. This paper aims to spell out the differences between the conceptual rationale and empirical studies on EU IAs. It illustrates how most of the prospects on IAs have been partly invalidated by five years of practice. It weighs the balance between political science and risk economics in the evaluation of IAs. The distinction is based on the fact that political scientists look at IAs in terms of process, function and organisational checks and balances, whereas for economists the focus is on quality of data, pricing techniques for non-market values (e.g. Value of Statistical Life and carbon pricing) and integration of macroeconomic modelling. The paper suggests that not enough research has been carried out in the area of risk economics. Section 2 gives an account of the rationales of IAs. Section 3 explains what has been achieved to date by reviewing both empirical evaluations of IAs in the first five years of their existence and the analytical challenges which emerge from subsequent developments in the IA system. Section 4 discusses which rationales for IA have been corroborated by empirical studies and provides examples of areas of risk economics which may need to be considered in future research. Section 5 draws conclusions and infers on future developments of EU IAs. 2. Rationales for IA Although it would be inappropriate to speak of any shared theory of IA, the several conceptual contributions have increased the significance of and expectations towards this policy appraisal instrument. There has been a recent explosion of research exploring different rationales for the existence of IA. Concepts of accountability, organisational structures, regulatory philosophies and policy learning underpin most of the rationales explained below. For this reason, it can be inferred that much of the conceptual literature on EU IAs stems out of the political sciences and related sub-disciplines, including public administration, administrative law and public policy analysis. This section reviews some of the rationales for IAs as they originated from the first five years of IA exercise in the EU. The rationales could be seen as conceptual objectives for the existence of IA. These will be then tested in the next section when reviewing empirical studies on IA.

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2.1. Improving regulatory quality One explanation for the existence of IA is that put forward by Radaelli (2001), concerning the concept of quality of regulation. The key concept is that IAs are instruments aimed at improving the quality of regulation. Radaelli (2003) acknowledges the volatility of the concept, but believes that it is possible to measure the regulatory quality and quality of IAs through a set of indicators (Radaelli and De Francesco 2007). The measurement of regulatory quality is strictly dependent upon context because different actors, such as politicians, civil servants, experts, firms and citizens intend and measure quality in different ways. The concept of regulatory quality implies that IAs can generate positive values and it has the merit of conforming to the policy literature on ‘better regulation’. However, as also discussed by Radaelli (2003), regulation is by definition open to different interpretations and in this regard the concept of quality is extremely difficult to measure in objective terms (Heritier 2004). So the question of whether IAs in their first five years have improved regulatory quality is extremely difficult to address and escapes because the notion of quality is likely to depend either on volatile, intangible factors. 2.2. Assessing sustainable development and climate change Can IAs completely identify economic, social and environmental impacts, to the extent that they are able to include an integrated sustainable development approach? The EU IA system includes sustainable development in its guidelines, and this has been officially coupled with competitiveness. Although no academic work has described IA as a tool dedicated exclusively to the cause of sustainable development, there is a debate as to its adequacy in covering sustainable development issues (Kirkpatrick and Lee 2004). With regard to sustainability of EU interventions towards third parties, namely the developing world, Adelle, Hertin, and Jordan (2006) argue that EU IAs do not work as effective instruments for the implementation of the EU commitments to promoting sustainability in the developing world. Some have argued that issues associated with sustainable development remain more central to the EU IA system than competitiveness (Chittenden, Ambler, and Xiao 2007). Two arguments go against this position. First, the integrated IA system does not replace the Sustainability IAs, which preceded the 2003 integrated system for trade policies. Second, IAs fail to appraise sustainable development indicators as it is demonstrated by empirical studies in Section 4.1. The role of assessment tools in relation to climate change is likely to be crucial, as policy-makers increasingly need to measure, report and verify initiatives aimed at reducing climate change within IA, weighing individual regulatory proposals according to the expected changes to levels of CO2 emissions. For instance, in the case of the third legislative package on energy liberalisation, the environmental impacts are presented with a general appreciation of the consequences of a single market on climate change (Torriti 2010). Are there ways to systematically appraise the impacts of climate change in IAs? Unlike practice in some of member states, the EU does not systematically use a price of CO2 in their IAs. For example, in 2007 the UK Government instituted in its IA guidelines a ‘Shadow Price of Carbon’ which measures each additional ton of greenhouse gas emitted for environmental impacts on climate change. The ‘Shadow Price of Carbon’ was set at £26.50/tCO2e at the time of writing this paper, and will rise annually to £60.80/

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tCO2e by 2050. Since no ‘Shadow Price of Carbon’ or any other equivalent parameters for valuing carbon emissions are included in the guidelines, it would be incorrect to state that IAs are designed in order to address climate change issues. 2.3. Facilitating the shift to risk-based regulation It has been argued that IAs might represent a shift from the precautionary principle to risk-based regulation (Löfstedt 2004). In the past, EU regulation has been characterised by the extensive adoption of precautionary measures aimed at preventing and mitigating hazards in various fields of risk regulation, including genetically modified foods, climate change and chemicals (Wiener 2006). Emerging technologies, for instance, have been a specific target area for the precautionary principle (Jasanoff 2005). Existing research considers the precautionary principle in terms of the final regulatory decision, which tends to be characterised by a preventative approach to the regulated subject (Goldstein 2002). Another approach would be to gather evidence as to risk-based techniques prior to the final regulatory decision. Did the use of risk analysis techniques in IAs impede precautionary measures and facilitate evidence-based policies? Is it possible to find elements of precautionary and risk-based tools of assessment in individual IA reports that deal with specific policy areas which are sensitive to the use of risk analysis? Löfstedt (2007) notices that the introduction of IA did not lead to a simultaneous withdrawal of the precautionary principle with attention dedicated to the use of risk–risk and risk–benefit techniques. There are at least three reasons why the shift did not happen. Firstly, it had proved extremely difficult to derive the means from the principle (Graham and Hsia 2002; Sandin 1999). IAs set out merely to assess the risks, as well as costs and benefits, and may include data from a risk assessment. The management of the risks, which may involve a precautionary position, takes place at a different level, on the sole basis of the existence or non-existence of data on risk. It is worth remembering that the scientific data from a risk assessment are often gathered by agencies, whereas the (risk) management on those data takes place at the policy level. Secondly, the EU IA integrated system includes, simultaneous environmental and social estimates with economic analysis. This means that the balance between precautionary measures aimed at protecting both human health and the environment, on one hand, and competitiveness and cost-effective measures, on the other hand, were often assessed on a case-by-case basis. On this point, the EU approach is far from being unique. The incorporation of socio-ethical values in IAs is a common topic for a number of national regulators who presently are experiencing a switch to different risk analysis practices in policy-making. Reviews of worldwide regulatory initiatives suggest that agencies which appear to have taken a much more holistic and systematic risk-based approach are more common in the UK, US and increasingly Australia and Canada. In these countries, the focus on risk management strategies is a means of orienting regulatory activities and organising governance attitudes and structures (Löfstedt 2004). There are European regulators or agencies adopting risk-based tools on an ad hoc basis by exhibiting a partial buy-in to broader risk philosophies. In Germany, for instance, with the work by the Occupational Health and Safety Committee there is evidence of a move in environmental and occupational health and safety regulation, to develop more systematic quality targets and evaluation techniques. The use of risk analysis techniques in EU IAs

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was rendered difficult by the complex multiplicity of regulatory philosophies that can be observed in legislative interventions. Thirdly, there is no policy paper to defend the view that the precautionary principle is directly in opposition to evidence based on risk analysis. The EC standpoint on this precautionary principle is stated in an apposite communication (EC 2000), which asserts that the adopted measures presuppose examination of the benefits and costs of action and lack of action. The use of the precautionary principle, as it is intended in the communication, does not preclude evidence-based approaches at all (Klinke and Renn 2002). Advocates of the precautionary principle have argued that precaution does not automatically mean banning substances or activities, but instead implies a gradual, step by step diffusion of risky activities or technologies until more knowledge and experience are accumulated. The 2009 EC guidelines provide an interesting account of how risk analysis should be employed. For instance, they are rather restrictive with regard to the monetarisation of the benefits associated with risk reductions in health and the environment (Value of Statistical Life). 2.4. Improving market competitiveness IAs may exist to improve market competitiveness. European institutions aim to understand whether it is appropriate to intervene in market structures and whether regulatory intervention is in effect needed. At the EU level, the concept that legislative action should be taken only when necessary was established at the EC in Stockholm (EC 2001). Competitiveness is considered as one of the main reasons for the implementation of IAs in EU regulatory decision-making in order to achieve macroeconomic goals as defined in the Lisbon agenda. IAs may facilitate competitiveness because regulatory intervention would be determined on the basis of the costs and benefits in the market. Scholars who embrace the idea that IA could be used to develop market competitiveness believe that economic techniques such as cost–benefit analysis play a pivotal role (Sunstein 2002a, 2002b; Viscusi 2006). Moreover, the presence of cost–benefit analysis in an IA is seen as an indicator of the drive for competitiveness of the EC (Vibert 2004). The performance of the EU internal market in the first five years of the IA system seems to discourage some of these arguments. 2.5. Improving regulatory competition IAs, in principle, could be used as an instrument for international regulatory competition. This point is slightly different from the previous one of market competitiveness. Whilst market competitiveness is about guaranteeing the correct running of the internal market, international regulatory competition is about institutional endeavour to obtain a more significant share of multi-national investments (Genschel and Plumper 1997). In the regulatory realm, public institutions compete in order to have the most attractive, investor-friendly and secure business environment. Having the best regulatory framework equals attracting investors as well as fostering the internal market. A functioning IA system may assure investors that regulatory interventions in the market are pondered and justified. The need for improving methods for regulatory competition was established in the Communication on Growth and Jobs (EC 2005), which asserted that the EU economy was lagging behind in achieving the

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goals of the Lisbon Agenda, partly due to excessive ‘red tape’. The political importance of policy appraisals like IAs has been highlighted by some authors (Turnpenny et al. 2009), who argue that these can be interpreted as a form of policy which enables one institution to gain recognition. Regarding the international role of IA, there are arguments for cooperation rather than competition. Five years of IA experience suggest that some degree of market failure will always be present. 2.6. Improving regulatory cooperation It is in the interest of some of the international competitors on regulation, namely the USA that the EU IA system is built to favour international collaboration. The Transatlantic Forum for Regulatory Cooperation and the Joint Report – both explained in Section 3.2 – are practical examples of such collaboration. In addition, there are various reasons why USA and EU should will to cooperate on IA. First, the EU needs to cooperate on regulatory affairs in order to maintain high trade revenues. The EU–US trading partnership, for instance is the largest bilateral trade and investment relationship in the world with trade flows across the Atlantic of around e620 billion per year and 14 million jobs depending on these bilateral investment flows, making up to 57% of world’s gross domestic product (GDP). Second, exchange of data is vital for improving regulatory decisions. Third, cooperation on IA would facilitate harmonised regulation to fight climate change. The EU has borrowed not only the notions of IA and ‘better regulation’ from the North American legal system (Wiener 2006), but also taken on board the idea that institutional features are influenced by inter-institutional dialogues (Alemanno and Wiener 2010). Indeed, it is extremely difficult to predict the extent to which EU and USA will cooperate in the future, given the change in leadership in the USA and the global economic downturn. Much will depend on factors which are out of IA control, such as the level of protectionism in internal markets and amount of priority given to climate change policies. However, IAs will be an instrument through which it might be possible to read the level of cooperation across the Atlantic (Torriti and Löfstedt 2010). 2.7. Creating the conditions for ‘policy learning’ IAs could be instituted in order to justify single policy intervention, but also to improve the reasoning and empirical grounding behind policy decisions. Majone (2003) looked at IA as an internal ‘policy learning’ process. The rationale for IA is to provide a framework that enables policy-makers to structure policies in terms of inputs and outputs. Uncertainty implies that the main aim of decision-making is not to optimise efficiency, but to improve the level of knowledge available for making policy decisions. Following Arrow’s (1962) definition of knowledge, Majone (2003) supports the view that internal dissemination of knowledge is a necessary step to reach procedural rationality. Ideas should not be considered in isolation, but should be related to other relevant ideas to see how they fit together in a coherent manner. Without going into further detail as to the meaning and implications of ‘policy learning’, it is incontrovertible that the diffusion of knowledge is likely to be one of the rationales for IAs. When looking at the evolution of IAs between 2003 and 2005 it is clear that there was no improvement at least in terms of policy options and quantification of costs and benefits. However, to neglect the ‘bad start’ for EU IAs would be to deny completely the usefulness of the ‘trial and error’ mechanism

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theorised by Majone’s (2003) policy-learning idea, which implies that the differences in quality between early days IAs and later versions may not undermine the learning purpose of why IAs were instituted. Recent studies explored the reliance of IAs on the knowledge and expertise of those individuals responsible for carrying them out (Hertin et al. 2008). On the one hand, the educational background of officials can determine the IA output. On the other hand, one reason for having IAs in the first place is precisely to improve the knowledge base of both policy decisions and policy-makers. 2.8. Widening the range of policy options The EC, in search of credibility and accountability, may adopt IA to improve the negative image of ‘bad’ regulation (Robinson 2004) by focusing on openness. In order to fill the democratic gap in policy-making, EU institutions are inclined to conceive policy instruments that increase the openness of their decisions (Richardson 2002). IA is one of these instruments for openness because its purpose is to prove that a range of different options have openly been taken into account before making the final policy decision (Pelkmans, Labory, and Majone 2000). It has been argued that understanding whether IAs appropriately weigh alternative policy options is fundamental to understand their existence (Baldwin 2004). In EC Communications and guidelines, the idea that IA can facilitate an ‘open-minded’ approach to policy options is repeated frequently. The expectations of the EC are that IAs will serve as a means to improve the openness of the EU decision-making process, in that it compels policy-makers to consider alternatives to regulation. IA involves the systematic examination of the alternative impacts arising from EC action or non-action and helps the regulator to specify policy or regulatory action. It defines what problem is to be addressed by regulation as well as its causes, and defines the objectives to focus on in solving the problem. The EC recommends identifying other policies and keeping an open mind about them. In internal working papers it is stated that: The earlier you are in the process and the less you have, the more important it is to look at a range of alternatives. Consultations with colleagues and interested parties can give you useful information. (EC 2003, 27)

As maintained by the EC, the examination of policy options leads directorate generals (DGs) to ‘think outside the box’. The ‘no policy change’ option, or nonaction, should always be considered, on the basis of what was agreed by the 2001 Stockholm EC (2001). In general, it is suggested ‘to keep an open mind and not to eliminate potentially effective responses to solve the problem’. More specifically, criteria of efficiency, effectiveness and consistency should be used to identify the most relevant options, which are to be analysed in more detail. The Mandelkern Group (2001), which pre-dates the constitution of the IA system, outlines a typology of possible alternatives. These include: do-nothing, incentive mechanisms, selfregulation, contractual policies, mechanisms to ensure the assumption of responsibility, mutual recognition and improving existing regulation. IA can be defined as a key to use in identifying issues for regulatory action, defining the objectives of regulation, and considering all the regulatory alternatives, including non-action. Turnpenny et al. (2008) argue that organisational traditions and the sectorisation of

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policy-making mean that IAs are often perceived as supports to, rather than drivers of, policy alternatives. 2.9. Including stakeholder opinions One of the possible reasons for the existence of IA may be as a result of stakeholder pressure to participate in policy-making. IAs are expected to enhance the level of transparency of the regulatory system (Radaelli 2003). The role of IA might be: To enhance the empirical basis of political decisions [. . .] and to make the regulatory process more transparent and accountable. (Radaelli 2003, 723)

They have the potential to increase the level of communication both within and outside regulatory bodies, i.e. to experts, stakeholders and the public, as explained below. According to the EU guidelines, IA increases internal communication, as it requires coordination among the different units or departments within regulatory institutions (EC 2009). Coordination between services is considered essential in order to avoid a single-sector ‘chimney’ approach and to anticipate and resolve potential issues as early as possible in the process. IAs also allow for the use of all relevant in-house knowledge and expertise. The lead DG is responsible for informing, consulting and coordinating with other interested DGs at a very early stage. IA also could create the grounds for professional consultation with experts and think tanks, as well as communication with stakeholders and the public. Technical aspects of regulatory impact assessment (RIA) can be addressed through consultation with experts in cost–benefit analysis and risk management. With this purpose, the EC decided to integrate the system of IA within the standard of minimum consultation, also producing guidelines on the use of science and expertise in the EU policy process (Meuwese 2008). IAs could also be intended to structure communication procedures with stakeholders. The EC officially consults stakeholders before making policy decisions, in order to gain a understanding of their concerns. The emphasis on stakeholder consultation appears to be due to two factors. First, learning from consultation is a mutual and collective process in which information is shared and new knowledge is developed among participants in both the public and private spheres (Milligan 2003). Secondly, appropriate communication with stakeholders on the action to be taken by the regulator may be perceived by stakeholders as reducing regulatory risk (Jacobs 1997). The less the regulatory environment is perceived as risky, the more likely it is that longer term investment will replace aggressive rent-seeking and short-term profit. 3. IA in practice: empirical studies on the first five years of IA 3.1. Empirical studies on IAs by the EC In addition to the literature which has interpreted the conceptual reasoning for the existence of IA, other studies focused on evaluating the content of sets of IAs in addition to research based on individual case studies, which is excluded from this review. Empirical evaluations of IA reports are in general characterised by negative appraisals of the IA system. These consist of evaluations of the procedural correctness, inclusion of sustainable development issues, technical correctness and risk economics.

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Two studies examined the procedural correctness of the first IAs produced by the EC. First, a pilot study by Lee and Kirkpatrick (2001) analysed a sample of 6 of the first 20 Extended IAs produced during 2003 by following a review based on variables set up by the authors. This pilot study used an integrated approach to look both at the quality of the content of individual Extended IAs and at the process followed. It develops a methodological review package for assessing the quality of IA reports and processes. However, it disregards criteria for selecting the cases to be used for analysis. Second, Meuwese (2008) on legal implications of IA when she makes use of case studies on IA of environmental policies. Three empirical studies examined the inclusion of sustainable development in early IAs. First, the Institute for European Environmental Policy’s report (IEEP 2004) reviewed how sustainable development considerations were addressed in the Extended IAs produced during 2003. One of the main findings was that none of the Extended IAs actually followed the EC’s guidelines. Second, Kirkpatrick and Franz (2007) used a scorecard method to assess the elements of sustainable development included in 13 IA documents from the year 2006 and concluded that there was an improvement in the inclusion of sustainable development in comparison with previous studies. Both the IEEP report and the study by Kirkpatrick and Franz (2007) are significant, since they provide empirical evidence from Extended IAs for a field – Sustainability IAs – where the literature is already quite extensive and includes several reviews of assessments methods and practice (Bond et al. 2001; De Bruijn and ten Heuvelhof 2002; Lee and Kirkpatrick 2001). Third, a study by Adelle, Hertin, and Jordan (2006) similarly evaluated the interaction between IAs and sustainable development, concluding that IAs did not sufficiently take into account sustainable development issues. This study is particularly pessimistic as it reveals that the IA procedure does not function as an effective instrument for the implementation of the Union’s commitment to promoting sustainability in non-European countries. While several IAs touch on external dimensions, it would be done in a vague and somewhat abstract manner, as it results in the evaluation score illustrated in Figure 1.

Figure 1. Degree of consideration given to the external dimension by EU IAs (Adelle et al. 2006).

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Studies looking at the technical correctness of the IA system focused on issues of quantification of the IAs, the consideration of policy alternatives. First, Vibert’s (2004) scorecard examined the first 20 EU assessments carried out in 2003 and looks at the way in which they quantify impact, their technical approaches and their outcomes. This work used a quantitative scorecard to examine the content of individual reports. It includes, for example, the number of IAs that quantify or monetise costs and benefits. The results rather predictably draw on competitiveness and lack of market analysis. Second, Lussis (2004) reviewed 13 IAs focusing on methodological inconsistencies. Third, Renda (2006) evaluated 70 Extended IAs produced by the EC between 2003 and 2005, using a scorecard as a benchmark. The scorecard entries included the level of quantification of the IAs, the consideration of policy alternatives and ex post assessment methodology. It was concluded that according to the scorecard, the IAs performance did not improve at all throughout the years. Fourth, a study by the Evaluation Partnership (2007) appraised a representative sample of 20 IAs, pointing out common problems with regard to (i) scope of application and proportionate analysis; (ii) timing and approach; (iii) quality control mechanisms and (iv) support and guidance. Two empirical studies in the area of risk economics are highlighted here. First, Torriti (2007a) examines 60 early days preliminary IAs with the view of understanding to what extent risk-benefit rationales are taken into account in the early stage of policy-making (Figure 2). Second, Nilsson et al. (2008) examine a set of IAs from EU (and also member states, including Germany, Sweden and UK) to understand how analytical tools, including cost–benefit analysis are embedded in IAs. They distinguish between ‘simple’, ‘formal’ and ‘advanced’ tools to conclude that the use of advanced analytical tools is very limited at the EC level (Table 1). 3.2. Developments in the IA system after the first five years After reviewing empirical evaluations of IAs in the first five years of their existence, the analytical challenges which emerge from more recent developments in the IA system will be examined in this sub-section. The reason for doing this is to give an indication of the context within which IA practice is currently operating. In turn, this informs the reader about the further need for empirical analysis on the risk economics aspects of IA reports. The 2009 IA guidelines suggest several areas which will need further theoretical and empirical investigation in the future. These relate to the quality of available

Figure 2. Risk analysis and IAs (Torriti 2007a).

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Table 1. Tool uses in IAs (Nilsson et al. 2008). Simple Germany Case A (Agriculture) Case B (Agriculture) Case C (Environment) Case D (Environment) Case E (Transport) Case F (Transport) Case G (Climate change) Sweden Case A (Biofuels) Case B (Environment) Case C (Climate change) Case D (Climate change) Case E (Transport) Case F (Transport) UK Case Case Case Case Case Case Case

A (Environment) B (Environment) C (Home office) D (Culture) E (Health) F (Trade & industry) G (Transport)

European Commission Case A (JLS) Case B (RELEX) Case C (ENV) Case D (DEV) Case E (ENV) Case F (JLS) Case G (MARKT) Case H (ENV) Case I (JLS) Case J (MARKT) Case K (JLS) Case L (EMPL) Case M (TAXUD) Case N (ENV) Case O (RTD) Case P (AGRI) Case Q (AGRI)

Formal

Advanced

A

A

A A

A E A A A E

A

A A E A

E

E E A E A

A A E

A A A

A A

E

E

A

A A A A A A A A A A A A A A A A A A

A A A A E A E A A E A A A E

E A A

Notes: A = active tool use, E = existing results.

data; the inclusion of economic criteria with environmental and social impacts; the embedment of the Standard Cost Model; and the integration of macroeconomic modelling in IAs. First, the choice of some of the examples of best practice in the guidelines is at times unfortunate. For instance, the guidelines make use of the IA on the introduction of biometrics for VISA system as a valuable case for presenting costs and benefits. However, the cost–benefit analysis on biometrics is funded on a rather

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Table 2. Financial costs of VIS As with biometrics (EPEC 2004). ‘VIS without biometrics’ One-off investment costs

Annual operational costs

‘VIS with biometrics’ One-off investment costs

Annual operational costs

Costs for the community

e30 million

e8 million

e93 million

Costs for the member states (national systems)

No estimates

No estimates

e186 million

e14–16 million e49 million

e246–256 million

e55–57 million

Total costs

dubious ‘cost transfer’ exercise. The estimate of the financial costs for member states associated with introducing biometrics (in Table 2) is based on the average of the only two member states that did provide cost estimates, i.e. Sweden and France. The starting assumption is that France is a large issuer of visas, whereas Sweden is smaller. This calculation of financial costs goes against the concept of differential costs and the estimate of expenses for individual national systems and is therefore statistically insignificant. The quality and availability of data remain the first problem with IAs and is dealt with in the next sections. Second, the guidelines stress the importance of maintaining an adequate balance between the three pillars of economic, social and environmental impacts. This is a laudable effort. However, more than 20 years of practice in cost–benefit analysis, Environmental and Health IAs prove that getting the balance is extremely problematic. The rationale behind estimating economic changes in the market rests in the concept of efficiency. However, the efficiency criteria on which IAs are based are relatively different from the Pareto efficiency criterion given by the Willingness to Pay – Willingness to Accept equation, because IAs are supposed to take into account not only changes in the market, but also in society and the environment. When efficiency conflicts with other values, it becomes problematic to create an overall economic criterion capable of integrating all values. In other words, IAs will probably never perform actual economic analysis based on the efficiency criterion, if they are also to take into account social, environmental and economic values. Third, the guidelines include the Standard Cost Model for assessing the administrative burdens imposed by new legislation. The endeavour of keeping red tape down is likely to be heavily challenged by a new wave of regulations on climate change and the financial market. Thus far the Standard Cost Model has been widely used for measuring and reducing administrative costs only ex post (Torriti 2007b). Its use in the ex ante phase of policy proposals has not been experimented and is likely to cause confusion among policy-makers. Fourth, the EC pursues the objective of addressing some of the macroeconomic impacts. This is an extremely interesting area both for researchers and policymakers. Although the quantification of macroeconomic impacts in IAs is faced with several constrictions and limits, it has the potential to overcome the excessive focus on microeconomics of IA. The technique of multiplying the costs and benefits experienced by an individual firm by the total population of firms, that is, using

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microeconomic instruments for macroeconomic purposes, has the merit of helping to provide actual figures for the impacts of legislative changes. This may produce a more rational approach to making decisions, as well as simplify the multiple factors that go into a legislative decision. However, there is a major problem with estimating costs and benefits for a distinct class of economic actors – or firms – and then expanding those findings to the whole population. IA and, even more aptly, cost–benefit analysis are instruments for judging efficiency in cases where the public sector supplies goods, or where the policies executed by the public sector influence the behavior of the private sectors and change the allocation of resources. 4. The gap between rationale for and practice in IA Theories are extremely important to understand the reasons why organisations and individuals behave in different ways. They become even more important when substantiated by significant evidence from several years of practice. In the case of the EU IA system, there has been a proliferation of conceptual approaches, if not theories. These might have improved the thinking around the structure, process and function of IAs in EU policy-making, but failed at least in four instances. First, the conceptual reasons why IAs have been instituted and employed in several occasions have been disjointed with empirical studies on IAs. Table 3 identifies which rationales (in Section 2) have been supported by empirical studies (Section 3). Table 3 summarises which empirical studies validate the rationales for IA. One basic observation is that the number of empirical studies is rather limited compared with rationales. On the one hand, this renders the validation exercise unbalanced. On the other hand, this speaks to the need for further empirical research on IA. From Table 3, it results that studies looking at sustainable development, market competitiveness, evidence-based risk regulation and policy learning have been backed by some empirical testing. However, conceptual approaches considering regulatory quality, regulatory competition and cooperation, openness and transparency as crucial issues for the existence or functioning of IAs have not been fully investigated empirically, perhaps for the intangibility of these arguments. Second, some of the empirical studies reviewed in Section 3 refer to the very early days, when the Commission IA system was still in its infancy. Indeed the extent to which the early days studies are as representative as the more recent ones is questionable. One might argue that the rationale for the existence of IA may not be empirically observed until much later or that empirical studies do not start from the same focus as conceptual research on the rationale for IA. This in itself is an issue which reveals the distance between rationale and practice. Third, much conceptual emphasis has been centred on political and policy aspects and less on the economic and risk rationale of IA. Surprisingly little work has been developed with regard to the apparent contradiction of using microeconomic tools for solving macroeconomic problems. The rationale of foreseeing how individual economic actors will react to legislative change and multiplying such effect for the affected population oversimplifies the complex market relations between economic (and non-economic) actors. Macroeconomic modelling has been used for decades in various spheres of EU policy-making, but its integration into microeconomic forecasting has always been problematic. Some serious research effort is needed around these topics.

Evidencebased riskregulation Regulatory quality

Hutter (2005); Löfstedt (2004); and Wiener and Rogers (2002) Radaelli (2003); Radaelli and De Francesco (2007)

Sustainable Adelle, Hertin, and Jordan (2006); Kirkpatrick and Franz development (2007) Nilsson et al. (2008) Market competitiveness Regulatory Torriti and Löfstedt (2010); Turnpenny et al. (2009); and competition and Wiener (2006) cooperation Majone (2003) Policy learning

Openness Löfstedt, (2007); Pelkmans, Labory, and Majone (2000); and Renda (2006); Turnpenny et al. (2008) Transparency Meuwese (2008); Milligan (2003); Radaelli (2003); and Renda (2006)

To shift the focus from precautionary principle to evidence-based risk-regulation To improve the quality of EC regulation and policy

To comprehend consequences on sustainable development and climate change To increase internal market competitiveness

To improve the credibility and accountability of the European regulator by widening the range of policy options To include stakeholder opinions in decision-making

To provide an analytical framework for structuring the formulation of policy-making

To improve the position of the EU in terms of regulatory competition and cooperation

Key concept

Source

Explanations

Table 3. Theoretical explanations for IAs and empirical studies.

To explore the concept of regulatory quality and measure the quality of IAs against a pre-defined benchmark To explore the role of economic techniques in environmental and socially relevant IAs To compare ex post the IA cost and benefit estimates with figures emerging from the market To explore comparatively the interactions involving IA with international governing bodies Time-scale analysis on the development of general procedures and technical content of IAs through an evaluation of the policy-learning curve To examine the use of policy alternatives in IA reports To examine the methods of consultation in IA reports

Yes

In part

Yes

Yes

No

Yes

Yes

To analyse the use of risk analysis in IAs within a specific policy area

Applicative research aspect

In part

In agreement with policy literature?

No

No

In part

No

Yes

In part

No

In part

Validated by empirical studies?

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Fourth, the conceptual perspectives on IA did not cover crucial aspects which should be associated with empirical studies. The existing European literature on risk economics comprises at least three topic areas which are strictly related to IA systems. First, the literature on the Value of Statistical Life focuses on methods to monetise non-market values associated with reductions in health and environmental risks which are vital to any discussion on the significance and rationale of policy appraisals (Alberini, Hunt, and Markandya 2006; Johannesson, Johansson, and Lofgren 1997; Miller 2000). Second, the issue of methods to price carbon focuses on the risk tradeoffs of pricing carbon for climate change policies and beyond (Ekins 1994). Third, the topic of integrating macroeconomic modelling into microeconomic policy analysis tools has been looked at in the realm of energy policies (Barker, Ekins, and Foxon 2007). The imbalance between risk economics and politics is also noticeable when looking at the little attention dedicated to economic forecasting vis-à-vis economic foresight in policy appraisals, especially with regard to climate change (Dasgupta 2008) and energy policies (Lind 1995). Analytical instruments such as scenario analysis, which is shaping important policies on the transition to a low carbon economy have been disregarded by most of the IA literature. 5. Conclusions The first five years of EU IAs were also five years of research of the IA system. There are considerable discrepancies between the prescriptions of IAs in the literature and the complex realities of EC policy processes explained in the empirical studies. By highlighting the gap between rationale and empirical tests, the paper addresses the research community towards more consistent empirical studies on the IA system. This is a first attempt to reconcile theory and empirical studies on the EU IA system. For this reason, it is emphasised here that the empirical studies reviewed in Section 3 refer to the first five years of the IA system only. More studies might (and indeed) have been undertaken since. Section 3 makes an attempt to incorporate some of the recent developments in the EU IA system in order to provide an adequate context to the observations drawn in this study. In a recent paper, Radaelli (2009) reflects on the evolution of research on IA by observing asymmetries in demand and supply of research. This paper focused on some of the asymmetries within the supply side and the excessive conceptual perspectives, e.g. on rationales vis-à-vis empirical studies on IAs in the first five years of research on the IA system. Above all, the role of IA cannot be disentangled from its overall aims (Chittenden, Ambler, and Xiao 2007), which are ‘to enhance the empirical basis of political decisions and to make the regulatory process more transparent and accountable’ (EC 2009). In this regard, some of the conceptual literature went perhaps too far in foreseeing a diversity of possible developments originating from IAs. At the same time, the empirical literature has been rather severe in evaluating early days IAs. One should bear in mind that the assessment exercise implies not only an economic appraisal of the costs, benefits and risks produced on the market by a legislative proposal, but also some understanding of its social and environmental consequences. These goals are extremely challenging even for an omniscient policy-maker. For IAs to play a key role in decision-making, a primary issue is that they are carried out to a high technical standard. This will depend almost exclusively on the quality of the data on which IAs are performed.

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In practical terms, one suggestion put forward by these authors is that higher resources should be spent on the IA process, including forms of peer-reviewing drafts and modifying IA reports after amendments to the legislative proposal. Perhaps the EC should consider narrowing down the scope of its integrated IA system. Narrower IAs will facilitate technical quantification and monetisation of costs, benefits and risks and in turn will unlock specific research aimed at improving the quality of evidence-based policy-making. In turn, research needs to find a better balance between political science and economics in order to contribute in better ways to evaluate empirically the performance of IAs. A higher focus on risk economics aspects of IA would imply more attention to issues such as the quality of data, different techniques to derive the value of statistical life, methods to estimate the price of carbon and the integration of macroeconomic modelling. All these aspects are vital to a functioning IA system, particularly in areas such as energy and climate change, where the EU is called to make an additional analytical effort when drafting new legislation. Acknowledgements The research leading to this paper was partly funded by the GE Foundation.

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Miller, T. 2000. Variations between countries in values of statistical life. Journal of Transport Economics and Policy 34, no. 2: 169–88. Milligan, E. 2003. Using RIA as a tool in regulatory consultations and communications. Ottawa: The Regulatory Consulting Group. Nilsson, M., A. Jordan, J. Turnpenny, J. Hertin, B. Nykvist, and D. Russel. 2008. The use and non-use of policy appraisal in public policy making: An analysis of three European countries and the European Union. Policy Sciences 41, no. 4: 335–55. Pelkmans, J., S. Labory, and G. Majone. 2000. Better EU regulatory quality: Assessing current initiatives and new proposals. In Regulatory reform and competitiveness in Europe, Volume 1: Horizontal issues, ed. G. Galli and J. Pelkmans. Cheltenham: Edward Elgar. Radaelli, C. 2001. L’analisi di impatto della regolazione in prospettiva comparata [Regulatory impact analysis: A comparative prospective]. Catanzaro: Rubbettino editore. Radaelli, C. 2003. The diffusion of regulatory impact analysis: Best-practice or lesson-drawing? European Journal of Political Research 43: 723–47. Radaelli, C. 2009. Desperately seeking Impact Assessments: Diary of a reflexive researcher. Evaluation 15, no. 1: 31–48. Radaelli, C., and F. de Francesco. 2007. Regulatory quality in Europe. Manchester: Manchester University Press. Renda, A. 2006. Impact Assessment in the EU. The state of the art and the art of the state. Brussles: CEPS. Richardson, J. 2002. Policy-making in the EU. In European Union: Power and policy-making, ed. J. Richardson. London: Routledge. Robinson, W. 2004. ‘Bad’ EU regulation: Prevention and cure. PLC Global Counsel Dispute Resolution Handbook. www.practicallaw.com/A36167 (accessed November 7, 2011). Sandin, P. 1999. Dimensions of the precautionary principle. Human and Ecological Risk Assessment 5: 889–907. Sunstein, C. 2002a. The cost–benefit state: The future of regulatory protection. Chicago, IL: American Bar Association. Sunstein, C. 2002b. Risk and reason. Cambridge: Cambridge University Press. Torriti, J. 2007a. (Regulatory) Impact Assessments in the European Union: A tool for better regulation, less regulation or less bad regulation? Journal of Risk Research 10, no. 2: 239–76. Torriti, J. 2007b. The standard cost model: When better regulation fights against red-tape. In Better regulation, ed. S. Weatherill. Oxford: Hart. Torriti, J. 2010. Impact Assessment and the liberalisation of the EU energy markets: Evidence based policy-making or policy based evidence-making? Journal of Common Market Studies 48, no. 4: 1065–81. Torriti, J., and R. Löfstedt. 2010. The role of regulatory Impact Assessment in fighting climate change and the economic downturn: A EU-US perspective. European Journal of Risk Regulation 1, no. 3: 251–7. Turnpenny, J., M. Nilsson, D. Russel, A. Jordan, J. Hertin, and B. Nykvist. 2008. Why is integrating policy assessment so hard? A comparative analysis of the institutional capacities and constraints. Journal of Environmental Planning and Management 51, no. 6: 759–75. Turnpenny, J., C. Radaelli, A. Jordan, and K. Jacob. 2009. The politics of policy appraisal: Emerging trends and new directions. Journal of European Public Policy 16, no. 4: 640–53. Vibert, F. 2004. The EU’s new system of regulatory Impact Assessment – A scorecard. London: European Policy Forum. Viscusi, K. 2006. Monetizing the benefits of risk and environmental regulation. Working Paper 06–09, AEI Brookings Joint Center. http://www.aei-brookings.org/publications/ abstract.php?pid=1068. Wiener, J. 2006. Risk and regulation: Issue for discussion. Paper for the Working Party on Regulatory Reform and Management, Paris: OECD. Wiener, J., and M. Rogers. 2002. Comparing precaution in the United States and Europe. Journal of Risk Research 5, no. 4: 317–49.

Applied Financial Economics, 2012, 22, 1215–1232

The impact of banking and sovereign debt crisis risk in the eurozone on the euro/US dollar exchange rate Stefan Eichler Faculty of Business and Economics, TU Dresden, Muenchner Platz 1/3, Dresden 01062, Germany E-mail: [email protected]

I study the impact of financial crisis risk in the eurozone on the euro/US dollar exchange rate. Using daily data from 3 July 2006 to 30 September 2010, I find that the euro depreciates against the US dollar when banking or sovereign debt crisis risk increases in the eurozone. While the external value of the euro is more sensitive to changes in sovereign debt crisis risk in vulnerable member countries than in stable member countries, the impact of banking crisis risk is similar for both country blocs. Moreover, rising default risk of medium and large eurozone banks leads to a depreciation of the euro while small banks’ default risk has no significant impact, showing the relevance of systemically important banks with regards to the exchange rate. Keywords: exchange rates; eurozone; banking crisis; sovereign debt crisis; subprime crisis JEL Classification: F31; F32; E44; E52

I. Introduction The financial turmoil associated with the recent financial crisis threatens the stability of the Economic and Monetary Union (EMU). The subprime lending crisis, which was triggered in 2007 by global macroeconomic imbalances, a lack of policy co-ordination, insufficient banking regulation, excessive risk-taking of banks, and fast credit expansion driven by lax monetary policy, has brought many banks in the EMU to the verge of bankruptcy. The crisis had a major adverse impact on the economies of several EMU member countries including a reduction in lending growth, a decline in output and investment, an increase in unemployment, capital market

contagion and a rapid increase in sovereign debt (Cheung et al., 2010; De Grauwe, 2010; Gros and Alcidi, 2010; Eichler and Herrera, 2011). The costs of bank rescue programs and the associated rise in (implicit) government debt as well as record-high fiscal deficits produced by the crisis feed speculations about possible sovereign debt defaults in the EMU, particularly of the most vulnerable EMU member countries Greece, Ireland, Italy, Portugal and Spain. In May 2010, the Greek government was de facto illiquid and had to be bailed out by the EU/IMF rescue package. Later on, in November 2010, Ireland had to be bailed out and in May 2011 Portugal received a rescue package. The seemingly notorious need for emergency funds to bail out vulnerable

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online ß 2012 Taylor & Francis http://www.tandfonline.com http://dx.doi.org/10.1080/09603107.2011.646064

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1216 EMU member countries has led to the implementation of the (temporary) European Financial Stability Facility and plans to implement a permanent European Stabilization Mechanism. As an accompanying measure to support EMU banks and governments, the European Central Bank (ECB) relaxed the standards of its refinancing operations by extending the horizon and range of collateral and even offering full allotment for some refinancing operations (Mojon, 2010; Trichet, 2010; Eichler and Hielscher, 2011). In an effort to increase the liquidity in the interbank market, the ECB has cut overnight interest rates from around 4% in October 2008 to around 0.4% half a year later (Wyplosz, 2010). At the time of the writing of this article (November 2011), the crisis in the eurozone is not yet resolved. The austerity program of the Greek government did not lead to the desired results and it is proving difficult to get the Greek public debt back to a sustainable level. In November 2011, market participants also began to worry about the sustainability of the Italian government finances and yield spreads of Italian government bonds (as well as Greek, Portuguese and Spanish government bonds) over the (allegedly risk-free) German government bonds are rallying. Even the ultimate event of withdrawals from the eurozone is now discussed by policymakers and the risk of such withdrawals is perceived by financial markets (Eichler, 2011). All these financial problems associated with the recent financial crisis may have an impact on the external value of the euro – this is the focus of the present article. Several papers show that financial crises can lead to a significant depreciation of the domestic currency or even a currency crisis. In an effort to fight or prevent a banking crisis the central bank often acts as a lender of last resort, which may lead to inflation and currency depreciation (see, for example, Kaminsky and Reinhart, 1999; Miller, 2000). Central banks may also act as a lender of last resort for domestic governments troubled by unsustainable public debt levels or sovereign debt crises, which may also result in significant depreciations of the domestic currency (see, for example, Bauer et al., 2003; Dreher et al., 2006; Herz and Tong, 2008; Maltritz, 2008). One of the most important consequences of the financial crisis for the eurozone is the rapid increase in banking and sovereign debt crisis risk in the EMU. I contribute to the literature by analysing the impact of banking and sovereign debt crisis risk in the EMU on the euro/US dollar exchange rate. According to theory, a banking crisis in the EMU should weaken the external value of the euro since: the ECB may 1

reduce interest rates or expand the liquidity supply in order to avert bank defaults; bank runs may lead to a capital flight out of the eurozone; banks may shift assets outside the eurozone in order to meet non-euro obligations; or a possible vicious circle may emerge where higher banking crisis risk deteriorates the solvency of governments (guaranteeing bank rescue packages).1 Possible sovereign debt crises should also lead to euro depreciation as: the ECB may implement an expansionary monetary policy to reduce the real value of sovereign debt; a loss in investor confidence in sovereign bonds (such as in the case of Greece) may lead to capital withdrawals from the eurozone; or a vicious cycle may emerge where higher sovereign default risk deteriorates the solvency of banks (holding sovereign bonds). Using daily data in the period 3 July 2006 to 30 September 2010, I find that the euro depreciates against the US dollar when banking crisis risk increases (as measured by higher Credit Default Swap (CDS) premiums or lower stock returns of banks) and sovereign debt crisis risk increases (as measured by higher sovereign CDS premiums or bond yield spreads) in the EMU. Significant negative impacts of banking crisis risk can be found for vulnerable and relatively stable EMU member countries, while the negative impact of sovereign debt crisis risk on the external value of the euro is more pronounced for the five vulnerable EMU member countries Greece, Ireland, Italy, Portugal and Spain than for the relatively stable EMU countries Austria, Belgium, France, Germany and the Netherlands. Analysing the impact of CDS premiums of individual EMU banks I find that the euro depreciates against the US dollar when CDS premiums rise – significantly only for medium and large banks. This result suggests that foreign exchange market investors take only possible defaults of medium and large EMU banks into account when setting the euro/US dollar exchange rate since the bailout costs of the too big to fail banks would be much higher (and the implications for inflation and interest rates in the EMU more pronounced) than for small banks.

II. Hypotheses on the Impact of Banking and Sovereign Debt Crisis Risk on the Euro/US Dollar Exchange Rate The following derives from the literature hypotheses regarding the impact of banking and sovereign debt

Maltritz (2010) shows that higher payments for bank bailouts may increase sovereign default risk.

Financial crisis risk and the EUR/USD exchange rate crisis risk in the eurozone on (changes in) the euro/US dollar exchange rate. I expect that if foreign exchange market investors take the impact of a possible banking or sovereign debt crisis in the eurozone on the external value of the euro into account, the euro/ US dollar exchange rate should be affected by empirical measures capturing these crisis risks. In the following two sub-sections I derive first the theoretical hypothesis from the literature and then explain how I measure the crisis risks empirically. The impact of banking crisis risk on the euro/US dollar exchange rate A higher banking crisis risk in the EMU should lead to a depreciation of the euro against the US dollar for several reasons. First, the ECB may act as a lender of last resort for EMU banks in order to avert a banking crisis in the eurozone. Several theoretical papers show that the domestic currency may depreciate sharply when the central bank acts as a lender of last resort to fight a banking crisis (Diaz-Alejandro, 1985; Velasco, 1987; Miller, 2000). Some studies also present empirical evidence of currency depreciation or even currency crises after the outbreak of banking crises (Kaminsky and Reinhart, 1999; Glick and Hutchison, 2001). The goal of the ECB is not only to fight inflation (its primary goal) but also to ensure macroeconomic stability within the EMU (its secondary goal).2 Since the outbreak or aggravation of a banking crisis may lead to serious disruptions on output, wealth and the functioning of financial markets, the ECB may try to avert a crisis outbreak by reducing interest rates or increasing the liquidity supply for banks. Since August 2007, counterparty risk increased severely in the EMU interbank market as indicated by rising interest rate spreads between uncollateralized and collateralized transactions. Since then, the ECB has increasingly played the role as an intermediary between banks in need for liquidity and banks in excess of liquidity (Mojon, 2010). The ECB has made its refinancing operations more flexible (and expansionary) by extending the range of collateral admitted to access ECB liquidity and by increasing the horizon of its refinancing operations through up to 12 months. Moreover, in June, September and December 2009, the ECB offered three 12 month refinancing operations with full allotment in order to meet the excess liquidity demand of EMU banks (in particular the most

1217 vulnerable ones). The June 2009 operation alone led to a liquidity injection of 442 billion euros (Mojon, 2010). The large liquidity injections into the EMU banking sector led to a sharp reduction in money market rates. The European Overnight Interest Rate Average (EONIA) fell from around 4% in October 2008 to around 0.4% half a year later (Wyplosz, 2010). This monetary easing suggests that the ECB acts as a lender of last resort by ensuring the liquidity supply and reducing the capital costs of EMU banks. Higher banking crisis risk in the EMU may thus lead to lower interest rates and, according to uncovered interest parity, to a depreciation of the euro against the US dollar. Second, bank runs in the EMU may trigger capital flights out of the eurozone (Calvo, 1998; Miller, 1998), which may lead to a depreciation of the euro against the US dollar. Third, vulnerable EMU banks which are indebted in foreign currency may be forced to exchange assets into foreign currency thereby exerting pressure on the external value of the euro. Fourth, currency depreciation and banking crisis may also have common causes such as liquidity drains (Chang and Velasco, 2001) or boom-bust cycles in bank lending (McKinnon and Pill, 1997). Fifth, higher banking crisis risk may also have an impact on the euro/US dollar exchange rate by increasing sovereign debt crisis risk. The governments of most EMU countries have implemented rescue packages to support their vulnerable banking sectors. A deterioration of the solvency of the national banking sector thus increases the (implicit) sovereign debt, thereby exerting pressure on the external value of the euro (as outlined in the next sub-section).3 In order to quantify banking crisis risk in the EMU I use two measures: CDS premiums and stock returns of EMU banks. A CDS represents a financial instrument to hedge against the risk that a bank will default on its debt. Rising CDS premiums thus indicate a higher bank default risk. As I aim to measure the risk of a banking crisis for the vulnerable or stable EMU country bloc, I calculate the weighted average of CDS basis points for all banks of the respective country bloc DATASTREAM provides data for by using average bank assets in the period 2006–2010 as the weighting scheme. I use CDS with maturities of 1 year. Table A1 in the Appendix lists the banks included in the CDS premium index of each country bloc.4 I calculate a CDS-based banking crisis

2 The trade-off of the ECB to fight inflation and/or to ensure macroeconomic stability in the EMU may also be discussed in the context of second-generation currency crisis models (Obstfeld, 1994, 1996). 3 Maltritz (2010) shows for the case of the financial crisis in Hungary in 2008 that higher payments for bank bailouts may increase sovereign default risk. 4 As a robustness check I also use equal weighted CDS premium indices in Section III.

1218 risk index for the five vulnerable EMU member countries (Greece, Ireland, Italy, Portugal and Spain) and for the five relatively stable EMU member countries (Austria, Belgium, France, Germany and the Netherlands).5 As an alternative indicator of banking crisis risk I use bank stock returns. According to the efficient markets hypothesis, stock returns will reflect all available information about the stock market investors’ expectations on the bank’s present and future financial conditions. Higher (lower) bank stock returns reflect a better (worse) solvency of a bank and thus a lower (higher) default risk. Of course, from a behavioural finance point of view, one may argue that stock prices do not solely reflect information on the fundamental situation of banks but are also driven by bank managers (who may have an incentive to release manipulated company information in order to influence the bank’s share price and to benefit from rising values of their stock options) or the owners of the bank (who may manipulate the bank’s share price in order to prevent a hostile takeover attempt). However, these possible sources of bank share price movements should be short-lived and the efficient market hypothesis should be true, as generally accepted in the majority of papers (see, for example, Mathur and Sundaram, 1997). To measure stock returns of domestic banks, I employ data on the national banking stock sub-index of the Dow Jones Total Market Index for each country. I use real Gross Domestic Product (GDP) weights to calculate an aggregated bank stock return index for the five vulnerable and the five relatively stable EMU countries. Higher stock returns of the banking sector indicate lower banking crisis risk. Hypothesis 1: I expect that the euro will depreciate against the US dollar when banking crisis risk in the EMU increases (as indicated by higher CDS premiums or lower stock returns of the banking sector).

The impact of sovereign debt crisis risk on the euro/ US dollar exchange rate A rising sovereign debt crisis risk in the EMU may also lead to a depreciation of the euro against the US dollar. First, several theoretical papers show that if 5

S. Eichler central banks act as a lender of last resort for overindebted governments a sovereign debt crisis may lead to a depreciation of the domestic currency (Bauer et al., 2003; Benigno and Missale, 2004; Maltritz, 2008). There is also empirical evidence that a sovereign debt crisis can lead to currency depreciation (Reinhart, 2002; Dreher et al., 2006; Herz and Tong, 2008). Since the subprime crisis has spilled over to the EMU, the ECB’s policy of monetary easing is (implicitly) also directed towards supporting troubled EMU governments. For example, the ECB has begun to buy sovereign bonds of the five vulnerable EMU member countries under its new Securities Markets Programme (irrespective of the high sovereign default probability as expected by market participants) and now also accepts low-rated sovereign EMU bonds as collateral for its refinancing operations. This change in the ECB’s monetary policy suggests that the ECB uses monetary policy to reduce capital costs of EMU governments by supporting sovereign bond prices. The increase in money supply may reduce the real value of sovereign debt by increasing the inflation tax and, in turn, lead to a depreciation of the euro against the US dollar according to relative purchasing power parity.6 Equivalently, a reduction in interest rates (in order to reduce capital costs for EMU governments) may lead to euro depreciation according to uncovered interest parity. Second, expectations of (further) monetary easing in the future would lead to euro depreciation even today. Burnside et al. (2001) show that even expectations that future fiscal deficits would be (partly) financed using the inflation tax may lead to a devaluation of the domestic currency. Financial market measures which capture market participants’ assessment of sovereign default risk may therefore have a strong impact on the euro/US dollar exchange rate. Third, if investors lose confidence in EMU sovereign bonds, as it was observed in the case of Greek sovereign bonds in 2010, sell-offs may occur in government bond markets (thereby aggravating the sovereign debt crisis). These spillover effects may lead to capital withdrawals from the eurozone and a depreciation of the euro.7 Fourth, a sovereign debt crisis cum euro devaluation may also be triggered by a common factor. The global recession in 2009 led to an increase in sovereign debt crisis risk in the EMU

In the regression analysis, in Section III, I also use banking and sovereign debt crisis risk indices for the United States, which are calculated according to the EMU indices. 6 The ECB indicated several times that the expansion of money supply associated with the purchases of sovereign EMU bonds would be sterilized. However, the distinction between quantitative easing and the ECB’s recent monetary policy is only semantic as sterilization ‘means replacing overnight deposits with the central bank with one-week term deposits’ (Buiter and Rahbari, 2010). 7 Chui et al. (2002) employ an open economy version of the Diamond–Dybvig model to explain a liquidity-driven devaluation cum sovereign debt crisis.

Financial crisis risk and the EUR/USD exchange rate (due to lower tax revenues and higher public expenditures, for example) and devaluation pressure on the euro (due to a crunch in export revenues). Fifth, higher sovereign debt crisis risk may also exert pressure on the external value of the euro by increasing banking crisis risk. EMU banks hold considerable amounts of sovereign bonds of EMU governments. Higher sovereign debt crisis risk leads to a reduction in sovereign bond prices, thereby reducing the asset value (and increasing the default risk) of EMU banks, which may lead to a depreciation of the euro against the US dollar (as outlined in the previous sub-section).8 I employ sovereign CDS premiums and bond yield spreads to measure sovereign debt crisis risk in the EMU. According to the case of banks, higher sovereign CDS premiums indicate higher sovereign default risk. As an alternative measure I use sovereign bond yield spreads, which are calculated as the difference between the redemption yield on domestic sovereign bonds and the redemption yield on German sovereign bonds. Higher sovereign bond yield premiums compensate the investor for a possible default on the bond. I use data on sovereign CDS and bonds with a maturity of 10 years provided by DATASTREAM. To obtain an aggregate measure of sovereign debt crisis risk for the five vulnerable and the five relatively stable EMU member countries, I aggregate the country-specific sovereign CDS premiums and bond yield spreads using real GDP weights. Higher sovereign CDS premiums or bond yield spreads indicate higher sovereign debt crisis risk. Hypothesis 2: I expect that the euro will depreciate against the US dollar when sovereign debt crisis risk in the EMU increases (as indicated by higher CDS premiums or bond yield spreads of EMU governments).

III. Empirical Analysis I start the empirical analysis by illustrating the euro/ US dollar exchange rate and banking crisis risk (Fig. 1) or sovereign debt crisis risk (Fig. 2) in the EMU. The dotted line shows the euro/US dollar exchange rate defined as the amount of US dollars one must pay for one euro (measured on the right y-axis), the bold solid line shows the CDS measure for the five vulnerable EMU member countries and the light solid line shows 8

1219

Fig. 1. Banking sector CDS indices and the euro/US dollar exchange rate

Fig. 2. Sovereign CDS indices and the euro/US dollar exchange rate

the CDS measure for the five stable EMU countries (measured on the left y-axis). Overall, the figures show a negative correlation between the external value of the euro and financial crisis risk in the EMU. That is, according to my hypotheses, the euro depreciates against the US dollar (indicated by a falling exchange rate) when banking or sovereign debt crisis risk in the EMU increases (as indicated by rising CDS premiums). From July 2006 (the beginning of the observation period) to July 2008, the euro steadily gained in value against the US dollar, which suggests that foreign exchange market investors did not expect that the subprime crisis would have a major impact on the EMU economies. The first major depreciation of the euro occurred during autumn and winter 2008 when the banking crisis in the United States spilled over to the EMU after Lehman Brothers collapsed. The sharp increase of CDS premiums of stable and vulnerable EMU countries’ banks may have weakened the euro as foreign exchange market investors realized that the quantitative easing of the ECB

For an analysis of a debt crisis on bank stock prices see, for example, Mathur and Sundaram (1997).

S. Eichler

1220 (meant to rescue the EMU-based banks) would reduce interest rates in the EMU. Since the bank rescue packages of EMU governments have implications for (implicit) public debt levels, the higher banking crisis risk was also associated with higher levels of sovereign debt crisis risk, which put further pressure on the external value of the euro. During the second half of 2009, the euro gained in value against the US dollar as banking crisis risk decreased in the EMU after the financial stabilization measures began to calm investors’ fears of further bank collapses. From December 2009 to June 2010, the euro again depreciated against the US dollar as the banking sector problems, recessions, and the decline in asset prices fed speculations about the outbreak of a sovereign debt crisis in the EMU (as indicated by rising sovereign CDS premiums). The ECB purchases sovereign bonds of vulnerable EMU countries under its Securities Markets Programme with implications for possible increases of future inflation and investors still fear sovereign defaults which lead to capital withdrawals from vulnerable EMU member countries. The higher sovereign debt crisis risk led again to increases in banking crisis risk (due to banks’ holdings of sovereign EMU bonds), which further led to euro depreciation. In order to analyse the impact of banking and sovereign debt crisis risk on the euro/US dollar exchange rate in more depth, I apply a Generalized Autoregressive Conditional Heteroscedasticity(1,1) (GARCH(1,1)) model as outlined in Equation 19,10: þ 2 DSovEMU þ 3 DBankUS Det ¼ 0 þ 1 DBankEMU t t t
EUR USD þ 4 DSovUS þ  D i  i 5 t t t þ

5 X i¼1

5þi Deti þ

4 X j¼1

10þj Daydummiesjt þ"t ð1Þ

where 2 t2 ¼ ! þ "2t1 þ t1

ð2Þ

All variables are used in first differences. The mean equation (Equation 1) specifies that the day-over-day (dod) percentage change in the euro/US dollar exchange rate (defined as the change in the log amount of US dollars one must pay for one euro11),

Det , is regressed on a constant, 0 , the dod change in a banking crisis risk variable (CDS premiums or log stock prices of banks) for the EMU countries, , and the dod change in a sovereign debt DBankEMU t crisis risk variable (CDS premiums or bond yield spreads of governments) for the EMU countries, .12 In order to account for the impact of DSovEMU t financial crisis risk in the United States, I include the dod change in a banking crisis risk variable (CDS premiums or log stock prices of US banks), DBankUS t , and the dod change in the CDS premiums of the 13 United States government, DSovUS In order to t . control for the interest parity condition I include the dod change in the difference between 1-year nominal interest rates
for euro funds and US dollar funds, , taken from Garban Information  iUSD D iEUR t t Services. I also P include five lags of the exchange 5 rate change, and day dummies, i¼1 Deti , P4 Daydummies , in order to control for day-ofjt j¼1 the-week effects. In order to account for serial correlation and heteroscedasticity in the error term I use a GARCH(1,1) model. The conditional variance equation (Equation 2) specifies that the conditional variance, t2 , is a function of a constant term, !, the previous day’s squared residual from the mean equation (the Autoregressive Conditional Heteroscedastic (ARCH) term), "2t1 , and the previous day’s conditional variance (the GARCH term), 2 t1 . The estimations use daily data in the period 3 July 2006 to 30 September 2010. The descriptive statistics of all variables are reported in Table A2 in the Appendix. Before estimating the models, I test for unit roots in the variables using the Augmented Dickey–Fuller (ADF) and the Phillips–Perron (PP) tests. The test results are reported in Table A3 in the Appendix and indicate that all variables contain a unit root in levels. Consequently I use all variables in first differences in the estimations. For each estimated specification I report several regression diagnostics including the adjusted R-squared, the log likelihood, the p-value of the Ljung–Box Q-statistic with 20 lagged residuals (testing the null hypothesis of no serial correlation in the residuals), and the p-value of the ARCH Lagrange Multiplier (LM) test with 20

9 Analogously to the case of the EMU member countries, I use measures for banking and sovereign debt crisis risk for the United States in order to account for the impact of these crisis risks on the external value of the US dollar. 10 The majority of papers use GARCH models in order to model exchange rate movements (see, for example, MoralesZumaqueroa and Sosvilla-Rivero, 2011). 11 A rising exchange rate indicates an appreciation of the euro against the US dollar. 12 Depending on the specification, the banking and sovereign debt crisis risk variable is used for the vulnerable and/or the stable EMU member countries. 13 The financial crisis risk variables are defined according to the risk variables for the EMU outlined in Section II. The individual banks included in the bank CDS premium index are reported in Table A1 in the Appendix.

Financial crisis risk and the EUR/USD exchange rate

1221

Table 1. Regression results for the vulnerable EMU members (I) Mean equation CDS premiums banks (D) (vulnerable EMU members) Bank stock returns (vulnerable EMU members) CDS premiums governments (D) (vulnerable EMU members) Sovereign bond yield spreads (D) (vulnerable EMU members) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D) Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(II)

0.016** (2.17) 0.016*** (2.63)

0.000 (1.05) 0.035*** (3.14) 0.965*** (86.78) 0.076 4149.63 0.962 0.512 1109

(IV)

0.023*** (3.79) 0.080*** (6.46) 0.013** (2.55)

0.090*** (9.09) 0.008 (1.55) 0.003** (1.98)

0.003 (1.56) 0.007 (0.93) 0.128 (0.64) 0.013 (0.46) 0.008 (0.27) 0.003 (0.09) 0.003 (0.10) 0.016 (0.57) 0.014 (0.29) 0.033 (0.68) 0.044 (0.89) 0.012 (0.24) 0.005 (0.13)

(III)

0.021** (2.45) 0.004 (0.62) 0.144 (0.77) 0.009 (0.33) 0.017 (0.55) 0.006 (0.22) 0.003 (0.11) 0.016 (0.59) 0.032 (0.66) 0.043 (0.90) 0.053 (1.08) 0.024 (0.48) 0.008 (0.23) 0.000 (1.11) 0.033*** (3.17) 0.966*** (89.13) 0.105 4165.99 0.996 0.398 1109

0.012 (1.52) 0.142 (0.72) 0.010 (0.37) 0.007 (0.21) 0.005 (0.16) 0.002 (0.05) 0.019 (0.66) 0.008 (0.17) 0.027 (0.56) 0.040 (0.79) 0.002 (0.04) 0.009 (0.25) 0.000 (1.04) 0.035*** (3.11) 0.965*** (86.10) 0.068 4146.13 0.917 0.379 1109

0.007* (1.83) 0.022*** (3.49) 0.009 (0.79) 0.162 (1.05) 0.004 (0.13) 0.017 (0.60) 0.000 (0.01) 0.007 (0.23) 0.018 (0.60) 0.030 (0.63) 0.040 (0.83) 0.050 (1.07) 0.016 (0.35) 0.007 (0.21) 0.000 (1.57) 0.034*** (5.98) 0.966*** (187.12) 0.098 4162.61 0.989 0.373 1109

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

lagged residuals (testing the null hypothesis of no heteroscedasticity in the residuals). For all regression models, I find no evidence for serial correlation and heteroscedasticity in the residuals suggesting that the GARCH(1,1) model is appropriately specified.

Tables 1–3 present the estimation results using the financial crisis risk indices of the five vulnerable EMU countries (Table 1), the five stable EMU countries (Table 2), and both country groups (Table 3). In each specification I use the same set of

S. Eichler

1222 Table 2. Regression results for the stable EMU members (I) Mean equation CDS premiums banks (D) (stable EMU members) Bank stock returns (stable EMU members) CDS premiums governments (D) (stable EMU members) Sovereign bond yield spreads (D) (stable EMU members) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D) Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(II)

0.032*** (3.41) 0.022** (2.19)

0.000 (1.05) 0.036*** (2.99) 0.964*** (82.12) 0.053 4141.73 0.908 0.323 1109

(IV)

0.032*** (3.44) 0.067*** (6.51) 0.027*** (2.91)

0.068*** (6.56) 0.010* (1.73) 0.003 (1.25)

0.002 (1.06) 0.010 (1.34) 0.002 (1.03) 0.008 (0.29) 0.007 (0.22) 0.003 (0.10) 0.002 (0.05) 0.014 (0.49) 0.014 (0.29) 0.020 (0.41) 0.036 (0.71) 0.006 (0.12) 0.007 (0.20)

(III)

0.020** (2.31) 0.008 (1.15) 0.003 (1.37) 0.007 (0.25) 0.017 (0.56) 0.004 (0.13) 0.012 (0.41) 0.017 (0.62) 0.026 (0.54) 0.025 (0.51) 0.046 (0.93) 0.013 (0.27) 0.002 (0.06) 0.000 (1.14) 0.034*** (3.12) 0.965*** (87.96) 0.089 4154.80 0.979 0.249 1109

0.014* (1.76) 0.002 (1.02) 0.004 (0.14) 0.006 (0.22) 0.002 (0.09) 0.004 (0.13) 0.014 (0.51) 0.008 (0.16) 0.023 (0.48) 0.033 (0.67) 0.004 (0.08) 0.010 (0.28) 0.000 (1.02) 0.036*** (3.01) 0.964*** (81.32) 0.044 4140.46 0.895 0.390 1109

0.013** (2.48) 0.018** (2.00) 0.013* (1.73) 0.003 (1.38) 0.001 (0.03) 0.017 (0.57) 0.005 (0.18) 0.015 (0.50) 0.018 (0.65) 0.019 (0.40) 0.029 (0.60) 0.043 (0.87) 0.001 (0.02) 0.001 (0.04) 0.000 (1.11) 0.034*** (3.18) 0.965*** (88.93) 0.081 4153.59 0.979 0.288 1109

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

control variables as outlined in Equation 1. In each specification I use one banking crisis risk variable and

14

one sovereign debt crisis risk variable for the EMU and the United States.14

In order to avoid multicollinearity problems, I do not use two alternative measures of financial crisis risk (banking or sovereign debt crisis risk) in the EMU in one specification.

Financial crisis risk and the EUR/USD exchange rate

1223

Table 3. Regression results for the vulnerable and stable EMU members (I) Mean equation CDS premiums banks (D) (vulnerable EMU members) Bank stock returns (vulnerable EMU members) CDS premiums governments (D) (vulnerable EMU members) Sovereign bond yield spreads (D) (vulnerable EMU members) CDS premiums banks (D) (stable EMU members) Bank stock returns (stable EMU members) CDS premiums governments (D) (stable EMU members) Sovereign bond yield spreads (D) (stable EMU members) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D) Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(II)

0.013* (1.69) 0.014** (2.40)

0.071** (2.51) 0.011** (2.04)

0.000 (1.00) 0.034*** (3.10) 0.965*** (85.82) 0.077 4150.24 0.951 0.469 1109

0.091*** (3.59) 0.006 (1.11) 0.013 (0.09)

0.006 (1.13) 0.003 (1.51) 0.022** (2.52) 0.003 (0.37) 0.159 (0.85) 0.011 (0.40) 0.016 (0.52) 0.006 (0.21) 0.004 (0.13) 0.016 (0.57) 0.032 (0.67) 0.039 (0.82) 0.054 (1.09) 0.025 (0.51) 0.008 (0.23) 0.000 (1.12) 0.033*** (3.07) 0.966*** (86.37) 0.106 4167.00 0.993 0.376 1109

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

0.004 (0.88) 0.001 (0.04)

0.009 (0.39) 0.013 (1.32)

0.002 (1.32) 0.006 (0.76) 0.136 (0.68) 0.014 (0.49) 0.007 (0.24) 0.002 (0.08) 0.003 (0.11) 0.015 (0.53) 0.014 (0.30) 0.030 (0.61) 0.044 (0.88) 0.013 (0.27) 0.005 (0.13)

(IV)

0.022*** (3.00)

0.005 (0.36) 0.009 (0.98)

(III)

0.011 (1.47) 0.148 (0.74) 0.010 (0.35) 0.005 (0.17) 0.004 (0.13) 0.001 (0.03) 0.018 (0.63) 0.007 (0.15) 0.027 (0.55) 0.040 (0.80) 0.001 (0.01) 0.010 (0.26) 0.000 (0.98) 0.035*** (3.07) 0.965*** (84.88) 0.051 4146.80 0.911 0.380 1109

0.009 (1.55) 0.021** (2.28) 0.008 (1.19) 0.170 (0.92) 0.003 (0.12) 0.015 (0.51) 0.001 (0.04) 0.007 (0.24) 0.017 (0.62) 0.030 (0.62) 0.040 (0.84) 0.051 (1.03) 0.015 (0.29) 0.007 (0.18) 0.000 (1.08) 0.033*** (3.05) 0.966*** (85.33) 0.098 4163.84 0.985 0.382 1109

1224 Overall, the regression results confirm Hypothesis 1. The results in Tables 1 and 2 suggest that higher banking crisis risk in the eurozone (as indicated by higher CDS premiums of banks or lower bank stock returns) leads to a lower exchange rate, i.e. a depreciation of the euro against the US dollar. This result is significant at least at the 5% level for both alternative measures of banking crisis risk and for vulnerable and stable EMU member countries suggesting that the fragility of both country groups’ banks has contributed to a weakening of the euro in times of higher banking crisis risk. The impact of higher banking crisis risk on the exchange rate is fairly similar for vulnerable and stable EMU countries. A one SD increase in bank CDS premiums (D) yields a depreciation of the euro against the US dollar by 0.08%–0.12% for the vulnerable EMU countries and by around 0.11% for the stable EMU countries. Table 3 reports the estimation results when banking crisis risk indices for vulnerable and stable EMU member states are included in one specification. The results indicate that higher banking crisis risk in vulnerable EMU countries leads to a significant euro depreciation whereas banking crisis risk in stable EMU countries plays no significant role. Foreign exchange market investors seem to anticipate that more fragility in the banking sectors of the vulnerable EMU countries, such as Ireland or Spain, may have a higher impact on the external value of the euro than more fragility in the relatively stable EMU countries. A possible explanation for this finding may be that the ECB is more willing to support the vulnerable member states by reducing the interest rate since these countries may have less fiscal capacities to bail out their banks using public revenues than the more solvent governments of the relatively stable EMU countries. Since the multi-billion euro bailout plans and deposit guarantees may overtax the fiscal capacities of the vulnerable EMU countries, the ECB may reduce interest rates as banking crisis risk rises in these countries, which leads to a depreciation of the euro against the US dollar. The regression results in Tables 1 and 2 largely confirm Hypothesis 2. Higher sovereign debt crisis risk in the eurozone (as indicated by higher sovereign CDS premiums or bond yield spreads) leads to a depreciation of the euro against the US dollar for most specifications. Comparing both country groups yields that in vulnerable EMU countries higher sovereign debt crisis risk leads to larger euro depreciation than in stable EMU countries. A one SD increase in sovereign 15

S. Eichler CDS premiums (D) of vulnerable EMU countries yields a euro depreciation of 0.07%–0.09% against the US dollar. A one SD increase in stable EMU countries’ sovereign CDS premiums (D) leads to a euro depreciation of only 0.04%–0.05% against the US dollar. This finding is further underlined by the regression results of Table 3 where sovereign debt crisis risk indicators of vulnerable and stable EMU countries are simultaneously used in one specification. The results indicate that higher sovereign debt crisis risk in vulnerable EMU countries leads to a significant depreciation of the euro against the US dollar (for two out of four specifications), whereas sovereign debt crisis risk in stable EMU countries plays no significant role. These results suggest that rising sovereign debt crisis risk of the vulnerable EMU country bloc has a much more adverse impact on the external value of the euro than sovereign debt crisis risk in the relatively stable EMU countries. This seems reasonable as the financial problems in the public sectors of the vulnerable EMU countries are much more severe and may trigger much more money creation and interest rate reductions by the ECB, for example, than higher sovereign debt crisis risk in the relatively stable EMU countries. Sovereign debt crisis risk of vulnerable EMU countries seems to be more important for the determination of the euro/US dollar exchange rate than sovereign debt crisis risk of stable EMU countries since the dynamics of public finances in the vulnerable EMU countries are typically perceived as being less sustainable than in the stable EMU countries. A sovereign debt crisis is thus much more likely in Greece, Ireland, Italy, Portugal or Spain than in Austria, Belgium, France, Germany or the Netherlands. It is therefore not surprising that the Securities Markets Programme of the ECB is directed towards purchasing sovereign bonds of vulnerable EMU countries. Moreover, capital withdrawals of investors are more likely triggered by higher sovereign default risk of the vulnerable EMU countries, which are more likely to default according to the financial market figures used. The results of Tables 1 and 2 may also be used to study the sensitivity of the euro/US dollar exchange rate to changes in banking crisis risk versus changes in sovereign debt crisis risk. For the vulnerable EMU countries both financial crisis risks have quite a similar impact on the exchange rate.15 For the stable EMU member countries banking crisis risk has a much larger impact on the exchange rate than

For vulnerable EMU countries, a one SD increase in bank CDS premiums (D) leads to a depreciation of the euro by 0.08%– 0.12%. A one SD increase in sovereign CDS premiums (D) leads to a depreciation of the euro by 0.07%–0.09%.

Financial crisis risk and the EUR/USD exchange rate sovereign debt crisis risk.16 Foreign exchange market investors seem to expect that the governments of stable EMU countries have sufficient fiscal resources to cope with problems in their banking sectors and thus sovereign debt crisis risk in these countries is not perceived as a threat for the euro. For the vulnerable EMU countries the costs of bank rescue packages appear to be so high that banking and sovereign debt crisis risk are interrelated (a possible vicious circle as outlined in Section II). Depreciations of the euro against the US dollar thus seem to be triggered by both types of financial crisis risks of the vulnerable EMU countries and by banking crisis risk of the stable EMU countries. The control variables are largely insignificant. A notable example is the bank stock returns variable in the United States, which has a negative and significant coefficient for all specifications tested. This result suggests that the US dollar depreciates against the euro when banking crisis risk in the United States increases (as indicated by lower US bank stock returns). Similar to the eurozone, rising banking crisis risk is perceived as a threat for the dollar’s value. Contrary to the eurozone, however, sovereign debt crisis risk in the United States is not found to be significant, indicating that foreign exchange market investors do not perceive sufficient default risk for the US government to justify a negative impact on the dollar. As a robustness check for the regression results in Tables 1 to 3, I re-estimate all models using banking and sovereign debt crisis risk indices based on equal country weights (rather than GDP weights as used in the benchmark regressions). Overall, the results of the robustness checks (reported in Tables A4–A6 in the Appendix) confirm the findings of Tables 1 to 3 suggesting that the results are not driven by movements in financial crisis risk in large EMU countries. Default risk of individual EMU banks may also affect the external value of the euro. Table 4 reports the results for the regressions, testing the impact of individual banks’ CDS premiums on the euro/US dollar exchange rate. I use a GARCH(1,1) model and the same control variables as in specification (I) in Table 1 and 2. In Table 4 I report the estimated coefficient of a bank’s CDS premiums together with the associated t-value and significance level. I also report the standardized coefficient measuring the impact of a one SD increase in a bank’s CDS premiums on the exchange rate change. The last column reports the average asset value of the bank in the considered period 2006–2010. 16

1225 The results suggest that for most banks rising CDS premiums lead to a depreciation of the euro against the US dollar suggesting that foreign exchange market investors take the default risk of individual banks into account when setting the euro/US dollar exchange rate. I find that the standardized impact of CDS premiums on the exchange rate change is higher for medium and large banks than for small banks. I find that the largest negative impact of rising CDS premiums on the exchange rate comes from the largest banks in my sample, for example, Societe Generale, Deutsche Bank, Banco Bilbao Vizcaya Argentaria, Unicredito or Banco Santander, while the CDS premiums of the smallest banks in my sample, such as IKB, Fortis, Raiffeisen Zentralbank O¨sterreich or Erste Bank have no significant impact on the exchange rate. Figure 3 displays the standardized coefficient of individual banks’ CDS premiums (on the y-axis) together with the average asset value of banks (on the x-axis). This figure shows that the impact of rising CDS premiums of individual banks on the euro/US dollar exchange rate is greater, the larger the considered bank is. Foreign exchange market investors seem to believe that the default risk of large eurozone banks is of more importance for the external value of the euro than the default risk of small eurozone banks. This result seems reasonable as large banks are typically too big to fail and their bailouts require larger amounts of money (with implications for inflation and interest rates in the EMU), while bailouts of small banks are relatively cheap with minor implications for the external value of the euro. This result suggests that the default risk of systemically relevant banks is of importance for the euro/US dollar exchange rate, while the default risk of minor banks is not.

IV. Conclusions I found that higher levels of banking and sovereign debt crisis risk in the eurozone lead to a depreciation of the euro against the US dollar. While the impact of banking crisis risk on the exchange rate is similar for vulnerable and stable EMU member countries, the external value of the euro is much more sensitive to changes in sovereign debt crisis risk in vulnerable member countries than in stable member countries. Foreign exchange market investors appear to expect that the governments of stable EMU member

For stable EMU countries, a one SD increase in bank CDS premiums (D) leads to a depreciation of the euro by around 0.11%. A one SD increase in sovereign CDS premiums (D) leads to a depreciation of the euro by 0.04%–0.05%.

S. Eichler

1226 Table 4. Estimation results for CDS premiums of individual banks

Bank name Austria Erste Bank Raiffeisen Zentralbank O¨sterreich Belgium Dexia Fortis France BNP Paribas Credit Agricole Natixis Societe Generale Germany Commerzbank Deutsche Bank IKB Deutsche Industriebank The Netherlands ABN Amro ING Greece EFG Eurobank Ergas National Bank of Greece Ireland Allied Irish Banks Bank of Ireland Italy Banca Italease Banca Siena Banca PPO Italiana Banca PPO Di Milano Unicredito Italiano Portugal Banco Commercial Portugues Banco Espirito Santo Spain Banco Intl. Finance Banco Bilbao Vizcaya Argentaria Banco Popular Espaniol Banco Sabadell Banco Santander La Caja de Ahorros

Coefficient

t-value

Standardized coefficient

Assets of the bank

0.003 0.001

1.17 0.55

0.023 0.010

201.26 76.16

0.000 0.000

0.10 1.15

0.002 0.008

574.91 92.91

0.028 0.024 0.002 0.040

5.61*** 5.00*** 0.77 6.91***

0.099 0.103 0.019 0.153

2047.65 1552.34 446.15 1018.76

0.025 0.029 0.000

5.67*** 6.17*** 0.74

0.109 0.128 0.016

839.73 1493.51 35.50

0.016 0.024

4.34*** 4.48***

0.070 0.096

662.30 1317.83

0.001 0.000

1.51 1.34

0.034 0.015

83.88 113.22

0.002 0.002

2.06** 1.93*

0.027 0.026

173.51 186.49

0.001 0.025 0.018 0.020 0.025

2.05** 5.52*** 6.47*** 3.42*** 6.29***

0.068 0.113 0.109 0.089 0.134

22.17 188.96 45.74 43.15 951.80

0.009 0.008

4.78*** 4.52***

0.090 0.094

90.84 73.57

0.003 0.025 0.003 0.004 0.023 0.012

2.32** 6.53*** 1.94* 1.56 6.38*** 3.77***

0.051 0.140 0.032 0.033 0.128 0.104

50.95 496.76 109.84 77.68 970.17 71.20

Notes: t-statistics in parentheses are based on robust SEs. Asset values (in billion euros) are average for the period 2006–2010. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

countries have enough fiscal resources to cope with the problems associated with the subprime crisis, while the governments of vulnerable EMU member countries may face unsustainable sovereign debt dynamics and may have to be bailed out. This result is supported by the observation that the Securities Markets Programme of the ECB is directed towards sovereign bond purchases of vulnerable EMU countries with implications for the external value of the euro.

Moreover, I found that rising default risk of individual eurozone banks leads to a depreciation of the euro against the US dollar. This effect is significant only for medium and large banks, which suggests that foreign exchange market investors expect the bailout costs of too big to fail banks (and their implications for interest rates and inflation in the eurozone) to be a threat for the euro’s external value whereas the bailout costs of small banks have no significant impact.

Financial crisis risk and the EUR/USD exchange rate

Fig. 3. Impact of individual banks’ CDS premiums on the euro/US dollar exchange rate and bank size

This analysis used risk measures based on daily data. The results may therefore shed light on the short-term impact of financial crisis risks on the external value of the euro. An interesting extension of this article may be to analyse the medium and long term impacts of banking and sovereign debt crisis risk on the exchange rate using data on the fundamentals of the banking system (such as loan losses and bank defaults) or the sustainability of public finances (such as public debt levels or fiscal deficits) measured at quarterly or annual frequency. The results suggest that reducing the risk of banking and sovereign debt crises in the eurozone may help in preserving the external value of the euro. Time will show how the recent policy changes in the EMU, such as the changes in the ECB’s monetary policy, the implementation of the European Financial Stability Facility, or reform proposals (such as common eurozone bonds, sovereign debt restructuring mechanisms, or a European Monetary Fund) will influence the external value of the euro.

References Bauer, C., Herz, B. and Karb, V. (2003) The other twins: currency and debt crises, Review of Economics, 54, 248–67. Benigno, A. and Missale, P. (2004) High public debt in currency crises: fundamentals versus signaling effects, Journal of International Money and Finance, 23, 165–88. Buiter, W. H. and Rahbari, E. (2010) Greece and the fiscal crisis in the Eurozone, CEPR Policy Insight No. 51. Burnside, C., Eichenbaum, M. and Rebelo, S. (2001) Prospective deficits and the Asian currency crisis, Journal of Political Economy, 109, 1155–97. Calvo, G. A. (1998) Varieties of Capital-Market Crises, Macmillan, Basingstoke.

1227 Chang, R. and Velasco, A. (2001) A model of financial crises in emerging markets, Quarterly Journal of Economics, 116, 489–517. Cheung, W., Fung, S. and Tsai, S.-C. (2010) Global capital market interdependence and spillover effect of credit risk: evidence from the 2007–2009 global financial crisis, Applied Financial Economics, 20, 85–103. Chui, M., Gai, P. and Haldane, A. G. (2002) Sovereign liquidity crises: analytics and implications for public policy, Journal of Banking and Finance, 26, 519–46. De Grauwe, P. (2010) The financial crisis and the future of the eurozone, Bruges European Economic Policy Briefings No. 21/2010. Diaz-Alejandro, C. (1985) Good-bye financial repression, hello financial crash, Journal of Development Economics, 19, 1–24. Dreher, A., Herz, B. and Karb, V. (2006) Is there a causal link between currency and debt crises?, International Journal of Finance and Economics, 11, 305–25. Eichler, S. (2011) What can currency crisis models tell us about the risk of withdrawal from the EMU? Evidence from ADR data, Journal of Common Market Studies, 9, 719–40. Eichler, S. and Herrera, R. (2011) Extreme dependence with asymmetric thresholds: evidence for the European Monetary Union, Journal of Banking and Finance, 35, 2916–30. Eichler, S. and Hielscher, K. (2011) Does the ECB act as a lender of last resort during the subprime lending crisis? Evidence from monetary policy reaction models, Journal of International Money and Finance. DOI: 10.1016/j.jimonfin.2011.11.009. Glick, R. and Hutchison, M. (2001) Banking and currency crises: how common are twins?, in Financial Crises in Emerging Markets (Eds) R. Glick and M. M. Spiegel, Cambridge University Press, Cambridge, pp. 35–69. Gros, D. and Alcidi, C. (2010) The crisis and the real economy, Intereconomics, 45, 4–10. Herz, B. and Tong, H. (2008) Debt and currency crises – complements or substitutes?, Review of International Economics, 16, 955–70. Kaminsky, G. L. and Reinhart, C. M. (1999) The twin crises: the causes of banking and balance-of-payments problems, American Economic Review, 89, 473–500. Maltritz, D. (2008) Modelling the dependency between currency and debt crises: an option based approach, Economics Letters, 100, 344–7. Maltritz, D. (2010) A compound option approach to model the interrelation between banking crises and country defaults: the case of Hungary 2008, Journal of Banking and Finance, 34, 3025–36. Mathur, I. and Sundaram, S. (1997) Reaction of bank stock prices to the multiple events of the Brazilian debt crisis, Applied Financial Economics, 7, 703–10. McKinnon, R. I. and Pill, H. (1997) Credible economic liberalizations and overborrowing, American Economic Review, 87, 189–93. Miller, V. J. (1998) Domestic bank runs and speculative attacks on foreign currencies, Journal of International Money and Finance, 17, 331–8. Miller, V. (2000) Central bank reactions to banking crises in fixed exchange rate regimes, Journal of Development Economics, 63, 451–72. Mojon, B. (2010) The 2007–2009 financial crisis and the European Central Bank, Open Economies Review, 21, 175–82.

S. Eichler

1228 Morales-Zumaqueroa, A. and Sosvilla-Rivero, S. (2011) The euro and the volatility of exchange rates, Applied Financial Economics, 21, 1235–53. Obstfeld, M. (1994) The logic of currency crises, Cahiers Economiques et Mone´taires, 43, 189–213. Obstfeld, M. (1996) Models of currency crises with selffulfilling features, European Economic Review, 40, 1037–47. Reinhart, C. M. (2002) Default, currency crises, and sovereign credit ratings, World Bank Economic Review, 16, 151–70.

Trichet, J.-C. (2010) State of the union: the financial crisis and the ECB’s response between 2007 and 2009, Journal of Common Market Studies, 48, 7–19. Velasco, A. (1987) Financial crises and balanceof-payments crises: a simple model of the Southern Cone experience, Journal of Development Economics, 27, 263–83. Wyplosz, C. (2010) The eurozone in the current crisis, ADBI Working Paper No. 207.

Appendix

Table A1. Domestic banks included in the national CDS premiums indices

Table A2. Descriptive statistics Variable

Country

Included domestic banks

Vulnerable EMU countries Greece EFG Eurobank Ergas,a National Bank of Greecea Ireland Allied Irish Banks, Bank of Ireland Italy Banca Italease, Banca di Siena, Banca Populare Italiana, Banca Populare di Milano, Unicredito Italiano Portugal Banco Commercial Portugues, Banco Espirito Santo Spain Banco Intl. Finance, Banco Bilbao Vizcaya Argentaria, Banco Popular Espanol, Banco Sabadell, Banco Santander, La Caja de Ahorros Stable EMU countries Austria Erste Bank, Raiffeisen Zentralbank O¨sterreich Belgium Dexia, Fortis France BNP Paribas, Credit Agricole, Natixis, Societe Generale Germany Commerzbank, Deutsche Bank, IKB Deutsche Industriebank The Netherlands ABN Amro, ING United States Bank of America, Citigroup, Goldman Sachs, JP Morgan, Merrill Lynch, Morgan Stanley, Wachovia, Wells Fargo Note: aIncluded through September 2009.

Mean

US dollar/euro 5.9E05 exchange rate (D log) Euro to US dollar 0.002 interest rate spread (D) Vulnerable EMU countries CDS premiums 0.213 banks (D) Bank stock 4.3E04 returns 0.228 CDS premiums governments (D) 0.002 Sovereign bond yield spreads (D) Stable EMU countries CDS premiums 0.074 banks (D) Bank stock 3.4E04 returns CDS premiums 0.060 governments (D) 4.3E04 Sovereign bond yield spreads (D) United States CDS premiums 0.074 banks (D) Bank stock 2.3E04 returns CDS premiums 0.042 government (D)

SD

Minimum Maximum

0.007

0.038

0.046

0.114

0.800

0.710

5.137 62.781

46.194

0.094

0.189

5.425 86.404

43.906

0.648

0.244

3.574 26.498

29.415

0.115

0.171

1.855 10.129

11.070

0.279

0.242

0.022

0.048

0.027

0.038

15.049 198.252 207.630 0.199

0.219

2.040 19.657

19.005

0.036

Financial crisis risk and the EUR/USD exchange rate

1229

Table A3. Unit root test results Variable US dollar/euro exchange rate (log) Euro to US dollar interest rate spread Vulnerable EMU countries CDS premiums banks Bank stock index (log) CDS premiums governments Sovereign bond yield spreads Stable EMU countries CDS premiums banks Bank stock index (log) CDS premiums governments Sovereign bond yield spreads United States CDS premiums banks Bank stock index (log) CDS premiums government

ADF

PP

1.844 (31.934)*** 2.052 (7.823)***

1.878 (31.941)*** 2.051 (48.924)***

0.381 (27.881)*** 1.085 (15.658)*** 0.221 (7.213)*** 1.707 (7.299)***

0.517 (27.756)*** 1.137 (30.937)*** 0.158 (27.679)*** 0.970 (33.534)***

1.431 (27.472)*** 1.233 (16.128)*** 1.144 (28.088)*** 1.690 (13.351)***

1.530 (27.219)*** 1.272 (31.134)*** 1.081 (28.086)*** 2.887** (57.877)***

2.141 (7.789)*** 1.769 (6.815)*** 1.402 (12.329)***

2.241 (26.866)*** 1.919 (38.363)*** 1.477 (34.989)***

Notes: Test statistics for variables in levels and first differences (in parentheses); For the order of the autoregressive correction for the ADF test, we use the modified Akaike Information Criterion (AIC). ** and *** denote significance at the 5 and 1% levels, respectively.

Table A4. Regression results for the vulnerable EMU members (equal weighted index) (I) Mean equation CDS premiums banks (D) (vulnerable EMU members) Bank stock returns (vulnerable EMU members) CDS premiums governments (D) (vulnerable EMU members) Sovereign bond yield spreads (D) (vulnerable EMU members) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D)

(II)

0.007* (1.80) 0.013*** (3.22)

(IV)

0.010*** (4.35) 0.068*** (6.01) 0.009** (2.40)

0.076*** (7.31) 0.006** (2.35) 0.005*** (2.92)

0.004*** (2.62) 0.010 (1.25) 0.146 (0.73)

(III)

0.016* (1.88) 0.007 (0.90) 0.178 (0.94)

0.015 (1.39) 0.179 (1.15)

0.005** (2.26) 0.016* (1.86) 0.010 (1.39) 0.202 (1.08) (continued )

S. Eichler

1230 Table A4. Continued

Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(I)

(II)

(III)

(IV)

0.011 (0.41) 0.008 (0.24) 0.001 (0.02) 0.002 (0.07) 0.018 (0.63) 0.014 (0.29) 0.032 (0.66) 0.045 (0.90) 0.011 (0.23) 0.00473 (0.13)

0.005 (0.18) 0.015 (0.51) 0.003 (0.10) 0.000 (0.00) 0.015 (0.55) 0.039 (0.81) 0.049 (1.03) 0.051 (1.04) 0.020 (0.41) 0.00969 (0.27)

0.006 (0.16) 0.007 (0.24) 0.008 (0.28) 0.002 (0.07) 0.019 (0.62) 0.009 (0.17) 0.030 (0.62) 0.040 (0.87) 0.005 (0.10) 0.00805 (0.25)

0.002 (0.06) 0.017 (0.56) 0.001 (0.04) 0.000 (0.01) 0.016 (0.57) 0.037 (0.76) 0.049 (1.02) 0.048 (0.98) 0.016 (0.33) 0.00865 (0.24)

0.000 (1.03) 0.035*** (3.09) 0.965*** (83.41) 0.049 4144.112 0.971 0.678 1109

0.000 (1.11) 0.035*** (3.25) 0.965*** (88.96) 0.082 4159.678 0.996 0.394 1109

0.000 (1.39) 0.036*** (5.87) 0.964*** (178.44) 0.039 4139.883 0.937 0.585 1109

0.000 (1.13) 0.035*** (3.21) 0.965*** (87.49) 0.078 4157.990 0.993 0.394 1109

(III)

(IV)

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

Table A5. Regression results for the stable EMU members (equal weighted index) (I) Mean equation CDS premiums banks (D) (stable EMU members) Bank stock returns (stable EMU members) CDS premiums governments (D) (stable EMU members) Sovereign bond yield spreads (D) (stable EMU members) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D)

(II)

0.016* (1.76) 0.012** (2.02)

0.014*** (3.35) 0.071*** (7.04) 0.014** (2.46) 0.014*** (2.84) 0.005*** (3.17)

0.004* (1.74) 0.013 (1.60) 0.231 (1.11)

0.070*** (6.66)

0.019** (2.22) 0.009 (1.19) 0.227 (1.19)

0.015 (1.41) 0.245 (1.56)

0.013** (2.26) 0.017** (1.96) 0.011 (1.53) 0.242 (1.27) (continued )

Financial crisis risk and the EUR/USD exchange rate

1231

Table A5. Continued

Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(I)

(II)

(III)

(IV)

0.005 (0.16) 0.006 (0.19) 0.004 (0.13) 0.004 (0.12) 0.014 (0.49) 0.021 (0.43) 0.027 (0.55) 0.040 (0.79) 0.008 (0.15) 0.00267 (0.07)

0.004 (0.13) 0.016 (0.51) 0.003 (0.09) 0.012 (0.40) 0.017 (0.63) 0.029 (0.58) 0.025 (0.51) 0.045 (0.90) 0.011 (0.22) 0.00271 (0.07)

0.001 (0.04) 0.008 (0.26) 0.005 (0.17) 0.005 (0.15) 0.014 (0.45) 0.012 (0.23) 0.028 (0.60) 0.037 (0.81) 0.006 (0.13) 0.00779 (0.24)

0.000 (0.01) 0.018 (0.58) 0.001 (0.05) 0.013 (0.43) 0.018 (0.64) 0.020 (0.41) 0.027 (0.55) 0.041 (0.83) 0.002 (0.03) 0.00231 (0.06)

0.000 (1.00) 0.036*** (2.91) 0.964*** (77.87) 0.026 4134.497 0.899 0.593 1109

0.000 (1.10) 0.034*** (3.20) 0.965*** (89.26) 0.084 4153.273 0.980 0.258 1109

0.000 (1.28) 0.036*** (6.03) 0.964*** (183.15) 0.021 4135.293 0.886 0.684 1109

0.000 (1.01) 0.034*** (3.19) 0.965*** (88.69) 0.080 4152.976 0.979 0.368 1109

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

Table A6. Regression results for the vulnerable and stable EMU members (equal weighted index) (I) Mean equation CDS premiums banks (D) (vulnerable EMU members) Bank stock returns (vulnerable EMU members) CDS premiums governments (D) (vulnerable EMU members) Sovereign bond yield spreads (D) (vulnerable EMU members) CDS premiums banks (D) (stable EMU members) Bank stock returns (stable EMU members) CDS premiums governments (D) (stable EMU members) Sovereign bond yield spreads (D) (stable EMU members)

0.006* (1.85) 0.044** (2.40) 0.012*** (2.87)

(II)

(IV)

0.009*** (3.11) 0.052*** (3.15) 0.008** (2.13) 0.005* (1.74) 0.002 (0.44)

0.001 (0.10) 0.007 (1.29)

(III)

0.030* (1.78) 0.008 (1.55)

0.004* (1.82) 0.027 (1.62)

0.010* (1.84)

0.008 (1.33) (continued )

S. Eichler

1232 Table A6. Continued (I) CDS premiums banks (D) (United States) Bank stock returns (United States) CDS premiums government (D) (United States) Interest rate spread (euro interest rate minus US dollar interest rate) (D) Exchange rate change (lagged 1 day) Exchange rate change (lagged 2 days) Exchange rate change (lagged 3 days) Exchange rate change (lagged 4 days) Exchange rate change (lagged 5 days) Dummy Monday Dummy Tuesday Dummy Wednesday Dummy Thursday Constant Variance equation ! "2t1 2 t1

Adjusted R-squared Log-likelihood Ljung–Box Q-statistics (p-value) ARCH LM test (p-value) Number of observations

(II)

0.004* (1.94) 0.008 (1.02) 0.147 (0.74) 0.012 (0.43) 0.006 (0.19) 0.001 (0.03) 0.003 (0.09) 0.017 (0.59) 0.016 (0.34) 0.030 (0.62) 0.046 (0.91) 0.014 (0.29) 0.004 (0.10) 0.000 (0.98) 0.035*** (3.06) 0.965*** (82.51) 0.051 4145.056 0.957 0.641 1109

(III)

(IV)

0.005*** (2.70) 0.020** (2.23) 0.004 (0.54) 0.182 (0.97) 0.007 (0.28) 0.015 (0.48) 0.005 (0.17) 0.004 (0.12) 0.015 (0.54) 0.035 (0.73) 0.040 (0.83) 0.051 (1.03) 0.021 (0.43) 0.008 (0.21) 0.000 (1.12) 0.034*** (3.22) 0.965*** (88.85) 0.093 4162.587 0.993 0.333 1109

Notes: t-statistics in parentheses are based on robust SEs. *, ** and *** denote significance at the 10, 5 and 1% levels, respectively.

0.014 (1.27) 0.192 (1.23) 0.005 (0.15) 0.006 (0.21) 0.008 (0.28) 0.001 (0.04) 0.018 (0.57) 0.006 (0.12) 0.028 (0.58) 0.039 (0.85) 0.001 (0.02) 0.010 (0.32) 0.000 (1.35) 0.036*** (5.85) 0.964*** (179.14) 0.038 4141.540 0.919 0.606 1109

0.018** (2.02) 0.009 (1.25) 0.213 (1.15) 0.000 (0.02) 0.017 (0.56) 0.000 (0.00) 0.004 (0.15) 0.016 (0.57) 0.029 (0.60) 0.041 (0.85) 0.046 (0.93) 0.010 (0.20) 0.004 (0.11) 0.000 (1.10) 0.034*** (3.15) 0.965*** (86.78) 0.087 4160.365 0.989 0.415 1109

Applied Mathematical Finance, Vol. 19, No. 1, 59–95, February 2012

The Implied Market Price of Weather Risk WOLFGANG KARL HÄRDLE & BRENDA LÓPEZ CABRERA Ladislaus von Bortkiewicz Chair of Statistics, Humboldt University of Berlin, Berlin, Germany

(Received 7 May 2010; in revised form 28 February 2011) ABSTRACT Weather derivatives (WD) are end-products of a process known as securitization that transforms non-tradable risk factors (weather) into tradable financial assets. For pricing and hedging non-tradable assets, one essentially needs to incorporate the market price of risk (MPR), which is an important parameter of the associated equivalent martingale measure (EMM). The majority of papers so far has priced non-tradable assets assuming zero or constant MPR, but this assumption yields biased prices and has never been quantified earlier under the EMM framework. Given that liquid-derivative contracts based on daily temperature are traded on the Chicago Mercantile Exchange (CME), we infer the MPR from traded futures-type contracts (CAT, CDD, HDD and AAT). The results show how the MPR significantly differs from 0, how it varies in time and changes in sign. It can be parameterized, given its dependencies on time and temperature seasonal variation. We establish connections between the market risk premium (RP) and the MPR. KEY WORDS: CAR process, CME, HDD, seasonal volatitity, risk premium

1. Introduction In the 1990s weather derivatives (WD) were developed to hedge against the random nature of temperature variations that constitute weather risk. WD are financial contracts with payments based on weather-related measurements. WD cover against volatility caused by temperature, rainfall wind, snow, and frost. The key factor in efficient usage of WD is a reliable valuation procedure. However, due to their specific nature one encounters several difficulties. First, because the underlying weather (and indices) is not tradable and second, the WD market is incomplete, meaning that the WD cannot be cost-efficiently replicated by other WD. Since the largest portion of WD traded at Chicago Mercantile Exchange (CME) is written on temperature indices, we concentrate our research on temperature derivatives. There have been basically four branches of temperature derivative pricing: actuarial approach, indifference pricing, general equilibrium theory or pricing via no arbitrage arguments. While the actuarial approach considers the conditional expectation of the pay-off under the real physical measure discounted at the riskless rate (Brix et al., 2005), the indifference pricing relies on the equivalent utility principle (Barrieu and El Karoui, 2002; Brockett et al., 2010) and the general equilibrium theory assumes Correspondence Address: Brenda López Cabrera, Ladislaus von Bortkiewicz Chair of Statistics, Humboldt University of Berlin, Berlin, Germany. Tel: +49(0)30 2093 1457 Email: [email protected] 1350-486X Print/1466-4313 Online/12/010059–37 © 2012 Taylor & Francis http://dx.doi.org/10.1080/1350486X.2011.591170

60 W. K. Härdle and B. López Cabrera investors’ preferences and rules of Pareto optimal risk allocation (Cao and Wei, 2004; Horst and Mueller, 2007; Richards et al., 2004). The Martingale approach, although less demanding in terms of assumptions, concentrates on the econometric modelling of the underlying dynamics and requires the selection of an adequate equivalent martingale measure (EMM) to value the pay-offs by taking expectations (Alaton et al., 2002; Benth, 2003; Benth and Saltyte-Benth, 2007; Benth et al., 2007; Brody et al., 2002; Huang-Hsi et al., 2008; Mraoua and Bari, 2007). Here we prefer the latter approach. First, since the underlying (temperature) we consider is of local nature, our analysis aims at understanding the pricing at different locations around the world. Second, the EMM approach helps identify the market price of risk (MPR), which is an important parameter of the associated EMM, and it is indispensable for pricing and hedging non-tradable assets. The MPR can be extracted from traded securities and one can use this value to price other derivatives, though any inference about the MPR requires an assumption about its specification. The MPR is of high scientific interest, not only for financial risk analysis, but also for better economic modelling of fair valuation of risk. Constantinides (1987) and Landskroner (1977) studied the MPR of tradable assets in the Capital Asset Pricing Model (CAPM) framework. For pricing interest rate derivatives, Vasicek (1977) assumed a constant market price of interest rate, while Hull and White (1990) used the specification of Cox et al. (1985). In the oil market, Gibson and Schwartz (1990) supposed an intertemporal constant market price of crude oil conveniences yield risk. Benth et al. (2008) introduced a parameterization of the MPR to price electricity derivatives. In the WD framework, Cao and Wei (2004) and Richards et al. (2004) studied the MPR as an implicit parameter in a generalization of the Lucas’ (1978) equilibrium framework. They showed that the MPR is not only statistically significant for temperature derivatives, but also economically large as well. However, calibration problems arise with the methodology suggested by Cao and Wei (2004), since it deals with a global model like the Lucas’ (1978) approach while weather is locally specified. Benth and Saltyte-Benth (2007) introduced theoretical ideas of equivalent changes of measure to get no arbitrage futures/option prices written on different temperature indices. Huang-Hsi et al. (2008) examined the MPR of the Taiwan Stock Exchange Capitalization-Weighted Stock Index ((the mean of stock returns – risk-free interest rate)/SD of stock returns) and used it as a proxy for the MPR on temperature option prices. The majority of temperature pricing papers so far has priced temperature derivatives assuming 0 or constant MPR (Alaton et al., 2002; Cao and Wei, 2004; Dorfleitner and Wimmer, 2010; Huang-Hsi et al., 2008; Richards et al., 2004), but this assumption yields biased prices and has never been quantified earlier using the EMM framework. This article deals exactly with the differences between ‘historical’ and ‘risk neutral’ behaviours of temperature. The contribution of this article is threefold. First, in contrast to Campbell and Diebold (2005), Benth and Saltyte-Benth (2007) and Benth et al. (2007), we correct for seasonality and seasonal variation of temperature with a local smoothing approach to get, independently of the chosen location, the driving stochastics close to a Gaussian Process and with that being able to apply pricing technique tools of financial mathematics (Karatzas and Shreve, 2001). Second and the main contribution, using statistical modelling and given that liquid derivative contracts based on daily

The Implied Market Price of Weather Risk 61 temperature are traded on the CME, we imply the MPR from traded futures-type contracts (CAT/HDD/CDD/AAT) based on a well-known pricing model developed by Benth et al. (2007). We have chosen this methodology because it is a stationary model that fits the stylized characteristics of temperature well; it nests a number of previous models (Alaton et al., 2002; Benth, 2003; Benth and Saltyte-Benth, 2005, 2007; Brody et al., 2002; Dornier and Querel, 2007); it provides closed-form pricing formulas; and it computes, after deriving the MPR, non-arbitrage prices based on a continuoustime hedging strategy. Moreover, the price dynamics of futures are easy to compute and require only a one-time estimation. Our implied MPR approach is a calibration procedure for financial engineering purposes. In the calibration exercise, a single date (but different time horizons and calibrated instruments are used) is required, since the model is recalibrated daily to detect intertemporal effects. Moreover, we use an economic and statistical testing approach, where we start from a specification of the MPR and check consistency with the data. The aim of this analysis is to study the effect of different MPR specifications (a constant, a (two) piecewise linear function, a time-deterministic function and a ‘financial-bootstrapping’) on the temperature futures prices. The statistical point of view is to beat this as an inverse problem with different degrees of smoothness expressed through the penalty parameter of a smoothing spline. The degrees of smoothness will allow for a term structure of risk. Since smoothing estimates are fundamentally different from estimating a deterministic function, we also assure our results by fitting a parametric function to all available contract prices (calendar year estimation). The economic point of view is to detect possible time dependencies that can be explained by investor’s preferences in order to hedge weather risk. Our findings that the MPR differs significantly from 0 confirm the results found in Cao and Wei (2004), Huang-Hsi et al. (2008), Richards et al. (2004) and Alaton et al. (2002), but we differ from them, by showing that it varies in time and changes in sign. It is not a reflection of bad model specification, but data-extracted MPR. This contradicts the assumption made earlier in the literature that the MPR is 0 or constant and rules out the ‘burn-in’ analysis, which is popular among practitioners since it uses the historical average index value as the price for the futures (Brix et al., 2005). This brings significant challenges to the statistical branch of the pricing literature. We also establish connections between the market risk premium (RP) (a Girsanovtype change of probability) and the MPR. As a third contribution, we discuss how to price over-the-counter (OTC) temperature derivatives with the information extracted. Our article is structured as follows. Section 2 presents the fundamentals of temperature derivatives (futures and options) and describes the temperature data and the temperature futures traded at CME, the biggest market offering this kind of product. Section 3 is devoted to explaining the dynamics of temperature data – the econometric part. The temperature model captures linear trend, seasonality, mean reversion, intertemporal correlations and seasonal volatility effects. Section 4 – the financial mathematics part – connects the weather dynamics with the pricing methodology. Section 5 solves the inverse problem of determining the MPR of CME temperature futures using different specifications. Section 1 introduces the estimation results and test procedures of our specifications applied into temperature-derivative data. Here we give (statistical and economic) interpretations of the estimated MPR. The pricing of

62 W. K. Härdle and B. López Cabrera OTC temperature products is discussed at the end of this section. Section 6 concludes the article. All computations in this article were carried out in Matlab version 7.6 (The MathWorks, Inc., Natick, MA, USA). To simplify notation, dates are denoted with yyyymmdd format. 2. Temperature Derivatives The largest portion of futures and options written on temperature indices is traded on the CME. Most of the temperature derivatives are written on daily average temperature indices, rather than on the underlying temperature by itself. A call option written on futures  F(t,τ1 ,τ2 ) with exercise time t ≤ τ1 and delivery over a period [τ1 , τ2 ] will pay max F(t,τ1 ,τ2 ) − K, 0 at the end of the measurement period [τ1 , τ2 ]. The most common weather indices on temperature are Heating Degree Day (HDD), Cooling Degree Day (CDD) and Cumulative Averages (CAT). The HDD index measures the temperature over a period [τ1 , τ2 ], usually between October and April: HDD(τ1 , τ2 ) =



τ2

max(c − Tu , 0) du,

(1)

τ1

where c is the baseline temperature (typically 18◦ C or 65◦ F) and Tu = (Tu,max + Tu,min )/2 is the average temperature on day u. Similarly, the CDD index measures the temperature over a period [τ1 , τ2 ], usually between April and October: CDD(τ1 , τ2 ) =



τ2

max(Tu − c, 0) du.

(2)

τ1

The HDD and the CDD index are used to trade futures and options in 24 US cities, 6 Canadian cities and 3 Australian cities. The CAT index accounts the accumulated average temperature over [τ1 , τ2 ]: CAT(τ1 , τ2 ) =



τ2

Tu du.

(3)

τ1

The CAT index is the substitution of the CDD index for 11 European cities. Since max(Tu − c, 0) − max(c − Tu , 0) = Tu − c, we get the HDD–CDD parity: CDD(τ1 , τ2 ) − HDD(τ1 , τ2 ) = CAT(τ1 , τ2 ) − c(τ2 − τ1 ).

(4)

Therefore, it is sufficient to analyse only HDD and CAT indices. An index similar to the CAT index is the Pacific Rim Index, which measures the accumulated total of 24-hr average temperature (C24AT) over a period [τ1 , τ2 ] days for Japanese cities: C24AT(τ1 , τ2 ) =



τ2 τ1

u du, T

(5)

 u = 1 24 Tui dui and Tui denotes the temperature of hour ui . A difference of where T 24 1 the CAT and the C24AT index is that the latter is traded over the whole year. Note that

The Implied Market Price of Weather Risk 63 temperature is a continuous-time process even though the indices used as underlying for temperature futures contracts are discretely monitored. As temperature is not a marketable asset, the replication arguments for any temperature futures contract do not hold and incompleteness of the market follows. In this context, any probability measure Q equivalent to the objective P is also an EMM and a risk neutral probability turns all the tradable assets into martingales after discounting. However, since temperature futures/option prices dynamics are indeed tradable assets, they must be free of arbitrage. Thanks to the Girsanov theorem, equivalent changes of measures are simply associated with changes of drift. Hence, under a probability space (, F, Q) with a filtration {Ft }0≤t≤τmax , where τmax denotes a maximal time covering all times of interest in the market, we choose a parameterized equivalent pricing measure Q = Qθ that completes the market and pin it down to compute the arbitrage-free temperature futures price: F(t,τ1 ,τ2 ) = E Qθ [Y |Ft ],

(6)

where Y refers to the pay-off from the temperature index in Equations (2)–(5). The MPR θ is assumed to be a real-valued, bounded and piecewise continuous function. We later relax that assumption, by considering the time-dependent MPR θt . In fact, the MPR can depend on anything that can affect investors’ attitudes. The MPR can be inferred from market data. The choice of Q determines the RP demanded for investors for holding the temperature derivative, and opposite, having knowledge of the RP determines the choice of the risk-neutral probability. The RP is defined as a drift of the spot dynamics or a Girsanov-type change of probability. In Equation (6), the futures price is set under a risk-neutral probability Q = Qθ , thereby the RP measures exactly the differences between the risk-neutral F(t,τ1i ,τ2i ,Q) (market prices) and the temperature market probability predictions Fˆ (t,τ i ,τ i ,P) (under P): 1

2

RP = F(t,τ1i ,τ2i ,Q) − Fˆ (t,τ1i ,τ2i ,P) .

(7)

Using the ‘burn-in’ approach of Brix et al. (2005), the futures price is only the historical average index value, therefore there is no RP since Q = P. 2.1 Data We have temperature data available from 35 US, 30 German, 159 Chinese and 9 European weather stations. The temperature data were obtained from the National Climatic Data Center (NCDC), the Deutscher Wetterdienst (DWD), Bloomberg Professional Service, the Japanese Meteorological Agency (JMA) and the China Meteorological Administration (CMA). The temperature data contain the minimum, maximum and average daily temperatures measured in degree Fahrenheit for US cities and degree Celsius for other cities. The data set period is, in most of the cities, from 19470101 to 20091231. The WD data traded at CME were provided by Bloomberg Professional Service. We use daily closing prices from 20000101 to 20091231. The measurement periods for the

64 W. K. Härdle and B. López Cabrera different temperature indices are standardized to be as each month of the year or as seasonal strips (minimum of 2 and maximum of 7 consecutive calendar months). The futures and options at the CME are cash settled, that is, the owner of a futures contract receives 20 times the index at the end of the measurement period, in return for a fixed price. The currency is British pounds for the European futures contracts, US dollars for the US contracts and Japanese Yen for the Asian cities. The minimum price increment is 1 index point. The degree day metric is Celsius or Fahrenheit and the termination of the trading is two calendar days following the expiration of the contract month. The accumulation period of each CAT/CDD/HDD/C24AT index futures contract begins with the first calendar day of the contract month and ends with the calendar day of the contract month. Earth Satellite Corporation (ESC) reports to CME the daily average temperature. Traders bet that the temperature will not exceed the estimates from ESC. 3. Temperature Dynamics In order to derive explicitly no arbitrage prices for temperature derivatives, we need first to describe the dynamics of the underlying under the physical measure. This article studies the average daily temperature data (because most of the temperature derivative trading is based on this quantity) for US, European and Asian cities. In particular, we analyse the weather dynamics for Atlanta, Portland, Houston, New York, Berlin, Essen, Tokyo, Osaka, Beijing and Taipei (Table 1). Our interest in these cities is because all of them with the exception of the latter two are traded at CME and because a casual examination of the trading statistics on the CME website reveals that the Atlanta HDD, Houston CDD and Portland CDD temperature contracts have relatively more liquidity. Most of the literature that discuss models for daily average temperature and capture a linear trend (due to global warming and urbanization), seasonality (peaks in cooler winter and warmer summers), mean reversion, seasonal volatility (a variation that varies seasonally) and strong correlations (long memory); see, for example, Alaton et al. (2002), Cao and Wei (2004), Campbell and Diebold (2005) and Benth et al. (2007). They differ from their definition of temperature variations, which is exactly the component that characterizes weather risk. Here we show that an autoregressive (AR) model AR of high order (p) for the detrended daily average temperatures (rather than the underlying temperature itself) is enough to capture the stylized facts of temperature. We first need to remove the seasonality in mean t from the daily temperature series Tt , check intertemporal correlations and remove the seasonality in variance to deal with a stationary process. The deterministic seasonal mean component can be approximated with Fourier-truncated series (FTS): t = a + bt +

L  l=1

cl cos



2π (t − dl ) , l · 365

(8)

where the coefficients a and b indicate the average daily temperature and global warming, respectively. We observe low temperatures in winter times and high temperatures in summer for different locations. The temperature data sets do not deviate from its

19480101–20081204 19730101–20090831 19480101–20080527 19700101–20090731 19700101–20081204 19490101–20081204 19730101–20090604 19480101–20081204 19920101–20090806 19730101–20090831

Period 61.95 (61.95, 61.96) 12.72 (12.71, 12.73) 9.72 (9.71, 9.74) 10.80 (10.79, 10.81) 68.52 (68.51, 68.52) 53.86 (53.86, 53.87) 16.78 (16.77, 16.79) 55.35 (55.35, 55.36) 23.32 (23.31, 23.33) 16.32 (16.31, 16.33)

aˆ (CI)

cˆ 1 (CI) 18.32 (18.31, 18.33) 14.93 (14.92, 14.94) 9.75 (9.74, 9.77) 8.02 (8.01, 8.03) 15.62 (15.62, 15.63) 21.43 (21.42, 21.44) 11.61 (11.60, 11.62) 14.36 (14.36, 14.37) 6.67 (6.66, 6.68) 10.38 (10.37, 10.38)

bˆ (CI) −0.0025 (−0.0081, 0.0031) 0.0001 (−0.0070, 0.0073) −0.0004 (−0.0147, 0.0139) −0.0020 (−0.0134, 0.0093) −0.0006 (−0.0052, 0.0039) −0.0004 (−0.0079, 0.0071) −0.0021 (−0.0109, 0.0067) −0.0116 (−0.0166, −0.0065) 0.0023 (−0.0086, 0.0133) −0.0003 (−0.0085, 0.0079)

−165.02 (−165.03, −165.02) −169.59 (−169.59, −169.58) −164.79 (−164.81, −164.78) −161.72 (−161.73, −161.71) −165.78 (−165.79, −165.78) −156.27 (−156.27, −156.26) −153.57 (−153.58, −153.56) −155.58 (−155.58, −155.57) −158.67 (−158.68, −158.66) −153.52 (−153.53, −153.52)

dˆ 1 (CI)

Notes: CI, Confidence interval. All coefficients are non-zero at 1% significance level. CIs are given in parentheses. Dates given in yyyymmdd format. The daily temperature is measured in degree Celsius, except for American cities measured in degree Fahrenheit.

Atlanta Beijing Berlin Essen Houston New York Osaka Portland Taipei Tokyo

City

Table 1. Coefficients of the Fourier-truncated seasonal series of average daily temperatures in different cities.

The Implied Market Price of Weather Risk 65

66 W. K. Härdle and B. López Cabrera mean level and in most of the cases a linear trend at 1% significance level is detectable as it is displayed in Table 1. Our findings are similar to Alaton et al. (2002) and Benth et al. (2007) for Sweden; Benth et al. (2007) for Lithuania; Campbell and Diebold (2005) for the United States; and Papazian and Skiadopoulos (2010) for Barcelona, London, Paris and Rome. In our empirical results, the number of periodic terms of the FTS series varies from city to city, sometimes from 4 to 21 or more terms. We notice that the series expansion in Equation (8) with more and more periodic terms provides a fine tuning, but this will increase the number of parameters. Here we propose a different way to correct for seasonality. We show that a local smoothing approach does that job instead, but with less technical expression. Asymptotically they can be approximated by FTS estimators. For a fixed time point s ∈ [1, 365], we smooth s with a Local Linear Regression (LLR) estimator:

s = arg min e, f

365   t=1

2 T¯ t − es − fs (t − s) K



 t−s , h

(9)

where T¯ t is the mean of average daily temperature in J years, h is the smoothing parameter and K(·) denotes a kernel. This estimator, using Epanechnikov Kernel, incorporates an asymmetry term since high temperatures in winter are more pronounced than in summer as Figure 1 displays in a stretch of 8 years plot of the average daily temperatures over the FTS estimates. After removing the LLR-seasonal mean (Equation (9)) from the daily average temperatures (Xt = Tt − t ), we apply the Augmented Dickey–Fuller (ADF) and the Kwaitkowski–Phillips–Schmidt–Shin (KPSS) tests to check whether Xt is a stationary process. We then plot the Partial Autocorrelation Function (PACF) of Xt to detect possible intertemporal correlations. This suggests that persistence of daily average is captured by AR processes of higher order p:

Xt+p =

p  i=1

βi Xt−i + εt , εt = σt et , et ∼ N(0, 1),

(10)

with residuals εt . Under the stationarity hypothesis of the coefficients βs and the mean zero of residuals εt , the mean temperature E [Tt ] = t . This is different to the approach of Campbell and Diebold (2005), who suggested to regress deseasonalized temperatures on original temperatures. The analysis of the PACFs and Akaike’s information criterion (AIC) suggests that the AR(3) model in Benth et al. (2007) explains the temperature evolution well and holds for many cities. The results of the stationarity tests and the coefficients of the fitted AR(3) are given in Table 2. Figure 2 illustrates that the ACFs of the residuals εt are close to 0 and according to Box-Ljung statistic the first few lags are insignificant, whereas the ACFs of the squared residuals εt2 show a high seasonal pattern. We calibrate the deterministic seasonal variance function σt2 with FTS and an additional generalized autoregressive conditional heteroskedasticity (GARCH) (p, q) term:

The Implied Market Price of Weather Risk 67 (b) Temperature in °C

Temperature in °F

(a) 80 60 40 20 20000101

20030101

20080501

0

20030101

10 0 −10

20060101

10 0 −10 20060101

20080501

20060101

20080501

20060101

20080501

20060101

20080501

(f) Temperature in °F

Temperature in °F

20030101 Time

(e) 80 60 40 20000101

20030101

20060101

20080501

80 60 40 20 20000101

20030101

Time

Time

(h) Temperature in °F

(g) Temperature in °C

20080501

20

Time

30 20 10 0 20000101

20030101

20060101

80 60 40 20 20000101

20080501

20030101

Time

Time

(j) Temperature in °C

(i) Temperature in °C

20060101

30

20000101

20080501

20030101 Time

(d)

20

−20 20000101

20

20000101

Temperature in °C

(c) Temperature in °C

20060101

Time

30

30 20 10 20000101

20030101 Time

20060101

30 20 10 0 20000101

20080501

20030101 Time

Figure 1. A stretch of 8 years plot of the average daily temperatures (black line), the seasonal component modelled with a Fourier-truncated series (dashed line) and the local linear regression (grey line) using Epanechnikov Kernel. (a) Atlanta, (b) Beijing, (c) Berlin, (d) Essen, (e) Houston, (f) New York, (g) Osaka, (h) Portland, (i) Taipei and (j) Tokyo.

σˆ t2

=c+

L   l=1



2lπ t c2l cos 365

∼ iid N(0, 1).





2lπ t + c2l+1 sin 365



2 , ηt + α1 (σt−1 ηt−1 )2 + β1 σt−1

(11)

68 W. K. Härdle and B. López Cabrera Table 2. Result of the stationary tests and the coefficients of the fitted AR(3). ADF–KPSS City Atlanta Beijing Berlin Essen Houston New York Osaka Portland Taipei Tokyo

AR(3)

CAR(3)

τˆ

kˆ

β1

β2

β3

α1

α2

α3

λ1

λ2,3

−55.55+ −30.75+ −40.94+ −23.87+ −38.17+ −56.88+ −18.65+ −45.13+ −32.82+ −25.93+

0.21∗∗∗ 0.16∗∗∗ 0.13∗∗ 0.11∗ 0.05∗ 0.08∗ 0.09∗ 0.05∗ 0.09∗ 0.06∗

0.96 0.72 0.91 0.93 0.90 0.76 0.73 0.86 0.79 0.64

−0.38 −0.07 −0.20 −0.21 −0.39 −0.23 −0.14 −0.22 −0.22 −0.07

0.13 0.05 0.07 0.11 0.15 0.11 0.06 0.08 0.06 0.06

2.03 2.27 2.08 2.06 2.09 2.23 2.26 2.13 2.20 2.35

1.46 1.63 1.37 1.34 1.57 1.69 1.68 1.48 1.63 1.79

0.28 0.29 0.20 0.16 0.33 0.34 0.34 0.26 0.36 0.37

−0.30 −0.27 −0.21 −0.16 −0.33 −0.32 −0.33 −0.27 −0.40 −0.33

−0.86 −1.00 −0.93 −0.95 −0.87 −0.95 −0.96 −0.93 −0.90 −1.01

Notes: ADF, augmented Dickey–Fuller; KPSS, Kwiatkowski–Phillips–Schmidt–Shin; AR, autoregressive process; CAR, continuous autoregressive model. ADF and KPSS statistics, coefficients of the AR(3), CAR(3) and eigenvalues λ1,2,3 , of the matrix A of the CAR(3) model for the detrended daily average temperatures for different cities. + 0.01 critical values, ∗ 0.1 critical value, ∗∗ 0.05 critical value, ∗∗∗ 0.01 critical value.

The Fourier part in Equation (11) captures the seasonality in volatility, whereas the GARCH part captures the remaining non-seasonal volatility. Note again that more and more periodic terms in Equation (11) provide a good fitting but this will increase the number of parameters. To avoid this and in order to achieve positivity of the variance, Gaussian risk factors and volatility model flexibility in a continuous time, we propose the calibration of the seasonal variance in terms of an LLR:

arg min g,h

 365  2  2 t−s εˆ t − gs − hs (t − s) K , h

(12)

t=1

where εˆ t2 is the mean of squared residuals in J years and K(·) is a kernel. Figure 3 shows the daily empirical variance (the average of squared residuals for each day of the year), the fittings using the FTS-GARCH(1,1) and the LLR (with Epanechnikov kernel) estimators. Here we obtain the Campbell and Diebold’s (2005) effect for different temperature data, high variance in winter to earlier summer and low variance in spring to late summer. The effects of non-seasonal GARCH volatility component are small. Figure 4 displays the ACFs of temperature residuals εt and squared residuals εt2 after dividing out the deterministic LLR seasonal variance. The ACF plots of the standardized residuals remain unchanged but now the squared residuals presents a non-seasonal pattern. The LLR seasonal variance creates almost normal residuals and captures the peak seasons as Figure 5 in a log Kernel smoothing density plot shows against a Normal Kernel evaluated at 100 equally spaced points. Table 3 presents the calibrated coefficients of the FTS-GARCH seasonal variance estimates and the

The Implied Market Price of Weather Risk 69

2

0 −0.01 −0.02 −0.03

ACF εt − Beijing

ACF ε t − Atlanta

0.05 0 −0.05

0.01

200 400 600 800 1000

0 −0.01 −0.02 −0.03

0

200 400 600 800 1000

0

0.02

0.02

0.02

0.02

−0.01

0 −0.01

0.01 0 −0.01

−0.02

−0.02

−0.02

−0.03

−0.03

−0.03

200 400 600 800 1000

0

200 400 600 800 1000

0.03

0.1

0.02

−0.05 −0.1

−0.02

Lag 0.1

0.01 0

0.05

−0.01

0

−0.02

−0.05

−0.03

−0.03 200 400 600 800 1000

0 200 400 600 800 1000

0.04

0.04

0.03

0.03

0.02

2

0 −0.01 −0.02

0.02 0.01 0 −0.01

0.02

0.02

0.01

−0.01

Lag

200 400 600 800 1000

0

−0.02 −0.03 0

200 400 600 800 1000

0

Lag

Lag

0.03

0.01

−0.01

−0.03 0

200 400 600 800 1000

0.03

0

0.1

200 400 600 800 1000 Lag

0.03

0.1

0.02

2

0 −0.01

0.05 0 −0.05

−0.02

0.01 0

2

0.01

ACF εt − Tokyo

ACF ε t − Taipei

0.02

200 400 600 800 1000 Lag

0.03

−0.02

−0.03 0

0

Lag

−0.02

−0.03

200 400 600 800 1000

2

0.05

ACF ε t − Osaka

0.05

0.01

−0.1 0

Lag

ACF εt − Portland

Lag

ACF ε t − Portland

0

200 400 600 800 1000

2

0

2

0 −0.01

0

Lag

0.15

0.05

200 400 600 800 1000

ACF εt − New York

0.01

−0.03

−0.03 0

ACF εt − New York

ACF ε t − Houston

0.02

0 −0.01 −0.02

Lag

0.03

0.01

2

2

0

0.01

ACF ε t − Essen

0.03

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Figure 2. The ACF of residuals εt (left panels) and squared residuals εt2 (right panels) of detrended daily temperatures for different cities.

descriptive statistics for the residuals after correcting by the FTS-GARCH and LLR seasonal variance. We observe that independently of the chosen location, the driving stochastics are close to a Wiener process. This will allow us to apply the pricing tools of financial mathematics.

70 W. K. Härdle and B. López Cabrera (b) 70

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Figure 3. The daily empirical variance (black line), the Fourier-truncated (dashed line) and the local linear smoother seasonal variation using Epanechnikov kernel (grey line) for different cities. (a) Atlanta, (b) Beijing, (c) Berlin, (d) Essen, (e) Houston, (f) New York, (g) Osaka, (h) Portland, (i) Taipei and (j) Tokyo.

4. Stochastic Pricing Model Temperatures are naturally evolving continuously over time, so it is very convenient to model the dynamics of temperature with continuous-time stochastic processes, although the data may be on a daily scale. We therefore need the reformulation of the underlying process in continuous time to be more convenient with market definitions.

0.03

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The Implied Market Price of Weather Risk 71

0.01 0 −0.01 −0.02

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400 600 Lag

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Figure 4. The ACF of residuals εt (left panels) and squared residuals εt2 (right panels) of detrended daily temperatures after dividing out the local linear seasonal variance for different cities.

We show that the AR(p) (Equation (10)) estimated in Section 3 for the detrended temperature can be therefore seen as a discretely sampled continuous-time autoregressive (CAR) process (CAR(p)) driven by a one-dimensional Brownian motion Bt (though the continuous-time process is Markov in higher dimension) (Benth et al., 2007): dXt = AXt dt + ep σt dBt ,

(13)

72 W. K. Härdle and B. López Cabrera (a)

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Figure 5. The log of Normal Kernel (∗ ) and log of Kernel smoothing density estimate of residuals after correcting FTS (+) and locar linear (o) seasonal variance for different cities (a) Atlanta, (b) Beijing, (c) Berlin, (d) Essen, (e) Houston, (f) New York, (g) Osaka, (h) Portland, (i) Taipei and (j) Tokyo.

where the state vector X t ∈ Rp for p ≥ 1 is a vectorial Ornstein–Uhlenbeck process, namely, the temperatures after removing seasonality at times t, t − 1, t − 2, t − 3, . . . ; ek denotes the kth unit vector in Rp for k = 1, . . . , p; σt > 0 is a deterministic volatility (real-valued and square integrable function); and A is a p × p matrix: ⎛

⎜ ⎜ A=⎜ ⎜ ⎝

0 0 .. .

1 0

0 1 .. .

0 ... ... −αp −αp−1 . . .

... 0 ... 0

⎞ 0 0⎟ .. ⎟ .⎟ ⎟, 1⎠

0 0 0 0 −α1

(14)

with positive constants αk . Following this nomenclature, Xq(t) with q = 1, . . . , p is the qth coordinate of Xt and by setting q = 1 is equivalent to the detrended temperature time series X1(t) = Tt − t . The proof is derived by an analytical link between X1(t) , X2(t) and X3(t) and the lagged temperatures up to time t − 3. X1(t+3) is approximated by Euler discretization. Thus for p = 1, Xt = X1(t) and Equation (13) becomes dX1(t) = −α1 X1(t) dt + σt dBt ,

(15)

which is the continuous version of an AR(1) process. Similarly for p = 2, assume a time step of length 1 dt = 1 and substitute X2(t) iteratively to get X1(t+2) ≈ (2 − α1 )X1(t+1) + (α1 − α2 − 1)X1(t) + σt et ,

(16)

21.51 3.89 5.07 4.78 23.61 22.29 3.34 12.48 3.50 3.80

cˆ 1

18.10 0.70 0.10 0.00 25.47 13.80 0.80 1.55 1.49 0.01

cˆ 2

7.09 0.84 0.72 0.42 4.49 3.16 0.80 1.05 1.59 0.73

cˆ 3 2.35 −0.22 0.98 0.63 6.65 3.30 −0.57 1.42 −0.38 −0.69

cˆ 4 1.69 −0.49 −0.43 −0.20 −0.38 −0.47 −0.27 −1.19 −0.16 −0.33

cˆ 5 cˆ 7 −0.68 −0.14 0.06 −0.06 −2.67 2.04 −0.07 0.34 −0.17 −0.14

cˆ 6 −0.39 −0.20 0.45 0.17 1.00 0.80 −0.18 0.46 0.03 −0.14 0.24 −0.11 0.16 0.05 0.68 0.11 0.01 −0.40 −0.09 0.26

cˆ 8 −0.45 0.08 0.22 0.17 −1.56 0.01 −0.03 0.45 −0.18 −0.13

cˆ 9 272.01 219.67 224.55 273.90 140.97 367.38 105.32 67.10 181.90 137.93

JB 3.98 3.27 3.48 3.65 3.96 3.43 3.37 3.24 3.26 3.45

Kurt

FTS

−0.70 −0.28 −0.05 −0.05 −0.60 −0.23 −0.11 0.06 −0.39 −0.10

Skew

εˆ t σˆ t

253.24 212.46 274.83 251.89 122.83 355.03 101.50 75.01 169.41 156.58

JB

Corrected residuals

3.91 3.24 3.51 3.61 3.87 3.43 3.36 3.27 3.24 3.46

Kurt

LLR

with

−0.68 −0.28 −0.08 −0.08 −0.57 −0.22 −0.11 0.02 −0.37 −0.13

Skew

Notes: FTS, Fourier-truncated series; JB, Jarque Bera; LLR, local linear regression; GARCH, generalized autoregressive conditional heteroskedasticity. Seasonal variance estimate of {ci }9i = 1 fitted with an FTS and statistics – Skewness (Skew), kurtosis (Kurt) and JB test statistics – of the standardized residuals with seasonal variances fitted with FTS-GARCH and with LLR. Coefficients are significant at 1% level.

Atlanta Beijing Berlin Essen Houston New York Osaka Portland Taipei Tokyo

City

Coefficients of the FTS

Table 3. Coefficients of the FTS-GARCH seasonal variance.

The Implied Market Price of Weather Risk 73

74 W. K. Härdle and B. López Cabrera where et = Bt+1 − Bt . For p = 3, we have: X1(t+1) − X1(t) = X2(t) dt, X2(t+1) − X2(t) = X3(t) dt, X3(t+1) − X3(t) = −α3 X1(t) dt − α2 X2(t) dt − α1 X3(t) dt + σt et , . . .,

(17)

X1(t+3) − X1(t+2) = X2(t+2) dt, X2(t+3) − X2(t+2) = X3(t+2) dt, X3(t+3) − X3(t+2) = −α3 X1(t+2) dt − α2 X2(t+2) dt − α1 X3(t+2) dt + σt et , substituting into the X1(t+3) dynamics and setting dt = 1: X1(t+3) ≈ (3 − α1 ) X1(t+2) + (2α1 − α2 − 3) X1(t+1) + (−α1 + α2 − α3 + 1) X1(t) .          β1

β2

(18)

β3

Please note that this corrects the derivation in Benth et al. (2007) and Equation (18) leads to Equation (10) (with p = 3). The approximation of Equation (18) is required to compute the eigenvalues of matrix A. The last columns of Table 2 display the CAR(3)parameters and the eigenvalues of the matrix A for the studied temperature data. The stationarity condition  t is 2fulfilled since the⊤ eigenvalues  of A have negative real parts and exp {A(s)} ep ep exp A⊤ (s) ds converges as t → ∞. the variance matrix 0 σt−s By applying the multidimensional Itˆo Formula, the process in Equation (13) with s Xt = x ∈ Rp has the explicit form Xs = exp {A(s − t)} x + t exp {A(s − u)} ep σu dBu for s ≥ t ≥ 0. Since dynamics of temperature futures prices must be free of arbitrage under the pricing equivalent measure Qθ , the temperature dynamics of Equation (13) becomes for s ≥ t ≥ 0: dXt = (AXt + ep σt θt )dt + ep σt dBtθ , Xs = exp {A(s − t)} x +



s

exp {A(s − u)} ep σu θu du + t



s t

exp {A(s − u)} ep σu dBuθ . (19)

By inserting Equations (1)–(3) into Equation (6), Benths et al. (2007) explicity calculated the risk neutral prices for HDD/CDD/CAT futures (and options) for contracts traded before the temperature measurement period, that is 0 ≤ t ≤ τ1 < τ2 :

The Implied Market Price of Weather Risk 75

FHDD(t,τ1 ,τ2 ) = FCDD(t,τ1 ,τ2 ) =



τ2

υt,s ψ

τ1



τ2

υt,s ψ



c − m{t,s,e⊤1 exp{A(s−t)}Xt } υt,s

m{t,s,e⊤1 exp{A(s−t)}Xt } − c

υt,s  τ2  = u du + at,τ1 ,τ2 Xt + τ1

FCAT(t,τ1 ,τ2 )





ds,

ds, (20)

τ1

θu σu at,τ1 ,τ2 ep du

t

τ1

+





τ2

τ1

  −1 θu σu e⊤ exp {A(τ2 − u)} − Ip ep du, 1A

−1 [exp {A(τ2 − t)} − exp {A(τ1 − t)}]; Ip is a p × p identity matrix; with at,τ1 ,τ2 = e⊤ 1A ψ(x) = x(x) + ϕ(x) ( denotes the standard normal cumulative distribution s  2 2 exp {A(s − t)} Xt ; υt,s = t σu2 e⊤ exp {A(s − t)} ep du; function (cdf)) with x = e⊤ 1 1 s and m{t,s,x} = s + t σu θu e⊤ 1 exp {A(s − t)} ep du + x. The solution to Equation (20) depends on the assumed specification for the MPR θ. In the next section, it is shown that different assumed risk specifications can lead into different derivative prices. The model in Benth et al. (2007) nests a number of previous models (Alaton et al., 2002; Benth, 2003; Benth and Saltyte-Benth, 2005; Brody et al., 2002); it generalizes the Benth and Saltyte-Benth (2007) and Dornier and Querel (2007) approaches and is a very well studied methodology in the literature (Benth et al., 2011; Papazian and Skiadopoulos, 2010; Zapranis and Alexandridis, 2008). Besides it gives a clear connection between the discrete- and continuous-time versions, it provides closedform non-arbitrage pricing formulas and it requires only a one-time estimation for the price dynamics. With the time series approach (Campbell and Diebold, 2005), the continuous-time approaches (Alaton et al., 2002; Huang-Hsi et al., 2008), neural networks (Zapranis and Alexandridis, 2008, 2009) or the principal component analysis approach (Papazian and Skiadopoulos, 2010) are not easy to compute price dynamics of CAT/CDD/HDD futures and one needs to use numerical approaches or simulations in order to calculate conditional expectations in Equation (6). In that case, partial differential equations or Monte Carlo simulations are being used. For option pricing, this would mean to simulate scenarios from futures prices. This translates into intensive computer simulation procedures.

5. The Implied Market Price of Weather Risk For pricing and hedging non-tradable assets, one essentially needs to incorporate the MPR θ which is an important parameter of the associated EMM and it measures the additional return for bearing more risk. This section deals exactly with the differences between ‘historical’ (P) and ‘risk-neutral’ (Q) behaviours of temperature. Using statistical modelling and given that liquid-derivative contracts based on daily temperatures are traded on the CME, one might infer the MPR (the change of drift) from traded (CAT/CDD/HDD/C24AT) futures–options-type contracts.

76 W. K. Härdle and B. López Cabrera Our study is a calibration procedure for financial engineering purposes. In the calibration exercise, a single date (but different time horizons and calibrated instruments are used) is required, since the model is recalibrated daily to detect intertemporal effects. Moreover, we use an economic and statistical testing approach, where we start from a specification of the MPR and check consistency with the data. By giving assumptions about the MPR, we implicitly make an assumption about the aggregate risk aversion of the market. The risk parameter θ can then be inferred by finding the value that satisfies Equation (20) for each specification. Once we know the MPR for temperature futures, then we know the MPR for options and thus one can price new ‘non-standard maturities’ or OTC derivatives. The concept of implied MPR is similar to that used in extracting implied volatilities (Fengler et al., 2007) or the market price of oil risk (Gibson and Schwartz, 1990). To value temperature derivatives, the following specifications of the MPR are investigated: a constant, a piecewise linear function, a two-piecewise linear function, a time-deterministic function and a ‘financial-bootstrapping’ MPR. The statistical point of view is to beat this as an inverse problem with different degrees of smoothness expressed through the penalty parameter of a smoothing spline. The economic point of view is to detect possible time dependencies that can be explained by investor’s preferences in order to hedge weather risk. In this article we concentrate on contracts with monthly measurement length periods, but similar implications apply for seasonal strip contracts. We observe different temperature futures contracts i = 1, . . . , I with measurement periods t ≤ τ1i < τ2i and τ2i ≤ τ1i+1 traded at time t, meaning that contracts expire at some point in time and roll over to another contract. Therefore, i = 1 denotes contract types with measurement period in 30 days, i = 2 denotes contract types in 60 days and so on. For example, a contract with i = 7 is six months ahead from the trading day t. For United States and Europe, the number of temperature futures contracts is I = 7 (April–October or October–April), while for Asia I = 12 (January–December). The details of the temperature futures data are displayed in Table 4. To simplify notation, dates are written in yyyymmdd format. 5.1 Constant MPR for Each Contract per Trading Date Given observed temperature futures market prices and by inverting Equation (20), we imply the MPR θu for i = 1, . . . , I futures contracts with different measurement time horizon periods [τ1i , τ2i ], t ≤ τ1i < τ2i and τ2i ≤ τ1i+1 traded at date t. Our first assumption is to set, for the ith contract, a constant MPR over [t, τ2i ], that is, we have that θu = θti : i θˆt,CAT



= arg min F θti

+



τ2i τ1i

CAT(t,τ1i ,τ2i )

−1 σˆ u e⊤ 1A



−



τ2i

τ1i

ˆ u du − at,τ i ,τ i Xt − θti  1 2

   exp A(τ2i − u) − Ip ep du

2

,



τ1 t

σˆ u at,τ1i ,τ2i ep du

20070316 20070316 20070316 20070316 20070316 20070316 20070427 20070427 20070427 20070427 20070427 20081027 20081027 20081027 20081027 20081027

Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Berlin-CAT Tokyo-C24AT Tokyo-C24AT Tokyo-C24AT Tokyo-C24AT Tokyo-C24AT

20070401 20070501 20070601 20070701 20070801 20070901 20070501 20070601 20070701 20070801 20070901 20090301 20090401 20090501 20090601 20090701

τ1 20070430 20070531 20070630 20070731 20070831 20070930 20070531 20070630 20070731 20070831 20070930 20090331 20090430 20090531 20090630 20090731

τ2

Measurement period

288.00 457.00 529.00 616.00 610.00 472.00 457.00 529.00 616.00 610.00 472.00 450.00 592.00 682.00 818.00 855.00

CME 363.00 502.11 571.78 591.56 566.14 414.33 506.18 571.78 591.56 566.14 414.33 118.32 283.18 511.07 628.24 731.30

MPR = 0 291.06 454.91 630.76 626.76 636.22 472.00 457.52 534.76 656.76 636.22 472.00 488.90 563.27 696.31 835.50 706.14

Constant MPR

Futures prices F(t,τ1 ,τ2 ,θ)ˆ

362.90 494.20 574.30 583.00 580.70 414.80 494.20 574.30 583.00 580.70 414.80 305.00 479.00 623.00 679.00 812.00

I(τ1 ,τ2 )

Realized Tt

Notes: CME, Chicago Mercantile Exchange; MPR, market price of risk. Weather futures at CME; futures prices F(t,τ ,τ ,θˆ ) from CME; estimated prices with MPR = 0; constant MPR for different controls per trading 1 2 date (constant MPR). Source: Bloomberg Professional Service (Weather futures).

t

Contract type

Trading date

Table 4. Weather futures and futures prices listed on date (yyyymmdd) at CME.

The Implied Market Price of Weather Risk 77

78 W. K. Härdle and B. López Cabrera ⎛

i = arg min ⎝FHDD(t,τ1i ,τ2i ) − θˆt,HDD θti



τ2i

τ1i

⎡

υt,s ψ ⎣

ˆ 1{t,s,e⊤ exp{A(s−t)}X } c−m t

1

υt,s

⎞2

⎤

⎦ ds⎠ ,

(21)

s 2 ˆ 1{t,s,x} = s + θti t σu e⊤ with m 1 exp {A(s − t)} ep du + x, υt,s , ψ(x) and x defined as in i Equation (20). The MPR for CDD futures θˆt,CDD is equivalent to the HDD case in Equation (21) and we will therefore omit CDD parameterizations. Note that this specification can be seen as a deterministic time-varying MPR θti that varies with date for any given contract i, but it is constant over [t, τ2i ].

5.2 One Piecewise Constant MPR A simpler MPR parameterization is to assume that it is constant across all time horizon contracts priced in a particular date (θt ). We therefore estimate this constant MPR for all contract types traded at t ≤ τ1i < τ2i , i = 1, . . . , I as follows: θˆt,CAT = arg min θt

+



I  i=1

τ2i

τ1i

FCAT(t,τ1i ,τ2i ) −

−1 σˆ u e⊤ 1A

θˆt,HDD = arg min θt



I  i=1

⎛





τ2i τ1i

ˆ u du − aˆ t,τ i ,τ i X t − θt  1 2

   exp A(τ2i − u) − Ip ep du

⎝FHDD(t,τ i ,τ i ) − 1

2



τ2i

τ1i

⎡

υt,s ψ ⎣

2

!

τ1i t

σˆ u aˆ t,τ1i ,τ2i ep du

,

ˆ 2{t,s,e⊤ exp{A(s−t)}X } c−m 1

υt,s

t

⎤

⎞2

⎦ ds⎠ ,

(22)

s 2 ˆ 2{t,s,x} = s + θt t σu e⊤ with m 1 exp {A(s − t)} ep du + x and υt,s , ψ(x) and x as defined in Equation (20). This ‘one piecewise constant’ MPR specification (θt ) is solved by means of the ordinary least squares (OLS) minimization procedure and differs from θti in Equation (21) because for all traded contracts at date t, we get only one MPR estimate (instead of i estimates) at time t, that is, θt is constant over [t, τ2I ].

5.3 Two Piecewise Constant MPR Assuming now that, instead of one constant MPR per trading day, we have a step function with a given jump point ξ (take e.g. the first 150 days before the beginning of the measurement period), so we have that θˆt = I (u ≤ ξ ) θt1 + I (u > ξ ) θt2 . The two piecewise constant function θˆt with t ≤ τ1i < τ2i is estimated with the OLS minimization procedure as follows:

The Implied Market Price of Weather Risk 79

fCAT (ξ ) = arg min

I 

1 2 θt,CAT ,θt,CAT i=1

− +

−

+

fHDD (ξ ) = arg min



!





ˆ u du − aˆ t,τ i ,τ i X t  1 2

   exp A(τ2i − u) − Ip ep du



−1 ξ ) σˆ u e⊤ 1A

⎛

⎝FHDD(t,τ i ,τ i ) − 1

t

τ1i



(23)

I (u > ξ ) σˆ u aˆ t,τ1i ,τ2i ep du

t

I (u >

s

τ2i

τ1i

τ1i





−1 ξ ) σˆ u e⊤ 1A

τ2i

I 

−



I (u ≤ ξ ) σˆ u aˆ t,τ1i ,τ2i ep du

t

I (u ≤

2 θt,CAT

CAT(t,τ1i ,τ2i )

τ1i

τ1i

1 ˆ 3{t,s,x} = s + θt,HDD m

+

F

τ2i

1 2 θt,HDD ,θt,HDD i=1

2 θt,HDD

!

1 θt,CAT



2



   exp A(τ2i − u) − Ip ep du

τ2i

τ1i

⎡

υt,s ψ ⎣

2

ˆ 3{t,s,e⊤ exp{A(s−t)}X } c−m t

1

υt,s

,

⎤

⎞2

⎦ ds⎠ ,

{A(s I (u ≤ ξ ) σu e⊤ exp − t)} e du + x p 1

s

I (u > t

ξ ) σ u e⊤ 1

exp {A(s − t)} ep du + x ,

2 and υt,s , ψ(x) and x as defined in Equation (20). In the next step, we optimized the value of ξ such as fCAT (ξ ) or fHDD (ξ ) is minimized. This MPR specification will vary according to the unknown ξ . This would mean that the market does a risk adjustment for contracts traded close or far from the measurement period.

5.4 General Form of the MPR per Trading Day Generalizing the piecewise continuous function given in the previous subsection, the (inverse) problem of determining θt with t ≤ τ1i < τ2i , i = 1, . . . , I, can be formulated via a series expansion for θt :

arg min γk

I  i=1



−

FCAT(t,τ1i ,τ2i ) − 

K τ2i 

τ1i

k=1



τ2i

τ1i

ˆt − ˆ u du − aˆ t,τ i ,τ i X  1 2

−1 γk hk (ui )σˆ ui e⊤ 1A





t

K τ1i 

γk hk (ui )σˆ ui aˆ t,τ1i ,τ2i ep dui

k=1

   exp A(τ2i − ui ) − Ip ep dui

2

,

80 W. K. Härdle and B. López Cabrera

arg min ak

I  i=1

⎛

⎝FHDD(t,τ i ,τ i ) − 1

2



τ2i

τ1i

⎡

υt,s ψ ⎣

ˆ 4{t,s,e⊤ exp{A(s−t)}X } c−m t

1

υt,s

⎤

⎞2

⎦ ds⎠ ,

(24)

s" 2 ˆ 4{t,s,x} = s + t K with m ˆ ui e⊤ k = 1 ak lk (ui )σ 1 exp {A(s − t)} ep dui + x and υt,s , ψ(x) and x as defined in Equation (20). hk (ui ) and lk (ui ) are vectors of known basis functions and may denote a B-spline basis for example. γ k and ak define the coefficients and K is the number of knots. This means that the inferred MPR is going to be a solution for an inverse problem with different degrees of smoothness expressed through the penalty parameter of a smoothing spline. The degrees of smoothness will allow for a term structure of risk. In other words, a time-dependent risk factor offers the possibility to have different risk adjustments for different times of the year.

5.5 Bootstrapping the MPR In this section we propose a bootstrapping technique to detect possible MPR timedependent paths of temperature futures contracts. More importantly, since these futures contract types have different measurement periods [τ1i , τ2i ] with τ1i < τ1i+1 ≤ τ2i < τ2i+1 , i = 1, . . . , I, and they roll over to another contracts when they expire at some point in time, it makes sense to construct MPR estimates from which we can price contracts with any maturity, without the need of external information. This ‘financial’ bootstrapping idea consists of estimating by forward substitution the MPR θti of the futures price contracts with the closest measurement period and placing it into the estimation for the next MPR θti+1 . We implement the estimation for CAT contracts, but the idea applies also for HDD/CDD contract types. First, for the first contract i = 1 1 is estimated from Equation (21): and t ∈ [τ11 , τ21 ], θˆt,CAT 

1 θˆt,CAT = arg min FCAT(t,τ11 ,τ21 ) − θt1

+



τ21 τ11

−1 σˆ u e⊤ 1A



τ21 τ11

ˆ t − θt1 ˆ u du − aˆ t,τ 1 ,τ 1 X  1 2

!

t

τ11

σˆ u aˆ t,τ11 ,τ21 ep du

2     1 . exp A(τ2 − u) − Ip ep du

(25)

1 is substituted in the period [τ11 , τ21 ] to get an estimate of Second, the estimated θˆt,CAT 2 : θˆt,CAT



2 = arg min FCAT(t,τ12 ,τ22 ) − θˆt,CAT 2 θt,CAT

−



τ22

τ12

−1 2 θt,CAT σˆ u e⊤ 1A





τ22 τ12

ˆt − ˆ u du − aˆ t,τ 2 ,τ 2 X  1 2



   exp A(τ22 − u) − Ip ep du

τ11 t

2

.

1 σˆ u aˆ t,τ12 ,τ22 ep du θˆt,CAT

(26)

The Implied Market Price of Weather Risk 81 1 2 in the period [τ11 , τ21 ] and θˆt,CAT in the period [τ12 , τ22 ] to estimate Then substitute θˆt,CAT 3 θˆt,CAT :



3 = arg min FCAT(t,τ13 ,τ23 ) − θˆt,CAT 3 θˆt,CAT

−



τ22

τ12



τ23

τ13

2 σˆ u aˆ t,τ13 ,τ23 ep du − θˆt,CAT

ˆ u du − aˆ t,τ 3 ,τ 3 X t −  1 2 

τ23 τ13

−1 3 θt,CAT σˆ u e⊤ 1A



t

τ11

1 σˆ u aˆ t,τ13 ,τ23 ep du θˆt,CAT

    exp A(τ23 − u) − Ip ep du

2

4 I , . . . , θˆt,CAT . In a similar way, one obtains the estimation of θˆt,CAT

5.6 Smoothing the MPR over Time Since smoothing individual estimates is different from estimating a deterministic function, we also assure our results by fitting a parametric function to all available contract prices (calendar year estimation). After computing the MPR θˆt,CAT , θˆt,HDD and θˆt,CDD for each of the previous specification and for each of the nth trading days t for different ith contracts, the MPR time series can be smoothed with the inverse problem points to find an MPR θˆu for every calendar day u and with that being able to price temperature derivatives for any date: ⎫2 ⎧ J n # n ⎨ ⎬ $2    αj j (ut ) , θˆt − f (ut ) = arg min arg min θˆt − ⎭ ⎩ αj f ∈Fj t=1

t=1

(27)

j=1

where j (ut ) is a vector of known basis functions, αj defines the coefficients, J is the number of knots, ut = t −  + 1 with increment  and n is the number of days to be smoothed. In our case, ut = 1 day and j (ut ) is estimated using cubic splines. Alternatively, one can first do the smoothing with basis functions of all available futures contracts: ⎫2 ⎧ J n  I ⎨ ⎬   βj j (ut ) , F(t,τ1i ,τ2i ) − arg min ⎭ ⎩ βj t=1 i=1

(28)

j=1

and then estimate the time series of θˆts s with the obtained smoothed futures prices F st,τ 1 ,τ I . ( 1 2) For example, for a constant MPR for all CAT futures contracts type traded over all ts with t ≤ τ1i < τ2i and τ2i ≤ τ1i+1 , we have:

.

82 W. K. Härdle and B. López Cabrera

s θˆt,CAT

= arg min s θt,CAT

+



τ2I

τ11



s FCAT(t,τ 1 I 1 ,τ2 )

−1 σˆ u e⊤ 1A



−



τ2I

τ11

ˆ u du − aˆ t,τ 1 ,τ I X t −  1 2

   exp A(τ2I − u) − Ip ep du

2

s θt,CAT

.

!

t

τ11

σˆ u aˆ t,τ11 ,τ2I ep du

(29)

5.7 Statistical and Economical Insights of the Implied MPR In this section, using the previous specifications, we imply the MPR (the change of drift) for CME (CAT/CDD/HDD/C24AT) futures contracts traded for different cities. Note that one might also infer the MPR from options data and compare the findings with prices in the futures market. Table 5 presents the descriptive statistics of different MPR specifications for BerlinCAT, Essen-CAT and Tokyo-C24AT daily futures contracts with t ≤ τ1i < τ2i traded during 20031006–20080527 (5102 contracts in 1067 trading days with 29 different measurement periods), 20050617–20090731 (3530 contracts in 926 trading dates with 28 measurement periods) and 20040723–20090831 (2611 contracts in 640 trading dates with 27 measurement periods). The MPR ranges vary between [–10.71, 10.25], [31.05, 5.73] and [–82.62, 52.17] for Berlin-CAT, Essen-CAT and Tokyo-C24AT futures contracts, respectively, whereas the MPR averages are 0.04, 0.00 and –3.08 for constant MPR for different contracts; –0.08, –0.38 and 0.73 for one piecewise constant; –0.22, –0.43 and –3.50 for two piecewise constant; 0.04, 0.00 and –3.08 for spline; and 0.07, 0.00 and –0.11 when bootstrapping the MPR. We observe that the two piecewise constant MPR function is a robust least square estimation, since its values are sensitive to the choice of ξ . Figure 6 shows the MPR estimates for Berlin-CAT futures prices traded on 20060530 with ξ = 62, 93, 123 and 154 and sum of squared errors equal to 2759, 14,794, 15,191 and 15,526. The line displays a discontinuity indicating that trading was not taking place (CAT futures are only traded from April to November and MPR estimates cannot be computed since there are no market prices). When the jump ξ is getting far from the measurement period, the value of the MPR θˆt1 decreases and θˆt2 increases, yielding a θˆt around 0. Table 5 also displays the estimates of the timedependent MPR (or spline MPR) from the bootstrapping technique. The spline MPR smooths the estimates over time and it is estimated using cubic polynomials with k equal to the number of traded contracts I at date t. The performance of the boostrapped MPR is similar to the constant MPR for different contracts per trading date estimates, suggesting that the only risk which the statistical model might imply is that the MPR will be equal at any trading date across all temperature contract types. The first panel in Figure 7 displays the Berlin-CAT, Essen-CAT and TokyoC24AT futures contracts traded at 20060530, 20060530 and 20050531, respectively. The second, third and fourth panels of Figure 7 show the MPR when it is assumed to be constant for different contracts per trading date, a two piecewise constant and the spline MPR. In the case of the constant MPR for different contracts per trading date, the lines overlap because the MPR for every contract i = 1, . . . , 12 is supposed to be constant over the period [t, τ2i ] at trading date t. The two piecewise constant function adjusts the risk according to the choice of ξ

Tokyo-C24AT 30 days

Essen-CAT 30 days (i = 1) 60 days (i = 2) 90 days (i = 3) 120 days (i = 4) 150 days (i = 5) 180 days (i = 6)

Berlin-CAT 30 days (i = 1) 60 days (i = 2) 90 days (i = 3) 120 days (i = 4) 150 days (i = 5) 180 days (i = 6)

Type

419

405

468

551

738

796

384

711

752

815

858

874

487

No. of contracts

WS (Prob) Min (Max)

WS (Prob) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD)

WS (Prob) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD)

Statistic

0.76 (0.61) −7.55 (0.17)

0.02 (0.11) −0.98 (0.52) 0.01 (0.12) −1.35 (0.62) 0.02 (0.14) −1.56 (0.59) −0.02 (0.19) −0.29 (0.51) 0.03 (0.05) −0.44 (0.13) 0.00 (0.07) −0.10 (0.57) −0.02 (0.06)

0.93 (0.66) −0.28 (0.12) 0.05 (0.07) −0.54 (0.84) 0.03 (0.13) −0.54 (0.82) 0.02 (0.09) −0.53 (0.84) 0.03 (0.14) −0.54 (0.84) 0.13 (0.09) −0.54 (0.82) 0.02 (0.12)

Constant

0.02 (0.13) −69.74 (52.17)

0.20 (0.34) −31.05 (5.73) −0.39 (1.51) −31.05 (5.73) −0.40 (1.56) −6.68 (5.14) −0.40 (0.88) −4.61 (1.44) −0.37 (0.51) −4.61 (1.44) −0.35 (0.49) −4.61 (1.44) −0.12 (0.45)

0.01 (0.08) −0.65 (0.11) −0.10 (0.12) −4.95 (8.39) −0.09 (1.06) −4.95 (8.39) −0.10 (1.07) −4.95 (4.56) −0.10 (0.94) −4.53 (4.56) −0.10 (0.95) −4.53 (4.56) −0.10 (0.95)

1 piecewise

0.01 (0.11) −69.74 (52.17)

0.01 (0.09) −1.83 (1.66) −0.43 (9.62) −1.83 (1.66) −0.46 (9.99) −5.40 (1.71) −0.40 (0.85) −6.60 (1.43) −0.41 (0.85) −6.60 (1.43) −0.30 (0.81) −4.25 (0.52) −0.21 (0.52)

0.00 (0.04) −1.00 (0.20) −0.17 (0.31) −10.71 (10.25) −0.17 (1.44) −10.71 (10.25) −0.17 (1.46) −7.71 (6.88) −0.19 (1.38) −8.24 (6.88) −0.20 (1.47) −8.24 (6.88) −0.14 (1.48)

2 piecewise

0.76 (0.61) −7.55 (0.17)

0.02 (0.11) −0.98 (0.52) 0.01 (0.12) −1.35 (0.62) 0.02 (0.14) −1.56 (0.59) −0.02 (0.19) −0.29 (0.51) 0.03 (0.05) −0.44 (0.13) 0.00 (0.07) −0.10 (0.57) −0.02 (0.06)

0.93 (0.04) −0.28 (0.12) 0.05 (0.07) −0.54 (0.84) 0.03 (0.13) −0.54 (0.82) 0.02 (0.09) −0.53 (0.84) 0.03 (0.14) −0.54 (0.84) 0.13 (0.09) −0.54 (0.82) 0.02 (0.12)

Bootstrap

Table 5. Statistics of MPR specifications for Berlin-CAT, Essen-CAT and Tokyo-C24AAT.

(Continued)

4.34 (0.96) −0.23 (−0.18)

1.63 (0.79) −0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

0.91 (0.66) 0.02 (0.21) 0.14 (0.06) 0.00 (0.21) 0.05 (0.05) 0.00 (0.21) 0.11 (0.07) 0.00 (0.21) 0.11 (0.09) −0.00 (0.20) 0.01 (0.07) −0.03 (0.12) 0.00 (0.03)

Spline

The Implied Market Price of Weather Risk 83

134

167

184

243

305

350

393

416

No. of contracts Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD) Min (Max) Med (SD)

Statistic −3.87 (2.37) −7.56 (0.14) −3.49 (2.47) −7.55 (1.02) −2.96 (2.65) −7.55 (1.02) −2.08 (2.74) −7.55 (1.02) −2.08 (2.71) −7.39 (1.26) −2.08 (2.74) −7.39 (1.26) −3.00 (2.86) −7.39 (1.26) −3.00 (2.74) −7.39 (0.44) −4.24 (2.39)

Constant −0.33 (19.68) −69.74 (52.17) −0.23 (21.34) −69.74 (26.82) 0.04 (20.84) −69.74 (26.82) 1.26 (19.54) −51.18 (26.82) 1.26 (16.03) −51.18 (19.10) 3.66 (15.50) −24.88 (26.16) 13.69 (10.05) −24.88 (21.23) 3.46 (11.73) −24.88 (26.16) 8.48 (13.88)

1 piecewise −0.48 (20.46) −69.74 (52.17) −0.41 (21.63) −69.74 (38.53) −0.33 (20.22) −69.74 (48.32) −0.11 (19.69) −51.18 (48.32) 7.17 (17.02) −51.18 (48.32) 7.63 (17.69) −54.14 (39.65) 10.61 (14.63) −82.62 (42.14) −4.24 (37.25) −82.62 (42.14) −7.39 (40.76)

2 piecewise −3.87 (2.37) −7.56 (0.14) −3.49 (2.47) −7.55 (1.02) −2.96 (2.65) −7.55 (1.02) −2.08 (2.74) −7.55 (1.02) −2.08 (2.71) −7.39 (1.26) −2.08 (2.74) −7.39 (1.26) −3.00 (2.86) −7.39 (1.26) −3.00 (2.74) −7.39 (0.44) −4.24 (2.39)

Bootstrap

−0.20 (0.01) −0.18 (−0.13) −0.15 (0.01) −0.13 (−0.09) −0.11 (0.01) −0.10 (−0.06) −0.08 (0.01) −0.06 (−0.04) −0.05 (0.00) −0.05 (−0.04) −0.04 (0.00) −0.07 (−0.05) −0.06 (0.00) −0.07 (−0.07) −0.07 (0.00) −0.07 (−0.03) −0.05 (0.00)

Spline

Notes: WS, Wald statistics; SD, standard deviation. Futures contracts traded during (20031006–20080527), (20050617–20090731), and (20040723–20090630) respectively, with trading date before measurement period t ≤ τ1i < τ2i , i = 1 , . . . , I (where i = 1 (30 days), i = 2 (60 days), . . . , i = I (210 days)): the WS, the WS probabilities (Prob), Minimun (Min), Maximum (Max), Median (Med) and SD. MPR specifications: Constant for different contracts per trading date (Constant), 1 piecewise constant, 2 piecewise constant (ξ = 150 days), bootstrap and spline.

(i = 2) 60 days (i = 3) 90 days (i = 4) 120 days (i = 5) 150 days (i = 6) 180 days (i = 7) 210 days (i = 8) 240 days (i = 9) 270 days (i = 10)

Type

Table 5. (Continued).

84 W. K. Härdle and B. López Cabrera

The Implied Market Price of Weather Risk 85 4

Day: 20060530

(b) MPR of CAT prices

MPR of CAT prices

(a)

2 0 −2

50 100 150 200 250 300 No. of days before measurement period 1 0 −1 −2

Day: 20060530

50 100 150 200 250 300 No. of days before measurement period

Day: 20060530

0 −1 −2

50 100 150 200 250 300 No. of days before measurement period (d) 0.2

MPR of CAT prices

MPR of CAT prices

(c)

1

Day: 20060530

0.1 0 −0.1 −0.2

50 100 150 200 250 300 No. of days before measurement period

Figure 6. Two piecewise constant MPR with jumps ξ = (a) 62, (b) 93, (c) 123 and (d) 154 days for Berlin-CAT contracts traded on 20060530. The corresponding sum of squared errors are 2759, 14794, 15191 and 15526. When the jump ξ is getting far from the measurement period, the value of the MPR θˆt1 decreases and θˆt2 increases, yielding a θˆt around 0.

(in this case ξ = 150 days). The spline MPR smooths over time and for days without trading (see the case of Berlin-CAT or Essen-CAT futures), it displays a maximum, for example, in winter. A penalizing term in Equation (24) might correct for this. In all the specifications, we verified the discussion that MPR is different from 0 (as Cao and Wei (2004), Huang-Hsi et al. (2008), Richards et al. (2004) and Alaton et al. (2002) do) varies in time and moves from a negative to a positive domain according to the changes in the seasonal variation. The MPR specifications change signs when a contract expires and rolls over to another contract (e.g. from 210 to 180, 150, 120, 90, 60, 30 days before measurement period); they react negatively to the fast changes in seasonal variance σt within the measurement period (Figure 3) and to the changes in CAT futures volatility σt at,τ1 ,τ2 ep . Figure 8 shows the Berlin-CAT volatility paths for contracts issued before and within the measurement periods 2004–2008. We observed the Samuelson effect for mean-reverting futures: for contracts traded within the measurement period, CAT volatility is close to 0 when the time to measurement is large and it decreases up to the end of the measurement period. For contracts traded before the measurement period, CAT volatility is also close to 0 when the time to measurement is large, but increases up to the start of the measurement period. In Figure 9, two Berlin-CAT contracts issued on 20060517 but with different measurement periods are plotted: the longest the measurement period, the largest the volatility. Besides this, one observes the effect of the CAR(3) in both contracts when the volatility decays just before maturity of the contracts. These two effects are comparable with the study for Stockholm CAT futures in Benth et al. (2007); however, the deviations are less smoothed for Berlin.

−1

CAT spline−MPR

CAT 2OLS−MPR

100 200 300 No.of days before measurement period 0.4 0.2

0 100 200 300 No.of days before measurement period 100 50 0 −50

100 200 300 No.of days before measurement period

AAT-Tokyo prices

200 100 200 300 No.of days before measurement period 0.5 0 −0.5

0.5

0 100 200 300 No.of days before measurement period 50

0

−50

100 200 300 No.of days before measurement period

30 20 10

−2 −4 −6

−8 50 100 150 200 No.of days before measurement period

100 200 300 No.of days before measurement period 1

Day: 20050531 40

50 100 150 200 No.of days before measurement period AAT MPR constant

0

400

AAT 2OLS−MPR

1

600

AAT spline−MPR

100 200 300 No.of days before measurement period

CAT-Essen prices

200

CAT MPR constant

400

Day: 20060530 800

CAT 2OLS−MPR

600

CAT spline−MPR

Day: 20060530 800

CAT MPR constant

CAT-Berlin prices

86 W. K. Härdle and B. López Cabrera

20 0 −20 −40

50 100 150 200 No.of days before measurement period 0

−0.2

−0.4

50 100 150 200 No.of days before measurement period

Figure 7. Futures CAT prices (1 row panel) and MPR specifications: constant MPR for different contracts per trading day, two piecewise constant and spline (2, 3 and 4 row panel) for BerlinCAT (left), Essen-CAT (middle), Tokyo-AAT (right) of futures traded on 20060530, 20060530 and 20060531, respectively.

We investigate the proposition that the MPR derived from CAT/HDD/CDD futures is different from 0. We conduct the Wald statistical test to check whether this effect exists by testing the true value of the parameter n based on the sample estimate. In  the multivariate case, the Wald statistic for θt ∈ Ri t=1 is 1 (θˆt − θ0 )⊤ (θˆt − θ0 ) ∼ χp2 ,  2 (θˆt − θ0 ) ∼ N (0, Ii ),

where  is the variance matrix and the estimate θˆt is compared with the proposed value θ0 = 0. Using a sample size of n trading dates of contracts with t ≤ τ1i < τ2i , i = 1, . . . , I, we illustrate in Table 5 the Wald statistics  for all n previous MPR specifications. We reject H0 : θˆt = 0 under the Wald statistic θt ∈ Ri t=1 for all cases. Although the constant per trading day and general MPR specifications smooth deviations over time, the Wald statistic confirms that the MPR differs significantly from 0. Our results are robust to all specifications. Figure 10 shows the smoothing of MPR individuals (Equation (27)) for different specifications in 1 (20060530), 5 (20060522–20060530) and 30 trading days (20060417– 20060530) of Berlin-CAT futures, while the last panel in Figure 10 gives the results

The Implied Market Price of Weather Risk 87 (b)

12

12

10

10

8

8

CAT volatility

CAT volatility

(a)

6

6

4

4

2

2

2004

2005

2006

2007

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2008

Before the measurement period 2006

Before the measurement period

(d)

12

12

10

10

8

8

CAT volatility

CAT volatility

(c)

6

6

4

4

2

2

0 2004

2005

2006

2007

2008

0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Within the measurement period 2006

Within the measurement period

Figure 8. The Samuelson effect for Berlin-CAT futures explained by the CAT volatility σt at,τ1 ,τ2 ep (black line) and the volatility σt of Berlin-CAT futures (dash line) from 2004 to 2008 and 2006 for contracts traded before (a) and (b) and within (c) and (d) the measurement period.

when MPR estimates are obtained from smoothed prices using the calendar year estimation (Equation (29)). Both smoothing procedures lead to similar outcomes: notable changes in sign, MPR deviations are smoothed over time and the higher the number of calendar days, the closer the fit of Equations (27) and (29). This indicates that sample size does not influence the stochastic behaviour of the MPR. To interpret the economic meaning of the previous MPR results, recall, for example, the relationship between the RP (the market price minus the implied futures price with MPR equal to 0) and the MPR for CAT temperature futures: RPCAT =



τ1i t

θu σu at,τ1i ,τ2i ep du +



τ2i

τ1i

    −1 θu σu e⊤ exp A(τ2i − u) − Ip ep du, 1A

(30)

which can be interpreted as the aggregated MPR times the amount of temperature risk σt over [t, τ1i ] (first integral) and [τ1i , τ2i ] (second integral). By adjusting the MPR value, these two terms contribute to the CAT futures price. For temperature futures with values that are positive related to weather changes in the short term, this implies a

88 W. K. Härdle and B. López Cabrera (a) 14

(b) 5

12 4

8 6

AR(3) effect

CAT volatility

10 3

2

4 1 2 0 −10 −8 −6 −4 −2 Trading days prior to measurement period

0 −10 −8 −6 −4 −2 Trading days prior to measurement period

Figure 9. (a) The CAT term structure of volatility and (b) the autoregressive effect of two contracts issued on 20060517: one with whole June as measurement period (straight line) and the other one with only the 1st week of June (dotted line).

negative RP meaning that buyers of temperature derivatives expect to pay lower prices to hedge weather risk (insurance RP). In this case, θt must be negative for CAT futures, since σt and X t are both positive. Negative MPRs translate into premiums for bearing risk, implying that investor will accept a reduction in the return of the derivative equal to the right-hand side of Equation (30) in exchange for eliminating the effects of the seasonal variance on pay-offs. On the other side, positive RP indicates the existence of consumers, who consider temperature derivatives for speculation purposes. In this case, θt must be positive and implies discounts for taking additional (weather) risk. This rules out the ‘burn-in’ analysis of Brix et al. (2005), which seems to popular among practitioners since it uses the historical average index value as the price for the futures. The sign of MPR–RP reflects the risk attitude and time horizon perspectives of market participants in the diversification process to hedge weather risk in peak seasons. By understanding the MPR, market participants might earn money (by shorting or longing, according to the sign). The investors impute value to the weather products, although they are non-marketable. This might suggest some possible relationships between risk aversion and the MPR. The non-stationarity behaviour of the MPR (sign changes) is also possible because it is capturing all the non-fundamental information affecting the futures pricing: investors preferences, transaction costs, market illiquidity or other fractions like effects on the demand function. When the trading is illiquid the observed prices may contain some liquidity premium, which can contaminate the estimation of the MPR. Figure 11 illustrates the RP of Berlin-CAT futures for monthly contracts traded on 20031006–20080527. We observe RPs different from 0, time dependent, where positive (negative) MPR contributes positively (negatively) to futures prices. The mean for the constant MPR for the i = 1, . . . , 7th Berlin-CAT futures contracts per trading date is of size 0.02, 0.05, 0.02, 0.01, 0.10, 0.02 and 0.04, thus the terms in Equation (30)

MPR of smoothed prices 1 0 −1

0 100 200 300 400 No. of days before measurement period

0 100 200 300 400 No. of days before measurement period Smoothing bootstrap−MPR

0 100 200 300 400 No. of days before measurement period

5 0 −5

Smoothing spline−MPR

0 100 200 300 400 No. of days before measurement period

50 0 −50

MPR of smoothed prices

0 100 200 300 400 No. of days before measurement period

CAT MPR

CAT MPR

Smoothing OLS−MPR

0 100 200 300 400 No. of days before measurement period

CAT MPR

0 100 200 300 400 No. of days before measurement period

Smoothing OLS2−MPR

5 0 −5

20 0 −20

CAT MPR

Smoothing spline−MPR

4 2 0

0.5 0 −0.5

CAT MPR

0 100 200 300 400 No. of days before measurement period

CAT MPR

0 100 200 300 400 No. of days before measurement period

20 0 −20

CAT MPR

0 100 200 300 400 No. of days before measurement period

Smoothing OLS2−MPR

Smoothing bootstrap−MPR

Smoothing OLS−MPR

Smoothing constant−MPR

0 100 200 300 400 No. of days before measurement period

0.5 0 −0.5

CAT MPR

1 0 −1

100 0 −100

0.5 0 −0.5

0.2 0 −0.2

CAT MPR

CAT MPR CAT MPR

0 0 100 200 300 400 No. of days before measurement period 5 0 −5

CAT MPR

Smoothing OLS−MPR

Smoothing constant−MPR

0 100 200 300 400 No. of days before measurement period

CAT MPR

0.5

CAT MPR

0 100 200 300 400 No. of days before measurement period

0.5 0 −0.5

CAT MPR

Smoothing constant−MPR 1 0 −1

CAT MPR

CAT MPR

CAT MPR

The Implied Market Price of Weather Risk 89

0.5 0 −0.5

Smoothing OLS2−MPR

0 100 200 300 400 No. of days before measurement period Smoothing bootstrap−MPR

0 100 200 300 400 No. of days before measurement period Smoothing spline−MPR

0 100 200 300 400 No. of days before measurement period MPR of smoothed prices

0 100 200 300 400 No. of days before measurement period

Figure 10. Smoothing the MPR parameterization for Berlin-CAT futures traded on 20060530: the calendar year smoothing (black line) for 1 day (left), 5 days (middle) and 30 days (right). The last row gives MPR estimates obtained from smoothed prices.

contribute little to the prices compared to the seasonal mean t . The RPs are very small for all contract types, and they behave constant within the measurement month but fluctuate with σt and θt , leading to higher RPs during volatile months (winters or early summers). This suggests that the temperature market does the risk adjustment according to the seasonal effect, where low levels of mean reversion mean that volatility plays a greater role in determining the prices. Our data extracted MPR results can be comparable with Cao and Wei (2004), Richards et al. (2004) and Huang-Hsi et al. (2008), who showed that the MPR is not only different from 0 for temperature derivatives, but also significant and economically large as well. However, the results in Cao and Wei (2004) and Richards et al. (2004) rely on the specification of the dividend process and the risk aversion level, while the approach of Huang-Hsi et al. (2008) depends on the studied Stock index to compute the proxy estimate of the MPR. Alaton et al. (2002) concluded that the MPR impact is likely to be small. Our findings can also be compared with the MPR of other nontradable assets, for example, in commodities markets; the MPR may be either positive or negative depending on the time horizon considered. In Schwartz (1997), the calibration of futures prices of oil and copper delivered negative MPR in both cases. For electricity, Cartea and Figueroa (2005) estimated a negative MPR. Cartea and Williams (2008) found a positive MPR for gas long-term contracts and for short-term

90 W. K. Härdle and B. López Cabrera CAT−RP

(a)

40 20 0 −20 −40 −60 −80 20040101

20050101

20060101

20070101

20080101

Time

(e) CAT−RP

CAT−RP

(b) 100 0 −100 20040101

20050101

20060101 Time

0 −100 20040101

20070101 20080101

20050101

20070101 20080101

CAT−RP

(f) 100 0 −100 20040101

20050101

20060101

100 0 −100 20040101

20070101 20080101

20050101

Time

20060101

20070101 20080101

Time

(d)

(g) CAT−RP

CAT−RP

20060101 Time

(c) CAT−RP

100

100 0 −100 20040101

20050101

20060101

20070101 20080101

100 0 −100 20040101

Time

20050101

20060101

20070101 20080101

Time

Figure 11. Risk premiums (RPs) of Berlin-CAT monthly futures prices traded during (20031006– 20080527) with t ≤ τ1i < τ2i and contracts i = 1 (30 days), i = 2 (60 days),. . ., i = I (210 days) traded before measurement period. RPs of Berlin CAT futures for (a) 30 days, (b) 60 days, (c) 90 days, (d) 120 days, (e) 150 days, (f) 180 days and (g) 210 days.

contracts the MPR changes signs across time. Doran and Ronn (2008) demonstrated the need of a negative market price of volatility risk in both equity and commodityenergy markets (gas, heating oil and crude oil). Similar to weather, electricity, natural gas and heating oil markets show seasonal patterns, where winter months have higher RP. The only difference is that in temperature markets, the spot–futures relation is not clear since the underlying is not storable (Benth et al., 2008). 5.8 Pricing CAT–HDD–CDD and OTC Futures Once that market prices of traded derivatives are used to back out the MPR for temperature futures, the MPR for options is also known and thus one can price other temperature contract types with different maturity (weekly, daily or seasonal contracts) and over the counter OTC derivatives (e.g. Berlin-CDD futures or for cities without formal WD market). This method seems to be popular among practitioners in other markets. This section tests the MPR specifications to fit market prices in sample. The implied MPR (under multiple specifications) from monthly CAT futures in Section 5.7 are used to calculate theoretical CDD prices Equation (20) for Berlin, Essen and Tokyo. We then compute HDD futures prices from the HDD–CDD parity in Equation (4) and compare them with market data (in sample performance). Table 4 shows the CME futures prices (Column 5), the estimated risk-neutral prices with P = Q (MPR = 0), the estimated futures prices with constant MPR for different contracts per trading date and the index values computed from the realized temperature data I(τ1 ,τ2 ) . While

The Implied Market Price of Weather Risk 91 the inferred prices with constant MPR replicate market prices, the estimated prices with P = Q are close to the realized temperatures, meaning that the history is likely a good prediction of the future. Table 6 describes the root mean squared errors (RMSEs) of the differences between the market prices and the estimated futures prices, with MPR values implied directly from specific futures contract types and with MPR values extracted from the HDD/CDD/CAT parity method, over different periods and cities. The RMSE is defined as + , n  , (Ft,τ1i ,τ2i − Fˆ t,τ1i ,τ2i )2 , RMSE = -n−1 t=1

where Fˆ t,τ1i ,τ2i are the estimated futures prices and small RMSE values denote good measure of precision. The RMSE estimates in the case of the constant MPR for different CAT futures contracts are statistically significant enough to know CAT futures prices, but fail for HDD futures. Since temperature futures are written on different indices, the implied MPR will be then contract-specific hence requiring a separate estimation procedure. We argue that this inequality in prices results from additional premiums that the market incorporates to the HDD estimation, due to possible temperature market probability predictions operating under a more general equilibrium rather than nonarbitrage conditions (Horst and Mueller, 2007) or due to the incorporation of weather forecast models in the pricing model that influence the risk attitude of market participants in the diversification process of hedging weather risk (Benth and Meyer-Brandis, 2009; Dorfleitner and Wimmer, 2010; Papazian and Skiadopoulos, 2010). We investigate the pricing algorithm for cities without formal WD market. In this context, the stylized facts of temperature data (t , σt ) are the only risk factors. Hence, a natural way to infer the MPR for emerging regions is by knowing the MPR dependency on seasonal variation of the closest geographical location with formal WD market. For example, for pricing Taipei weather futures derivatives, one could take the WD market in Tokyo and learn the dependence structure by simply regressing the average MPR of Tokyo-C24AT futures contracts i over the trading period against the seasonal variation in period [τ1 , τ2 ]: τ

θˆτi1 ,τ2 = σˆ τ21 ,τ2 =

1 1  θˆ i , τ1 − t t t

τ2 1  σˆ 2 . τ2 − τ1 t=τ t 1

In this case, the quadratic function that parameterizes the dependence is θt = 4.08 − 2.19σˆ τ21 ,τ2 + 0.28σˆ τ41 ,τ2 , with R2adj = 0.71 and MPR increases by increasing the drift and volatility values (Figure 12). The dependencies of the MPR on time and temperature seasonal variation indicate that for regions with homogeneous weather risk there is some common market price of weather risk (as we expect in equilibrium).

20070401 20070501 20070601 20070701 20070801 20070901 20061101 20061201 20061101 20061201 20070401 20070501 20070601 20070701 20070801 20070901 20070401 20070501 20070601 20070701 20070801 20070901 20090301 20090401 20090501 20090601 20090701

Atlanta-CDD+ Atlanta-CDD+ Atlanta-CDD+ Atlanta-CDD+ Atlanta-CDD+ Atlanta-CDD+ Berlin-HDD∗ Berlin-HDD∗ Berlin-HDD+ Berlin-HDD+ Berlin-CAT+ Berlin-CAT+ Berlin-CAT+ Berlin-CAT+ Berlin-CAT+ Berlin-CAT+ Essen-CAT+ Essen-CAT+ Essen-CAT+ Essen-CAT+ Essen-CAT+ Essen-CAT+ Tokyo-C24AT+ Tokyo-C24AT+ Tokyo-C24AT+ Tokyo-C24AT+ Tokyo-C24AT+ 20070430 20070531 20070630 20070731 20070831 20070930 20061130 20061231 20061130 20061231 20070430 20070531 20070630 20070731 20070831 20070930 20070430 20070531 20070630 20070731 20070831 20070930 20090331 20090430 20090531 20090630 20090731

τ2 230 228 230 229 229 230 22 43 22 43 230 38 58 79 101 122 230 39 59 80 102 123 57 116 141 141 141

No. of contracts 15.12 20.56 18.52 11.56 21.56 17.56 129.94 147.89 39.98 57.89 18.47 40.38 10.02 26.55 34.31 32.48 13.88 52.66 15.86 16.71 31.84 36.93 161.81 112.65 81.64 113.12 78.65

MPR = 0 20.12 53.51 43.58 39.58 33.58 53.58 164.52 138.45 74.73 58.45 40.26 47.03 26.19 16.41 12.22 17.96 33.94 52.95 21.35 44.14 22.66 14.28 148.21 99.55 70.81 92.66 74.95

Constant 150.54 107.86 97.86 77.78 47.86 77.86 199.59 169.11 89.59 99.11 134.83 107.342 78.18 100.22 99.59 70.45 195.98 198.18 189.45 155.82 56.93 111.58 218.99 156.15 111.21 104.75 116.34

1 piecewise 150.54 107.86 97.86 77.78 47.86 77.86 199.59 169.11 89.59 99.11 134.83 107.34 78.18 100.22 99.59 70.45 195.98 198.188 189.45 155.82 56.92 111.58 218.99 156.15 111.21 110.68 3658.39

2 piecewise 20.15 53.52 44.54 39.59 33.59 53.54 180.00 140.00 74.74 58.45 40.26 47.03 26.20 16.41 12.22 17.96 33.94 52.95 21.38 44.14 22.66 14.28 148.21 99.55 70.81 92.66 74.95

Bootstrap

27.34 28.56 35.56 38.56 38.56 18.56 169.76 167.49 79.86 88.49 18.44 40.38 10.02 26.55 34.31 32.48 13.87 52.66 15.86 16.71 31.84 33.93 158.16 109.78 79.68 111.20 77.07

Spline

RMSE between estimated with MPR (θt ) and CME prices

Notes: RMSE, root mean squared error; MPR, market price of risk; CME, Chicago Mercantile Exchange. Futures prices with t ≤ τ1i < τ2i and the estimated futures with implied MPR under different MPR parameterizations (MPR = 0, constant MPR for different contracts (Constant), 1 piecewise constant MPR, 2 piecewise constant MPR, bootstrap MPR and spline MPR). + Computations with MPR implied directly from specific futures contract types (+ ) and ∗ through the parity HDD/CDD/CAT parity method(∗ ).

τ1

Contract type

Measurement period

Table 6. RMSE of the differences between observed CAT/HDD/CDD.

92 W. K. Härdle and B. López Cabrera

The Implied Market Price of Weather Risk 93 0.6

0.4 Apr

Aug

MPR

0.2

0

May Nov Jul Oct

−0.2

Jun Sep

−0.4 2.5

3 3.5 4 4.5 5 Average temperature variation in measurement month

5.5

Figure 12. The calibrated MPR as a deterministic function of the monthly temperature variation of Tokyo-C24AT futures from November 2008 to November 2009 (prices for 8 contracts were available).

6. Conclusions and Further Research This article deals with the differences between ‘historical’ and ‘risk-neutral’ behaviours of temperature and gives insights into the MPR, a drift adjustment in the dynamics of the temperature process to reflect how investors are compensated for bearing risk when holding the derivative. Our empirical work shows that independently of the chosen location, the temperature-driving stochastics are close to the Gaussian risk factors that allow us to work under the financial mathematical context. Using statistical modelling, we imply the MPR from daily temperature futures-type contracts (CAT, CDD, HDD, C24AT) traded at the CME under the EMM framework. Different specifications of the MPR are investigated. It can be parameterized, given its dependencies on time and seasonal variation. We also establish connections between the RP and the MPR. The results show that the MPRs–RPs are significantly different from 0, changing over time. This contradicts with the assumption made earlier in the literature that MPR is 0 or constant and rules out the ‘burn-in’ analysis, which is popular among practitioners. This brings significant challenges to the statistical branch of the pricing literature, suggesting that for regions with homogeneous weather risk there is a common market price of weather risk. In particular, using a relationship of the MPR with a utility function, one may link the sign changes of the MPR with risk attitude and time horizon perspectives of market participants in the diversification process to hedge weather risk. A further research on the explicit relationship between the RP and the MPR should be carried out to explain possible connections between modelled futures prices and their deviations from the futures market. An important issue for our results is that the econometric part in Section 2 is carried out with estimates rather than true values. One thus deals with noisy observations, which are likely to alter the subsequent estimations and test procedures. An alternative to this is to use an adaptive local parametric estimation procedure, for example, in Mercurio and Spokoiny (2004) or Härdle et al. (2011).

94 W. K. Härdle and B. López Cabrera Finally, a different methodology, but related to this article, would be to imply the pricing kernel of option prices. Acknowledgement We thank Fred Espen Benth and two anonymous referees for several constructive and insightful suggestions on how to improve the article.

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Journal of Risk Research Vol. 11, No. 7, October 2008, 905–923

The influence of mood on the willingness to take financial risks John E. Grablea* and Michael J. Roszkowskib a

Institute of Personal Financial Planning, Kansas State University, Kansas, USA; bOffice of Institutional Research, La Salle University, Philadelphia, USA The purpose of this study was to determine whether support could be found for either the Affect Infusion Model or the Mood Maintenance Hypothesis regarding how mood influences financial risk tolerance. An ordinary least-squares regression model was used to determine if people who exhibited a happy mood at the time they completed a survey scored differently than those who were not happy. In a sample (n5460) of employed mid-western respondents between the ages of 18 and 75 years, being in a happy mood was positively associated with having a higher level of financial risk tolerance, holding biopsychosocial and environmental factors constant. Support for the Affect Infusion Model was obtained. Keywords: risk tolerance; mood; emotions; risk-as-feelings; risky decisions

Introduction Mood is a transient generalized affective state (Watson and Vaidya 2003) that can exert a dramatic influence on almost all aspects of a person’s daily life. According to Sizer (2000, 762), ‘Moods affect a wide range of our thoughts, feelings and attitudes in ways that are not constrained by subject matter or inferential rules’. Although the psychological literature recognizes that a person’s emotional state impacts decision making in general as well as on a variety of consumer behaviors (Bagozzi, Gopinath, and Nyer 1999; Lerner, Han, and Keltner 2007; Lerner and Keltner 2000; Lerner, Small, and Loewenstein 2004; Luomala and Laaksonen 2000; Mellers, Schwartz, and Ritov 1999; Schwarz and Clore 1996; Slovic et al. 2004), there still exists considerable debate among economists regarding the role that mood plays in the way in which consumers make financial decisions and how this affects financial markets (Ackert, Church, and Deaves 2003; Clarke and Statman 1998; Olson 2006). Until recently, the cognitively based utility theory of risk tolerance dominated financial services research. A shortcoming associated with the traditional economic utility approach is that the theory fails to adequately explain many financial attitudes and behaviors, such as shifting of risk-aversion preferences when questions with similar payoffs are framed differently (Slovic et al. 2004). The role of affective states as a factor influencing behavior is seldom examined within economic utility frameworks. Over the past several decades, researchers have begun to examine the impact moods have on the way people perceive risk (Johnson and Tversky 1983) and how individuals make risky decisions when in different states of mind (Hirshleifer and Shumway 2003; Hockey et al. 2000). These researchers acknowledge that responses to risky situations and circumstances are a result of both analytical (i.e. cognitive) *Corresponding author. Email: [email protected] ISSN 1366-9877 print/ISSN 1466-4461 online # 2008 Taylor & Francis DOI: 10.1080/13669870802090390 http://www.informaworld.com

906 J.E. Grable and M.J. Roszkowski and affective (e.g. emotional) influences (Schunk and Betsch 2006; Townsend 2006; Wang 2006), but they differ in their conceptualizations of the role of specific emotional states, and unfortunately the predictions based on different models are contradictory and the results are inconsistent. Two competing possibilities have been proposed to explain how, theoretically, mood can alter one’s willingness to accept risk. The one theory is termed the Affect Infusion Model (AIM), while the other is called the Mood Maintenance Hypothesis (MMH). Perplexingly, these two approaches lead to opposing predictions about the influence that positive and negative moods will have on risk tolerance. According to AIM, a positive mood is expected to increase risk tolerance, whereas a negative mood should lower it (Forgas 1995), because selective attention and priming (Rusting and Larsen 1995, for example) causes the subjective probabilities to be construed differently. When in a good mood, the individual tends to focus on positive cues in the environment. Conversely, a bad mood shifts one’s attention to the negative features of the situation. On the other hand, MMH, advanced by Isen and her colleagues (Isen and Labroo 2003; Isen and Patrick 1983), suggests that a good mood will lead to greater caution, whereas a bad mood will encourage greater recklessness. According to this theory, people in a good mood want to remain in that state, so they are unwilling to take risks that could potentially result in losses that would shift them into a bad mood. However, when in a bad mood, they will behave less cautiously in the hopes of taking a chance and obtaining a reward, which would put them back into a good mood. The purpose of the present study was to determine whether support could be found for either the AIM or MMH approach using data from surveys of adults completing a risk-tolerance questionnaire of the type used by financial advisors to gauge client risk tolerance to determine suitable investments. Little empirical research has been published on the topic with this type of participant. Understanding the role of moods on a person’s risk attitude is not insignificant. Obviously, such information can be used to inform consumer decisions related to the investment and allocation of assets. It can also be used to help consumers adopt realistic purchasing behaviors. Emotions and moods Definition of mood and emotion The terms ‘mood’ and ‘emotion’ are often used interchangeably, when in fact they are closely related but distinct phenomena (Beedie, Terry, and Lane 2005). Although the differences are subtle, the implications resulting from the distinction can be dramatic. Both emotions and moods fall within the theoretical realm of ‘affect’, which can be defined as ‘…the specific quality of goodness or badness (1) experienced as a feeling state (with or without consciousness) and (2) demarcating a positive or negative quality of a stimulus’ (Slovic et al. 2004). Thus, at the most general level, affective states of both sorts can be categorized into positive (pleasant) and negative (unpleasant) feelings. However, emotions are feelings about a particular circumstance or event (someone or something) that arise from cognitive appraisals of circumstances, whereas moods are more generalized non-specific states that are not directed at any

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particular target (Bagozzi, Gopinath, and Nyer 1999; Siemer 2005; Sizer 2000; Watson and Clark 1997). In other words, emotions are in reaction to specific stimuli, whereas moods are free-floating feelings that need not be linked to anything specific. Emotional states include specific feelings like anger, jealousy, fear and envy, while moods are general states of mind such as happy and sad. The dispositional theory of moods suggests that a person’s mood is temporary (Siemer 2005), but the duration of moods is longer than that of emotions. Moreover, moods tend to be unaffected by personal beliefs, and unlike emotions, moods are ‘not intentional mental states’ (Sizer 2000, 754). Sizer describes moods as follows: Moods are disengaged or disconnected from our beliefs and knowledge, demonstrating that they are not penetrable or influenced by the semantic contents of our representations. If one is depressed about everything, or things in general, then no particular piece of new information or change in belief is going to affect this underlying pervasive mood. Even if one is depressed and anxious about many different things – the noise outside the window, the pain in one’s temple, or the upcoming meeting – the underlying state of anxiety remains regardless of the content of the subject’s worrying. New information (that the noise outside was caused by the cat) does nothing to relieve the anxiety; it simply shifts it to a different focus. (760)

Role of mood in financial decisions Traditional (utility theory) models The specific role of mood in influencing attitudes and behaviors has received scant attention within the personal and consumer finance literature. This is primarily the result of the theoretical models used by those who study consumer attitudes and behaviors. Until recently, expected utility theory (or a conceptual offshoot) was the primary theoretical underpinning of nearly all personal and household finance research. Advocates of economic utility as a model of decision making assume that decisions are made logically using a reasoned processing method (i.e. rational/analytic system). The influence of emotions and moods in such decisions, by definition, is excluded. Consider expected utility theory as a framework for rational/analytic systems, e.g. Modern Portfolio Theory (MPT). According to Loewenstein and his associates (2001, 267), ‘economic utility theories posit that risky choice can be predicated by assuming that people assess the severity and likelihood of the possible outcomes of choice alternatives, albeit subjectively and possibly with bias or error, and integrate this information through some type of expectation-based calculus to arrive at a decision. Feeling triggered by the decision situation and imminent risky choice are seen as epiphenomenal – that is, not integral to the decision-making process’. MPT is based on the assumption that individual investors develop tradeoffs between risks and returns when creating portfolios of risky assets (Mayo 2000). MPT, according to Mayo, ‘indicates that investors require ever-increasing amounts of additional return for equal increments of risk to maintain the same level of satisfaction’ (184). The Capital Asset Pricing Model (CAPM) extends MPT by defining the relationship between risk and return as purely positive. In a CAPM framework, investors rationally obtain higher rates of return by taking greater risks. At the root of both MPT and CAPM is the assumption that portfolio asset allocation depends on an ‘individual’s willingness to bear risk’ (Mayo 2000, 189). Implicit in this assumption is that investors are economically rational when making tradeoffs between risk and return.1 In other words, within an MPT framework,

908 J.E. Grable and M.J. Roszkowski tradeoffs between risk and return are purely analytical with emotions playing almost no role in influencing behavior. More recent models Researchers have started to take steps to move beyond traditional economic utility theory modeling of risk tolerance. The blending of behavioral, psychological and economic theories has opened up new lines of research within the personal and household finance fields. These include the burgeoning fields of behavioral finance and household economics. It is now generally recognized in these disciplines that individuals can use two modes of thinking when assessing circumstances and evaluating risks: rational/analytic (cognitive) and experiential (affective) systems (Epstein 1994; Slovic et al. 2004; Wang 2006). Schunk and Betsch (2006) reported evidence suggesting that a preference for either a cognitive (rational/analytic) or an affective (experiential) mode for processing risk information may be an individual difference. Some people, whom they call ‘intuitive’ decision makers, are more likely to process risk information on the basis of affective states, whereas others, whom they call ‘deliberative’ decision makers, are more apt to process the information on a cognitive basis. They concluded the following: Our findings suggest that intuitive people use the affective risk information contained in the lotteries when making their decisions, which might lead to the risk attitude (i.e., a feeling of risk) becoming integrated in the judgment, resulting in risk-averse or riskseeking behavior. Deliberative people, on the contrary, seem to base their decisions on the stated values rather than on affect. It seems unlikely that deliberative people do not have any affective reactions to the lotteries, but they might therefore abstract from this affective information and might discount or neglect it when making their judgments (a process that requires time). (11)

Approaches to modeling risky decision making now exist that not only acknowledge that affect plays a role in the process, but in fact have the experiential system as the core process. Moods, in particular, are believed to influence the type and amount of risks people are willing to take. Slovic et al. (2004, 315) found that ‘people base their judgments on an activity or a technology not only on what they think about it but also on how they feel about it. If their feelings toward an activity are favorable, they are moved toward judging the risks as low and the benefits as high; if their feeling toward it are unfavorable, they tend to judge the opposite – high risk and low benefit. Under this model, affect comes prior to, and directs, judgments of risk and benefit’ (315). There is some evidence to suggest that collective mood or market sentiment impacts stock and bond market returns at the macro-level (Olson 2006). For instance, Clarke and Statman (1998) found that high returns in the stock market, over short periods of time, are associated with increased bullishness among investment newsletter writers. Clarke and Statman hypothesized that volatility in the markets increases bullishness. While not explicitly stated, their findings suggest that newsletter writers’ moods appear to impact risk-taking attitudes, and that the moods of investors change over short periods of time. The risk-as-feelings hypothesis A framework that combines the rational/analytic and experiential systems is the ‘risk-as-feelings’ hypothesis proposed by Loewenstein and his associates (2001). This

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model postulates that individuals evaluate risky situations using both cognitive and affective processes. In this framework, cognitive evaluations are based on subjective probability estimates and anticipated outcomes, whereas feelings about risk are influenced by factors such as vividness and mood. A unique feature of the model is the proposition ‘…that responses to risky situations (including decision making) result in part from direct (i.e. not cortically mediated) emotional influences…’ (270). In other words, according to Slovic et al. (2004), ‘affect influences judgment directly and is not simply a response to a prior analytic evaluation’ (315). Schwarz (2000) pointed out that even though emotion (i.e. mood) appears to affect a person’s judgment about future events directly, the relationship between affect and decision making is bidirectional. In other words, it is possible that outcomes associated with given behaviors can induce changes in emotions resulting from the gains and losses experienced with the risky behavior. Decisions that are influenced by emotions and moods tend to be easier, faster, and often more efficient than decisions made using a rational/analytic system. This is not to say that experiential decision systems that have an affect basis always lead to optimal financial risk choices. It is possible that a person’s mood can cause shortterm myopic decisions that do not account for later changes in emotions and circumstances. Loewenstein and his associates (2001) noted that affect can even play an important role in traditional risk-averse forward-looking decision making. If this is true, the role of moods and emotions in influencing rational/analytic decision systems becomes a topic of interest not only to behavioralists but economic rationalists as well. Determinants of risk tolerance Biopsychosocial and environmental factors Irwin (1993) presented a risk-taking behavioral model that can be used as a framework for understanding the determinants of risk tolerance and risk taking. Irwin suggested that both environmental and biopsychosocial factors can exert an influence on risk tolerance. Environmental factors, as defined by Irwin, include individual and family financial attributes. Examples of environmental factors are income, net worth and home ownership status. Biopsychosocial factors are those aspects of an individual’s life that reflect stable (perhaps immutable) individual differences. These factors include one’s demographic characteristics (e.g. racial background, age and gender) and deeply ingrained or inherent personality dimensions over which a person has little or no control. Examples of the latter are traits that result from a person’s social environment, attitudes, beliefs and psychosocial factors. In general, the literature to date confirms that both environmental and biopsychosocial factors play a role in the way a person evaluates financially risky situations (Callan and Johnson 2002; Coleman 2003; Goodall and Corney 1990; Grable and Joo 2004; Grable and Lytton 1998; Hawley and Fujii 1993–1994; Horvath and Zuckerman 1993; Huston, Chang, and Metzen 1997; Kennickell, StarrMcCluer, and Sunden 1997; Roszkowski 1999; Sung and Hanna 1996; Wang and Hanna 1997; Wong and Carducci 1991). Eleven environmental and biopsychosocial variables commonly emerge in research findings as being associated with financial risk-taking attitudes (see Bajtelsmit 2006 for a discussion of some of these factors).

910 J.E. Grable and M.J. Roszkowski Table 1. Factors affecting financial risk-tolerance attitudes. Variable Age* Gender Race/ethnic background Financial satisfaction Household income Net Worth Education Homeownership Marital status* Employment status Financial knowledge

Type

Characteristic

Relationship

Biopsychosocial Biopsychosocial Biopsychosocial Biopsychosocial Environmental Environmental Environmental Environmental Environmental Environmental Environmental

Younger Male Non-Hispanic White Higher Higher Higher Higher educational level Own home Single Employed full-time Higher

Positive Positive Positive Positive Positive Positive Positive Positive Positive Positive Positive

*Research findings are not consistent in the relationship to risk tolerance; curvilinear effects have sometimes been noted with age.

The relationships between these variables and financial risk tolerance are summarized in Table 1. Mood In a recent discussion of risk tolerance at Morningstar Forums (2007), a participant named Megan made the following astute observation: ‘My risk tolerance sometimes is high, sometimes is low (depends on my mood, hah)’. This is anecdotal evidence that an important factor missing in Table 1 is mood. A person’s mood is known to impact all types of daily decisions, including the type of clothes worn, food eaten, and participation in risky and non-risky activities (Ackert, Church, and Deaves 2003; Hirshleifer and Shumway 2003; Schwarz 2000). Therefore, it would be quite strange if mood did not exert some influence on risk tolerance, as Megan suggested. Megan failed to indicate, however, whether a good mood raised or lowered her risk tolerance. In general, there exists controversy about the way mood affects a person’s financial risk-tolerance (Hockey et al. 2000). Evidence in support of the Affect Infusion Model (AIM) Based on AIM, individuals who exhibit a positive mood when making a risky choice tend to be willing to take more risks than those with a negative or neutral mood. A number of studies have reported results supporting AIM (Chou, Lee, and Ho 2007; Deldin and Levin 1986; Fehr et al. 2007; Johnson and Kahneman 1983; Leith and Baumeister 1996; Mayer et al. 1992; Mittal and Ross 1998; Nygren et al. 1996; Pietromonaco and Rook 1987; Wegener, Petty, and Klein 1994; Williams 2004; Wright and Bower 1992; Yuen and Lee 2003). For instance, Wright and Bower found that cheerful people (i.e. those in a ‘happy’ mood) tend to be more optimistic in general, and that optimistic people are more likely to report higher probabilities for positive risk events and lower probabilities for negative risk events. They observed that mood states have a greater influence on judging events that were less

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frequent. Sizer (2000) added that people may be less cautious when in a happy mood because positive moods are associated with wide informational focusing and lessened concentration on details. According to Schwarz (2000, 433), ‘Individuals are likely to evaluate about any target more positively when they are in a happy rather than sad mood’. Schwarz (2000) recommended that researchers studying the role of moods on decision making do so by accounting for mediating factors, such as gender. In fact, Fehr et al. (2007) did find a substantial gender difference in the impact that a good mood exerts on subjective probability weighting. Females in a good mood assigned higher subjective probability weights under both gain and loss scenarios, consistent with AIM. As a group, men, on the other hand, were not influenced by good mood. One possible reason is that the men were more analytical in their approach to the task, with 40% reporting that they made their decisions on the basis of the lotteries’ expected payoffs. Among the women, only about 8% used expected value calculations to benchmark their decisions. This finding suggests that, in the terminology used by Schunk and Betsch (2006), women are more ‘intuitive’ while men are more ‘deliberative’. The males who used the expected value approach were especially resistant to the influence of moods, but even men who do not use this rule showed only a weak mood effect, so adherence to mechanical rules does not explain the sex difference entirely. While this finding may suggest that women were less rational in their approach to the task, their probability weighting function under a good mood was less S-shaped, indicating that when in a positive frame of mind, they made more rational decisions. Good mood had an especially strong influence on women when the stated probabilities were high for gains and low for losses. Another mediating factor may be age. According to Chou et al. (2007), most studies of mood and risk taking compare a good mood to a neutral mood, with relatively few comparing a neutral to a bad mood. Yuen and Lee (2003) found that people under an induced sad mood were less risk tolerant than people in a neutral mood, consistent with the predictions based on AIM. However, people in an induced happy mood were not more risk tolerant than people in a neutral mood. In other words, Yuen and Lee discovered that mood has an asymmetrical impact on risk tolerance. That is, the difference in risk taking between an induced negative mood and a neutral state was larger than the difference between an induced positive mood and the neutral state. Chou et al. (2007) suggested that the findings from the Yuen and Lee (2003) study may be due to the fact that the participants were young people, who have a tendency to focus more on the negative rather than the positive aspects of a situation. In Chou et al.’s study, young and old people were therefore compared to see if positive and negative moods have an asymmetrical impact on risk taking among older individuals as well. As in the earlier study (i.e. Yuen and Lee 2003), Chou et al. found that among the young, there was no difference in risk taking between positive versus neutral moods, but there was a difference between the negative and neutral states. In contrast, the opposite occurred among the older subjects. Namely, the difference in risk taking was greater between the positive and the neutral mood states than between the neutral and the negative state. If the neutral point is disregarded, then for both the young and old participants, greater risk taking was evident among those in a happy mood than those in a sad mood.

912 J.E. Grable and M.J. Roszkowski Mehra and Sah (2002) examined the theory of projection bias and moods. Projection bias suggests that individuals take actions today as if the circumstances used to make the decision will persist into the future. This is a bias because people’s preferences change over time; however, when making risky decisions, people tend to not account for these shifting preferences. Mehra and Sah hypothesized that individuals project their current mood into their visualization of the future. They found that ‘small fluctuations in investors’ discount factors induce large fluctuations in equity prices’ (883), and that, in general, positive moods (i.e. projections) lead to increases in equity prices. Again, these results are consistent with AIM. Two studies on the role of weather fluctuations on moods and equity prices also support the AIM hypothesis. Kamstra, Kramer, and Levi (2003) found the effects of a seasonal affective disorder (SAD) in stock market returns. Basically, SAD is a psychological condition where a reduction in the number of daylight hours is correlated with the onset of depressive symptoms. In such cases, fewer daylight hours result in increased levels of depression. Kamstra and his associates found a clear link between SAD symptoms and lower risk-taking behaviors. Hirshleifer and Shumway (2003) also discovered that the amount of sunshine in a given market influences moods, and that these moods in turn impact stock market returns. In their study, Hirshleifer and Shumway collected weather data in the major stock trading centers in 26 countries from 1982 to 1997. They concluded that ‘Sunshine is strongly significantly correlated with stock returns’ (1009), and that ‘People in good moods tend to generate more unusual associations, perform better in creative problemsolving tasks, and show greater mental flexibility’ (1012). Evidence in support of the Mood Maintenance Hypothesis (MMH) A smaller number of studies have reported support for MMH. Isen and Geva (1987) and Isen and Patrick (1983) report that positive moods produce risk-averse financial behaviors. A study by Kliger and Levy (2003) also used weather conditions as a proxy for mood, but unlike the results of Kamstra, Kramer, and Levi (2003), their findings were more in line with MMH. In real capital market decisions, investors were less risk tolerant under pleasant weather conditions (i.e. proxy for good mood) and more risk tolerant during unpleasant weather conditions (i.e. proxy for bad mood). Hockey and his associates (2000) found that risk-taking propensities were affected by a person’s level of fatigue, which induced a negative mood. When in a negative mood induced by fatigue, people exhibited increased levels of risk taking. According to Hockey et al., ‘Risky decisions are thought to be rejected under positive moods because the likely loss will upset the good mood state, whereas the likely gain from a low risk decision would serve to enhance or maintain it’ (824). Gaps in literature In summary, there is ample evidence to suggest that a person’s mood is related to the amount of risk they are willing to tolerate at any given time, but it is not clear what direction it will take. Given that nearly all of the previous literature is based on either macro-economic data or experiments using participants with induced mood states rather than naturally occurring moods, an apparent need exists to further study the influence of mood on risk-tolerance using individuals in a natural state of mind

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engaging in daily financial decision-making situations. Moreover, since nearly all studies to date have compared a positive to a neutral mood but relatively few studies have addressed the impact of negative mood on risk taking, there is value in further examining the effects of a negative mood on risk taking. Methodology An ordinary least-squares regression model was used to determine if people who exhibited a happy, neutral and gloomy mood at the time they completed a financial risk-tolerance quiz scored differently, holding other relevant factors constant. Participants Data for this study were obtained from a convenience sample of mid-western individuals who replied to a survey during spring 2005. The survey was sent to randomly selected, employed individuals from databases owned by the research team. Just over 1300 surveys were originally mailed (using the US postal service); 548 were returned. Thirty-six surveys were returned as undeliverable, while three were returned with missing data. Nine surveys were returned but not opened. The useable return response rate was calculated to be 38%. Given missing data, the sample size for this study was reduced to 460 respondents. The mean age of respondents was 44 years (SD512). Nine percent of respondents were never married, 4% were not married but living with a significant other, 4% were in a significant relationship, 63% were married, 8% were remarried, 1% were separated, 8% were divorced and 3% were widowed or other. Less than 1% of the sample was selfemployed. Six percent were employed part-time, while 85% were employed on a fulltime basis. One percent was either retired or a student; 7% were not employed. Over 92% of sample respondents were non-Hispanic Whites. Two percent were AfricanAmerican/Black, 2% were Hispanic/Latino, and 4% indicated another ethnic/racial background including Native American, Asian/Pacific Islander or Other. The sample was relatively homogenous and representative of the three communities from which data were collected. However, the sample was overrepresented by women. Seventy-one percent of respondents were women, while 29% were men. Moreover, on average the respondents were better educated and wealthier than the state and nation. For approximately 10%, a high school diploma or less was the highest educational attainment. Twenty-eight percent had some college or vocational training, 6% held an associate’s degree, 34% held a bachelor’s degree, and 22% earned a graduate or professional degree. The median household income, computed on grouped data, was $55,702. Outcome variable A 13-item risk-tolerance scale (Grable and Lytton 1999) was used as the dependent variable. For illustration, several sample items from the scale are shown below:

N

If you unexpectedly received $20,000 to invest, what would you do? a. Deposit it in a bank account, money market account, or an insured CD b. Invest it in safe high quality bonds or bond mutual funds c. Invest it in stocks or stock mutual funds

914 J.E. Grable and M.J. Roszkowski

N

When you think of the word ‘risk’ which of the following words comes to mind first? a. b. c. d.

N

Loss Uncertainty Opportunity Thrill

Some experts are predicting prices of assets such as gold, jewels, collectibles and real estate (hard assets) to increase in value; bond prices may fall; however, experts tend to agree that government bonds are relatively safe. Most of your investment assets are now in high interest government bonds. What would you do? a. Hold the bonds b. Sell the bonds, put half the proceeds into money market accounts, and the other half into hard assets c. Sell the bonds and put the total proceeds into hard assets d. Sell the bonds, put all the money into hard assets, and borrow additional money to buy more

N

Given the best and worst case returns of the four investment choices below, which would you prefer? a. b. c. d.

$200 gain best case; $0 gain/loss worst case $800 gain best case; $200 loss worst case $2600 gain best case; $800 loss worst case $4800 gain best case; $2,400 loss worst case

Possible scores on the scale can range from 13 (lowest risk tolerance) to 47 (highest risk tolerance). Scores in this study ranged from 14 to 34. The mean and standard deviation of the distribution were 23.16 and 4.10, respectively. The Cronbach’s alpha coefficient in this sample was 0.70, suggesting an adequate level of reliability for research. Grable and Lytton (1999, 2001) employed a principal components factor analysis in the development of the scale. The factor analysis resulted in three extracted factors: investment risk, risk comfort and experience, and speculative risk. The reliability of the overall instrument, using Cronbach’s alpha, has ranged from 0.70 to 0.85 (Yang 2004). The validity of the instrument has also been assessed. For example, Grable and Lytton (2001) compared the scale to the Survey of Consumer Finances (SCF) risk-assessment item. They found a modest positive correlation between the two measures (i.e. r50.54). Grable and Lytton (2003) also conducted a follow-up study of the scale’s criterion-related validity, finding that scores on the scale were positively related to the level of equity assets owned by individuals. Lower scale scores indicated an increased likelihood of holding cash or fixed-income assets. Control variables While the focus of the study was the relationship between mood and risk tolerance, 11 other independent variables (Table 2) were included in the model to serve as control variables for environmental and biopsychosocial influences on risk tolerance.

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Table 2. Control, independent, and dependent variables used in the regression model. Variable Control variables Age Gender (15male) Race/Ethnic Background (15non-Hispanic White) Household Income (15less than $20,000, 55$50,001–$60,000, 105over $100,000) Self-reported net worth (15in serious debt,105money left over) Educational status Some college or less Associate’s degree Bachelor’s degree Graduate degree Own home Married Employed full-time Financial knowledge (15the lowest level, 105the highest level) Financial satisfaction (15extremely unsatisfied,105extremely satisfied) Independent variables Mood Happy Neutral Gloomy Dependent variable Grable and Lytton risk tolerance questionnaire

Descriptive statistic M544.29 (SD512.00) 29% 94% M55.21 (SD52.48) M57.61 (SD52.64) 37% 6% 35% 22% 76% 64% 90% M56.42 (SD51.73) M55.62 (SD52.04)

38% 57% 5% M523.16 (SD54.10)

Age was measured at the interval level, while gender was coded 1 if male, otherwise 0. Non-Hispanic Whites, those who owned their own home, those who were married and those employed full-time were coded 1, otherwise 0. Those with an Associate’s, Bachelor’s, or Graduate degree were compared to those with some college education or less (i.e. the reference category). Household income was used as an interval variable. Financial satisfaction and financial knowledge were measured using 10point self-assessment scales that asked respondents to circle the number that represented how satisfied they were with their present financial situation and how knowledgeable they thought they were about personal finances compared to others, respectively. The scales were similar to ones used by Joo (1998) and Prawitz et al. (2006). Higher scores indicated increased satisfaction and knowledge. Self-reported net worth was measured using a 10-point scale originally designed by Porter (1990) and subsequently revised by Joo. Respondents were asked to indicate if they would be in serious debt (1), break even (5), or have money left over (10) if they sold all of their major possessions, investments, and other assets and paid off their debts. Independent variable A respondent’s affective state was assessed at the time the survey was completed. The mood measure included in the analysis was the first question on the survey. This

916 J.E. Grable and M.J. Roszkowski placement is crucial because earlier questions can affect the answer to the mood question (Kahneman and Krueger 2006; Kahneman et al. 2006). It consisted of a self-rating into one of three categories: happy, neutral and gloomy. The percentage of the sample in each mood was: 38% happy, 57% neutral and 5% gloomy. The respondents did not receive any immediate gratuity for participating; however, respondents could request results from the study. This aspect of the study is being made explicit, since it is possible that receipt of compensation might significantly lift a participant’s mood before completion. The small percentage of respondents in a gloomy mood posed a dilemma. Several options were considered: eliminate them from the analysis, combine gloomy with neutral, and run the analysis with three levels of mood. Given the purpose of the study, eliminating gloomy respondents was deemed a poor choice. The findings reported by Chou et al. (2007) and Yuen and Lee (2003) argue against option b. Option c appeared most reasonable given that a search of the literature on the distribution of happy, neutral and sad moods suggests that the frequency of the gloomy mood in our sample probably represents the actual distribution of gloomy moods in the population (Almeida, Wethington, and McDonald 2001; Crawford and Henry 2004; Kahneman et al. 2006; Kennedy-Moore 1992; Ram et al. 2005). In essence, few people are unhappy at any one moment for long. To illustrate, consider a Time magazine telephone poll conducted 13–14 December 2004 and reported in the 17 January 2005 issue (Wallis 2005). Among the 1009 adults surveyed by SRBI Public Affairs about how frequently they are happy, 78% reported being happy ‘most or all the time’, 16% answered ‘some of the time’, and only 5% said ‘not very often’. Respondents who were happy or gloomy were compared to those who were neutral in terms of mood (i.e. neutral was the reference category). It was hypothesized that respondents who were in a happy mood would exhibit a higher risk tolerance, holding all other factors constant, whereas those in a gloomy mood should score lower on the risk-tolerance scale. Data analysis method An ordinary least-squares regression was used to determine the relationship between mood and risk-tolerance, controlling for other variables related to risk tolerance. Multicollinearity was not an issue. All analyses were run using SPSS 15.0 for Windows. Results Descriptively, the mean risk-tolerance scores for the three mood states were as follows when uncontrolled for possible confounding variables: gloomy522.77 (SD54.34), neutral523.84 (SD50.18) and happy524.54 (SD54.68). To determine the independent contribution of mood on risk tolerance, an ordinary least-squares (OLS) multiple regression was employed. A number of regression analyses were conducted, but only the final regression model is shown in this paper. In addition to the regression model that is displayed, models were tested (not shown) that included curvilinear variables for age and income. The inclusion of these variables did not significantly improve the amount of explained variance in the model. Also, because others have found gender differences in the effect of mood on risk tolerance (i.e.

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females more subject to it), and possible asymmetry issues (e.g. Chou et al. 2007), four terms were created to account for possible interactions between gender and mood and age and mood. The regression models that included these interaction terms were over-specified. As such, the final regression model excludes these interactions. Results from the final multiple regression analysis, reported in Table 3, indicate that 10 of the control variables shown in Table 2 were significantly associated with financial risk tolerance. Age was negatively associated with risk tolerance, which means that younger respondents, on average, were willing to take more financial risk than older respondents. Males were found to be more risk tolerant than females. Household income and reported net worth were both positively related to risk tolerance. Respondents who held an Associate’s, Bachelor’s or Graduate degree were progressively more risk tolerant than the reference category (some college or less). Those who considered themselves to be more financially knowledgeable were more risk tolerant. Finally, financial satisfaction was associated with risk tolerance, but the relationship was negative (i.e. the less satisfied a respondent was with their current financial situation, the more likely they were to have a higher risk tolerance, on average). After controlling for these variables, mood was found to still be predictive of risk tolerance. The Beta (standardized regression coefficient) indicates that being in a happy mood was positively associated with having a higher level of financial risk tolerance as compared to the reference category (i.e. neutral mood) when holding the control variables constant. Conversely, participants who were in a gloomy mood Table 3. Results of regression analysis showing relationship between mood and risk tolerance, controlling for biopsychosocial and environmental variables. Variable Age Gender (15male) Race/ethnic background (15non-Hispanic White) Household income Self-reported net worth Educational status Associate’s degree Bachelor’s degree Graduate degree Homeownership (15own home) Marital status (15married) Employment status (15employed full-time) Financial knowledge Financial satisfaction Mood Happy Gloomy F55.91***, R250.18, adjusted R250.15. *p,0.05, **p,0.01, ***p,0.001.

b

SE

Beta

t

20.04 1.72 20.75

0.02 0.46 0.97

20.11 0.17 20.04

22.16* 3.78*** 20.76

0.31 0.21

0.11 0.10

0.17 0.13

1.73 2.25 3.02 0.03 20.55 0.99 0.49 20.29

0.75 0.77 0.82 0.65 0.48 0.70 0.13 0.13

0.18 0.24 0.29 0.01 20.06 0.07 0.20 20.13

2.30* 2.92** 3.68*** 0.04 21.15 1.43 3.80*** 22.22*

0.88 20.31

0.43 1.05

0.10 20.01

2.03* 20.30

2.85** 2.01*

918 J.E. Grable and M.J. Roszkowski exhibited lower risk-tolerance scores relative to those who were neutral, but this Beta was extremely small and failed to reach statistical significance. This suggests that a positive mood has greater bearing on risk tolerance than does a negative mood. The asymmetry observed in previous studies is suggested by the descriptive data but the small number of cases in a gloomy mood did not allow enough power to permit for an inferential analysis to establish that mood has an asymmetrical impact on risk taking (as suggested by Chou et al.). Discussion As shown in this and other studies, risk tolerance is related to relatively static biopsychosocial and environmental variables that served as the control variables in the present study. Younger respondents, males, and those with higher incomes and net worth were more risk tolerant than others. Financial knowledge and education were positively related to risk tolerance, whereas financial satisfaction was negatively associated with a person’s willingness to take financial risks. Most notably, results from this study show that transient states such as mood also have a bearing on risk tolerance that is as strong as some of the environmental and biopsychosocial variables. This too has been demonstrated in previous studies, but much of the previous literature devoted to assessing the relationship between mood and financial risk taking reported findings based on data collected at either the macro level (e.g. Hirshleifer and Shumway 2003) or via psychological experiments (e.g. Wright and Bower 1992) that looked at induced moods rather than naturally occurring ones. Little has been known about how mood affects the scores on a risk tolerance questionnaire of the type commonly used by financial advisors. Results from this study document that a client’s mood has a bearing on the score he or she obtains on such a measure. Test-takers who classified themselves as happy scored significantly higher relative to persons in a neutral state, even when holding all other known relevant factors constant. From an academic point of view, findings showing an association between mood and risk tolerance suggest that the newer models of risk taking, such as the risk-asfeelings framework, might offer insights into decision making involving risk that are not currently addressed using traditional economic utility theories. The results add further credence to the risk-as-feelings hypothesis since it appears that individual assessments of risk are influenced by the affect attribute of mood. More specifically, these findings are consistent with the preponderance of research published to date (Schwarz 2000) in offering support for the AIM of risk taking rather than the MMH. Currently, economic utility theory and financial decision-making models based on the theory (e.g. MPT and CAPM) do not account for inputs that might be considered ‘irrational’. Changing one’s taste for risk on different occasions because of mood is quite ‘irrational’ in an economic sense. However, this does appear to be the case, and current economic utility theory does not account for this anomaly, whereas the AIM and the risk-as-feelings hypothesis do. The data support Jackson’s (2006) contention that ‘cognitive and affective appraisals may interact; feelings about a risk object may infuse more formal and numeric appraisals’ (258). As Loewenstein et al. (2001) suggest, cognitive and experiential processes can operate side-by-side.

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These findings also have practical policy implications. Economic utility theory posits that individuals act rationally when making risk and return tradeoffs, and as such their performance on a risk-tolerance questionnaire should not be subject to the vagaries of mood. Results of this study indicate that this assumption may only partially be true. The data indicate that risk tolerance is a function, in part, of a person’s current affective state. Although longitudinal changes were not explicitly examined in this study, the data suggest that rather than assuming that risk tolerance is static and fixed, and then using this test score as a factor in the determination of the appropriate risk and return tradeoff, it may be more astute to assume that risk tolerance evaluations can change as a person’s mood changes. Being in a euphoric mood when taking such a test could result in an overestimation of an investor’s typical risk tolerance level. Someone who is happy when completing a risk tolerance questionnaire may be unknowingly projecting current mood into the future without realizing that because of this, the long-run level of risk tolerance is being compromised. In other words, clients in happy moods may be engaging in a projection bias, and it might be wise for such clients and their advisors to step back and reevaluate the level of risk offered by a service or product before purchase when in a more neutral state. Although the findings from this study are noteworthy, it is important to consider several caveats. First, the number of participants in a gloomy state was small, which limits the power of any analyses involving that group. That is, the small number of participants in a gloomy mood did not permit for statistical inferences on the asymmetry issue. Second, the mood measure consisted of a single item self classification, which can be questioned because single item scales tend to be less reliable than a composite based on a number of different items. Third, it cannot be ruled out that there may have been a self-section bias in the sample. For example, potential respondents who were very gloomy might have lacked the motivation to even complete the survey. Fourth, it would be useful to determine if the effects of mood identified in this study are unique to the middle US or whether these differences are geographically broader. Likewise, no international comparisons were possible. Fifth, the sample was limited in its racial and ethnic diversity. Since almost all respondents to the survey were non-Hispanic White, possible interactions between ethnicity and mood on risk tolerance could not be studied. Finally, research is needed to examine the interplay among risk tolerance and mood with other variables, such as self-esteem, investment choice and asset accumulation over time. The limitations inherent in this study provide ample opportunities for future research. Notes 1. In fact, much of the literature indicates that financial risk tolerance, which is defined as a person’s willingness to engage in ‘behaviors in which the outcomes remain uncertain with the possibility of an identifiable negative outcome’ (Irwin 1993, 11), is closely associated with financial behaviors as described by MPT and CAPM (Hariharan, Chapman, and Domian 2000; Irwin 1993; Morse 1998; Trone, Allbright, and Taylor 1996). For example, Hariharan and his associates found that ‘increased risk tolerance reduces an individual’s propensity to purchase risk-free assets’ (159). As predicted by MPT, high risk tolerance tends to be associated with the propensity to save (Cavanagh and Sharpe 2002; Chang

920 J.E. Grable and M.J. Roszkowski 1994; Chen and Finke 1996; Huston and Chang 1997), the likelihood of owning investment assets (Xiao 1996), and participating in retirement plans (Yuh and DeVaney 1996).

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China Journal of Accounting Studies, 2013 Vol. 1, No. 1, 47–61, http://dx.doi.org/10.1080/21697221.2013.781766

The risk premium of audit fee: Evidence from the 2008 financial crisis Tianshu Zhanga and Jun Huangb* a School of Accountancy, Shanghai Institute of Foreign Trade, People’s Republic of China; Institute of Accounting and Finance, Shanghai University of Finance and Economics, People’s Republic of China

b

This paper uses the 2008 financial crisis to examine the association between audit pricing and firm risk. The empirical analysis shows that when firm risk increased during the crisis, accounting firms charged more for their auditing services, supporting the risk premium of audit fee. An analysis of different industries presents a positive correlation between the audit fees and firm risk for export companies that were seriously shocked by the crisis. Further, compared with private firms, the audit fees of State-Owned Enterprises (SOEs) did not increase with firm risk under the crisis, due to the government’s bailout guarantee. Finally, the risk premium of audit fee was only found for companies audited by non-Big Four accounting firms. Keywords: audit fee; financial crisis; firm risk; risk premium

1. Introduction The recent recession in the US, beginning with the bankruptcy of Lehman Brothers in September 2008 and the subsequent collapse of the US sub-prime mortgage market, had a ripple effect around the world. At the micro level, firms became vulnerable as a result of the global credit squeeze; for example, more than 4900 firms in Guangdong province had gone bankrupt by the end of 2008 (Huang, 2009). As firm risk increases during a crisis, an interesting question is whether auditors pay attention to it and how this further relates to their provision of services. Unfortunately, the literature does not provide a satisfactory answer. In this paper, we use the 2008 financial crisis as an exogenous event to investigate the increased risk premium on audit fees. Since the pioneering research of Simunic (1980), the literature has explored various determinants of audit pricing, such as firm size, asset structure, business complexity and audit opinion (Anderson & Zeghal, 1994; DeFond, Francis, & Wong, 2000; Francis, 1984; Firth, 1985). Simunic (1980) argues that firm risk should be an important factor of audit fees, because it influences the amount of effort expended by accounting firms and the potential cost of a lawsuit. For example, auditors might implement more procedures and face a higher possibility of lawsuit for risky firms, which would incur a risk premium on audit fee (Li & Wu, 2004). However, whether audit pricing is associated with firm risk is still unclear. Studies in China and other countries draw inconsistent conclusions (Gul & Tsui, 1998; Seetharaman, Gul, & Lynn, 2002; Simunic & Stein, *Corresponding author. Email: [email protected] Paper accepted by Xi Wu. Ó 2013 Accounting Society of China

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1996; Wu, 2003; Zhang, Chen, & Wu, 2005; Zhu & Yu, 2004). A more important issue is that the endogeneity problem is ignored in the extant research, thus reducing the reliability of conclusions. The 2008 global financial crisis provides a good opportunity to examine the relationship between firm risk and audit fees. First, numerous companies have suffered in this global crisis. Owing to declining demand and tightened monetary policies, most firms’ operating risks have increased. This situation allows us to analyze how accounting firms respond to this increased risk, by examining changes in their audit fees. Second, as the crisis came as a shock, changes in audit fees should be a reaction to incremental risk, thus providing a natural experimental research setting and avoiding the endogeneity problem. Finally, because the financial crisis affected firms differently, we can deepen our analysis by comparing across industries, types of ownership, and auditors to improve our understanding of the risk premium on audit fee. Our paper makes several contributions to the literature. First, although firm risk is theoretically a predictor of audit fees, the evidence is far from conclusive. We examine the risk premium on audit pricing empirically, which increases our knowledge of the relationship between firm risk and audit fees. Second, by analyzing audit fees during the 2008 financial crisis, our analysis provides a new perspective on how the crisis influenced firm behavior and sheds light on the aftermath of this recession. Third, our research shows that state ownership lends an implicit guarantee to SOEs that influences the behavior of accounting firms, as shown by the effect of state ownership on the risk premium of audit fee. Last but not least, our paper has implications for research methodology. We employ the 2008 financial crisis as an exogenous event to investigate how audit fees change with incremental firm risk, thus eliminating the endogeneity problem in the analysis. The remainder of this paper proceeds as follows. Section 2 reviews the literature. Section 3 develops our hypotheses. Section 4 introduces the sample, data and model, and gives the summary statistics. Section 5 presents the empirical results about the risk premium of audit fee. Section 6 further analyzes how the risk premium is generated. Section 7 performs some robustness tests, and Section 8 concludes the paper. 2. Literature review Since Simunic (1980) first analyzed the determinants of audit fees, various factors relating to accounting firms’ charges have been investigated. (1) Firm size: audit fees are positively correlated with firm size (Simunic, 1980). (2) Asset structure: Simunic (1980) and Firth (1985) find that accounts receivables and inventories can explain audit pricing. (3) Business complexity: the number of subsidiaries is a determinant of audit fees (Anderson & Zeghal, 1994; Francis, 1984). (4) Audit opinion: Simunic (1980) finds that audit opinion influences auditors’ charges. (5) Accounting firm: the literature provides evidence that the characteristics of accounting firms, such as their size and reputation, are related to audit fees (Beatty, 1993; DeFond, Francis, & Wong, 2000; Francis, 1984). (6) Corporate governance: Abbott, Parker, Paters, and Raghunandan (2003) suggest that audit fees are affected by firms’ governance structure, such as the presence of an audit committee. Theoretically, firm risk should be an important factor on audit fees because it influences the amount of effort and the lawsuit cost of accounting firms. However, the empirical evidence is inconclusive. For example, Simunic (1980) and Simunic and Stein (1996) find that audit fees increase with firm risk. Gul and Tsui (1998)

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show that the risk measure of free cash flow positively correlates with audit pricing. Choi, Kim, Liu, and Simunic (2008) employ data from 15 countries and provide evidence that audit fees are higher under better legal regimes because of the enhanced possibility of lawsuit. Although the above analyses confirm that firm risk is positively related to audit fees, there are different findings. Francis (1984) argues that firm risk cannot explain audit pricing, based on an analysis of Australian companies. Employing Canadian data, Chung and Lindsay (1988) find that audit fees do not increase with firms’ operational risks. Further, the analysis of Seetharaman et al. (2002) shows that audit fees for listed companies in Britain are unrelated to risk. As for Chinese firms, the risk premium of audit fee is unclear. Zhang et al. (2005) find that auditors charge more when companies are burdened with higher loan guarantees. Using commercial banks as their sample, Liu and Zhou (2007b) document that risk measures such as customer concentration, the asset sensitivity gap, return on capital, and the capital adequacy rate are important determinants of audit fees. In contrast, the research of Liu, Sun, and Liu (2003), Wu (2003), Zhu and Yu (2004) show that firm risk cannot explain audit pricing, whether employing firm leverage or performance as the risk measure. Overall, the literature does not provide a clear picture of how audit fees are associated with firm risk. Moreover, the analyses ignore the fact that the relationship between firm risk and audit fees might be endogenous, which lowers the reliability of the related findings. We use the 2008 financial crisis as an exogenous event to examine the risk premium of audit fee after controlling for endogeneity. 3.

Hypothesis development

The audit fee paid to accounting firms is usually composed of three parts. The first is the fixed cost of carrying out the necessary audit processes and issuing an audit report. The second is the risk cost, defined as the expected loss due to audit failure, including the cost of a lawsuit and loss of reputation. The third is the accounting firm’s profit, determined by the local economy and market competition. Of the three components, the fixed cost and risk cost are related to firm risk. When companies experience high uncertainty, accounting firms should implement more auditing work to reduce the possibility of offering an incorrect audit opinion when financial reports are materially misrepresented. For example, they may implement more account receivable confirmations and inventory counts, which could increase the fixed cost of the audit. Moreover, the possibility of distress and bankruptcy is higher for risky companies, which increases the potential lawsuit cost and reputation loss to accounting firms, thus further raising the risk cost of the audit. Following the 2008 financial crisis, firms faced high risk due to low product demand and tightened bank credit. For example, sales went down, inventories were overstocked, and account receivables were difficult to collect. All these cast more doubt on firms’ futures. Furthermore, when firm performance declines under a crisis, managers have more incentives to manipulate earnings to ensure good compensation and beat analysts’ forecasts, resulting in a higher possibility of misrepresentation in financial statements. To avoid issuing an incorrect audit opinion, auditors might carry out more audit procedures and increase the scope of the audit, leading to a higher fixed cost of audit. Moreover, when companies are vulnerable to bankruptcy and the odds of accounting fraud increase after a crisis, accounting firms face a higher risk of lawsuit

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and would ask for more risk compensation.1 Based on this analysis, we propose our first hypothesis. Hypothesis 1: The audit fee increases with firm risk under the financial crisis.

Although the impact of the 2008 financial crisis was undoubtedly widespread, and a great many companies suffered as a result, the effects differed across industries. The crisis began with the collapse of the US sub-prime mortgage market and immediately spread to other countries. The downturn in the economic prospects of Western countries, with rising unemployment rates and decreasing consumption expenditure, led to a decline in product demand from emerging markets.2 The crisis seriously shocked export firms, and they are more likely to manipulate earnings. The possibility of bankruptcy for export firms is also higher, which increases the lawsuit risk. To avoid audit failure, accounting firms may need to implement additional audit procedures and require more risk compensation, resulting in higher audit fees. This leads to our second hypothesis. Hypothesis 2: The risk premium of audit fee is more significant for export firms than for non-export firms under the financial crisis.

One notable feature of China’s stock market is that SOEs account for a large proportion of listed companies (Liu, Sun, & Liu 2003). The level of state ownership was still as high as 50.2% at the end of 2006. Kornai (1988, 1993) argues that state ownership provides an implicit assurance to SOEs. Once SOEs fall into distress, the government is more likely to bail them out to avoid the prospect of much unemployment and society instability. The analysis of Faccio, McConnell, and Masulis (2006) provides supporting evidence for this argument. As government bailout reduces the possibility of a subsequent lawsuit, accounting firms should require less risk compensation. SOEs, therefore, should incur lower risk premiums than private firms. This leads to our third hypothesis. Hypothesis 3: The risk premium of audit fee is more significant for private firms than for SOEs under the financial crisis.

Finally, we discuss how auditors influence the risk premium on audit fee. The Big Four accountancy firms tend to produce higher-quality audits. DeAngelo (1981) and Dye (1993) state that to maintain their reputation, the Big Four have better control of the audit process and their audit quality is considered to be better. When firm risk increased under the financial crisis, the Big Four may have implemented stricter audit procedures and required more risk compensation, generating a significant risk premium. However, whether the Big Four have a higher quality audit in an emerging market such as China remains in question. First, Liu and Xu (2002) point out that compared to domestic accounting firms, the Big Four face a lower risk of lawsuit because of political privilege and public relationship building.3 Second, Chinese listed companies lack the demand for high-quality audit due to state ownership concentration, IPO market regulation, and weak protection of property rights (DeFond, Wong, & Li, 2000). Thus, the Big Four may have little incentive to provide good quality audits in China. Finally, recent empirical evidence shows that the audit quality of the Big Four is no better than that of non-Big Four and sometimes even worse (Guo, 2011; Liu & Zhou, 2007a). Therefore, we do not make a prediction about how a Big Four audit influences the risk

China Journal of Accounting Studies

51

premium of audit fee under the crisis, and empirically test this question in the later analyses. 4.

Research design

4.1. Sample The 2008 financial crisis began with the bankruptcy of Lehman Brothers in September 2008, and quickly spread to other countries. China’s economy also experienced a slowdown in 2008 because of the crisis. To promote economic growth, the Chinese government implemented an RMB4 trillion economic stimulus program and the economy started to recover in the second half of 2009. Thus, we choose all listed companies in 2008 as our sample. We also include 2007 data as a comparison to figure out the shock of the financial crisis. For consistency, we do not include data before 2007 because the financial reports of listed companies have changed considerably since China’s new accounting standards were issued in 2006 (Zhu, Zhao, & Sun, 2009). Finally, we exclude observations within two years of firms’ IPOs because audit fees are usually higher around the time of IPOs. 4.2.

Data

The audit fee data are taken from the CCER China Security Market Database. The financial data on listed companies are taken from the China Stock Market and Accounting Research Database (CSMAR). The WIND Database provides the information on firms’ ultimate owners. Finally, the export data come from the China Industrial Companies Database compiled by the Chinese National Bureau of Statistics. 4.3.

Model

We employ the following model to investigate the risk premium of audit fee under the crisis: Fee log ¼ a0 þ a1 ROA þ a2 Crisis þ a3 ROA PCrisis þ a4 Size P þ a5 Lev þ a6 Liquidity þa7 Diversify þ a8 Big4 þ a9 Age þ Industry þ Region þ e;

ð1Þ

where Fee_log is the natural logarithm of audit fee, adjusted by the annual inflation rate. Following Simunic (1980) and Francis (1984), we employ firm performance as our risk measure, defined as the ratio of net income to total assets (ROA). Crisis is a dummy variable, equal to one if the fiscal year is 2008 and zero otherwise. The control variables are as follows: Size is the natural logarithm of total assets; Lev denotes the ratio of debts to total assets; Liquidity is current assets divided by current liabilities; Diversify stands for the segment number; Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise; Age refers to the number of years since a firm was listed; and Industry and Region are the industry and region dummy variables, respectively. The interaction item ROACrisis examines how firm risk relates to audit fees under the crisis. If the coefficient α3 is significantly negative, it means that accounting firms charge more to firms with higher risk, supporting the risk premium of audit fee.

52 Table 1.

Zhang and Huang Summary statistics.

Variable

Obs.

Mean

Median

Std.

Min.

Max.

Fee ROA Crisis Size Lev Liquidity Diversify Big4 Age

2173 2173 2173 2173 2173 2173 2173 2173 2173

72.71 0.0320 0.5513 21.46 0.2322 1.457 2.341 0.0529 10.31

50.00 0.0334 1.00 21.41 0.2107 1.137 2.00 0.00 11.00

161.3 0.1232 0.4975 1.278 0.2108 1.348 1.334 0.2239 3.653

10.00 –0.9548 0.00 10.84 0.00 0.0386 1.00 0.00 3.00

5821 0.6256 1.00 27.35 1.821 12.44 9.00 1.00 19.00

Fee represents audit fees, in units of RMB10,000. ROA is defined as the ratio of net income to total assets. Crisis is a dummy variable, equal to one if the fiscal year is 2008 and zero otherwise. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed.

Table 2.

ROA Crisis Size Lev Liquidity Diversify Big4 Age

Pearson correlations. Fee

ROA

Crisis

Size

Lev

Liquidity

Diversify

Big4

–0.0012 (0.956) 0.0331 (0.123) 0.3579 (0.000) 0.0147 (0.493) –0.0699 (0.001) 0.0579 (0.007) 0.4361 (0.000) 0.0063 (0.770)

–0.0962 (0.000) 0.0553 (0.010) –0.2383 (0.000) 0.1415 (0.000) –0.0081 (0.707) 0.0206 (0.338) –0.0731 (0.001)

0.0225 (0.294) –0.0206 (0.337) 0.0121 (0.572) –0.0145 (0.498) 0.0149 (0.488) 0.0673 (0.002)

–0.0312 (0.156) –0.0833 (0.000) 0.1147 (0.000) 0.3496 (0.000) –0.0173 (0.422)

–0.3774 (0.000) 0.0055 (0.799) 0.0060 (0.781) 0.0727 (0.001)

–0.0687 (0.001) –0.0690 (0.001) –0.0847 (0.000)

0.0048 (0.824) 0.1789 (0.000)

–0.0045 (0.835)

Fee represents audit fees, in units of RMB10,000. ROA is defined as the ratio of net income to total assets. Crisis is a dummy variable, equal to one if the fiscal year is 2008 and zero otherwise. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed. P values are in parentheses.

4.4. Statistics Table 1 reports the descriptive statistics. The mean audit fee is RMB727,000, and the median is RMB500,000; however, there is significant variance among the audit fees of sample firms. For example, the minimum is RMB100,000 and the maximum is close to RMB60 million. The Crisis statistic shows that the mean value is 0.5513, indicating that crisis sample firms are closely matched to non-crisis sample firms.4 The table shows that the average ROA is 0.0320, and debts on average account for 23.33% of total assets. Further, the mean ratio of current assets to current liabilities is 1.457 and firms on average operate 2.341 segments. Interestingly, only 5.29% of firms employ Big Four auditors. Finally, the average number of years since listing is 10.31.

53

China Journal of Accounting Studies Table 3.

Results for the risk premium on audit fee. 2008 As the Crisis Period

Variables ROA Crisis ROACrisis Size Lev Liquidity Diversify Big4 Age Constant Industry Region Obs. Adj-R2

2009 As the Crisis Period

Coefficient (1)

P Value (2)

Coefficient (3)

P Value (4)

0.2991 0.0501⁄⁄⁄ –0.3422⁄⁄ 0.2395⁄⁄⁄ 0.1941⁄⁄⁄ –0.0282⁄⁄⁄ 0.0237⁄⁄⁄ 1.0289⁄⁄⁄ 0.0162⁄⁄⁄ 7.6829⁄⁄⁄

(0.556) (0.004) (0.015) (0.000) (0.000) (0.000) (0.001) (0.000) (0.000) (0.000)

0.2406 0.1402⁄⁄⁄ –0.5020⁄⁄ 0.1473⁄⁄⁄ 0.0836 –0.0297⁄⁄⁄ 0.0143 0.5563⁄⁄⁄ –0.0005 9.9713⁄⁄⁄

(0.191) (0.000) (0.037) (0.000) (0.280) (0.005) (0.187) (0.000) (0.902) (0.000)

Yes Yes 2173 0.62

Yes Yes 2299 0.19

The dependent variable is the natural logarithm of audit fee. ROA is defined as the ratio of net income to total assets. Crisis is a dummy variable, equal to one if it is the crisis period and zero otherwise. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed. Industry represents the industry dummy variables. Region represents the region dummy variables. We winsorize the variables ROA, Lev, and Liquidity at the top and bottom 0.5% to mitigate the effect of outliers. One, two, or three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

Table 2 reports the Pearson correlation coefficients among the variables. It shows that audit fees are higher for large and more complex firms. The ratio of current assets to current liabilities is negatively correlated with audit fees. Finally, the Big Four charge more for their auditing services. 5. Empirical analysis 5.1. The audit fee risk premium The first two columns in Table 3 report the regression results for model (1). We find that the coefficient of Crisis is 0.0501, significant at the 1% level, suggesting that audit fees increased after the financial crisis. However, as Crisis is a dummy variable, it might represent factors other than risk, so we employ the interaction item ROACrisis to examine the association between audit fees and incremental risk at the time of the crisis. We find that the coefficient of ROACrisis is –0.3422, significant at the 5% level.5 The result indicates that when firms face higher risk under the crisis, accounting firms charge more due to rising fixed and risk costs; that is, audit fees incur a risk premium. The regression also shows that audit fee correlates positively with firm size (Size), leverage (Lev), segment number (Diversify) and age (Age), but negatively with current assets (Liquidity). The Big Four charge higher fees than the non-Big Four. 5.2.

The difference between export and non-export firms

To compare the risk premium of audit fee between export and non-export firms, we calculate the ratio of export output to total output for each industry using the China

54 Table 4.

Zhang and Huang Results of the sub-group regression.

Variables ROA Crisis ROACrisis Size Lev Liquidity Diversify Big4 Age Constant Industry Region Obs. Adj-R2 F Test for interaction

Export firms (1)

Non-export firms (2)

SOEs (3)

Private firms (4)

Big Four (5)

Non-Big Four (6)

0.2661 (0.238) 0.0339 (0.255) – 0.7379⁄⁄⁄ (0.004) 0.2362⁄⁄⁄ (0.000) 0.1114 (0.164) –0.0195

0.0421 (0.840) 0.0342 (0.266) –0.0092

–0.0171 (0.928) 0.0453⁄⁄ (0.046) –0.1421

0.2500 (0.748) 0.0310 (0.245) –0.3419⁄⁄

–0.8256 (0.614) –0.0818 (0.589) –0.8070

0.2753 (0.568) 0.0446⁄⁄⁄ (0.009) –0.3042⁄⁄

(0.670) 0.4641⁄⁄⁄ (0.000) 0.7726 (0.188) 0.0971

(0.024) 0.2284⁄⁄⁄ (0.000) 0.1652⁄⁄⁄ (0.000) –0.0247⁄⁄⁄

(0.481) 0.0567 (0.210)

(0.000) 0.0244⁄⁄⁄ (0.000)

(0.968) 0.2714⁄⁄⁄ (0.000) 0.1743⁄ (0.066) –0.0195

(0.154) (0.136) 0.0349⁄⁄⁄ 0.0315⁄⁄ (0.006) (0.019) 1.0555⁄⁄⁄ 0.8928⁄⁄⁄ (0.000) (0.000) 0.0318⁄⁄⁄ 0.0172⁄⁄⁄ (0.000) (0.000) 7.0650⁄⁄⁄ 7.5024⁄⁄⁄ (0.000) (0.000) Yes Yes Yes Yes 638 698 0.61 0.62 Z Value = –2.02 (0.043)

(0.542) (0.042) 0.2813⁄⁄⁄ 0.1948⁄⁄⁄ (0.000) (0.000) 0.0422 0.1198⁄⁄ (0.605) (0.036) – – 0.0396⁄⁄⁄ 0.0305⁄⁄⁄ (0.000) (0.003) 0.0209⁄⁄ 0.0257⁄⁄ (0.020) (0.019) 0.9579⁄⁄⁄ 1.1080⁄⁄⁄ (0.000) (0.000) 0.0134⁄⁄⁄ 0.0273⁄⁄⁄ (0.000) (0.000) 6.8087⁄⁄⁄ 8.6402⁄⁄⁄ (0.000) (0.000) Yes Yes Yes Yes 2380 793 0.66 0.55 Z Value =1.70 (0.089)

0.0047 0.0162⁄⁄⁄ (0.788) (0.000) 3.0146⁄⁄ 7.9174⁄⁄⁄ (0.030) (0.000) Yes Yes Yes Yes 115 2058 0.54 0.44 Z Value =–0.27 (0.787)

The dependent variable is the natural logarithm of audit fee. ROA is defined as the ratio of net income to total assets. Crisis is a dummy variable, equal to one if the fiscal year is 2008 and zero otherwise. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed. Industry represents the industry dummy variables. Region represents the region dummy variables. We winsorize the variables ROA, Lev and Liquidity at the top and bottom 0.5% to mitigate the effect of outliers. P values are in parenthesis. One, two, or three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

Industrial Companies Database, and employ its median to divide the sample firms into two groups, export and non-export companies. We then run a regression for each group and the results are presented in columns (1) and (2) of Table 4. The number of observations is reduced because the China Industrial Companies Database only covers manufacturing firms, thus only industrial listed companies are included in the regression.6 We find that the coefficient of ROACrisis is significantly negative for export firms, but insignificant for non-export firms. We conduct an F test to compare the two interaction coefficients, and it is significant at the 5% level. The above finding suggests that the risk premium of audit fee is highly significant for export firms when they are seriously affected by the financial crisis, but is insignificant for non-export firms because they are less influenced during the crisis. The results for the control variables are similar to the previous regression.

China Journal of Accounting Studies 5.3.

55

The difference between SOEs and private firms

We further investigate how corporate ownership influences the risk premium of audit fee. Specifically, we divide the sample by SOEs and private firms and run sub-group regressions. Columns (3) and (4) in Table 4 display the results. The coefficient of Crisis is significantly positive in the SOE regression, indicating that the audit fee for SOEs increases after the crisis. However, as the variable Crisis does not represent firm risk precisely, we cannot be sure whether the increase in SOEs’ audit fees is due to higher risk under the crisis or to other factors. For example, the managers of SOEs may have a low incentive to bargain with accounting firms and thus SOEs’ audit fees may increase gradually. Thus, we examine the interaction item ROACrisis and find that it generates an insignificant coefficient in column (3), but a significantly negative coefficient in column (4). Further, the F test of the two interaction items is significant at the 10% level. The result indicates that state ownership provides an implicit bailout guarantee to SOEs, thus mitigating the risk premium on SOEs’ audit fees during the crisis. However, as the possibility of bailout is lower for private firms, accounting firms charge more when private firms’ risk increases under the crisis. The control variables generate qualitatively similar results. 5.4. The difference between Big Four and non-Big Four audited firms Finally, we examine the effect of auditors on the relationship between firm risk and audit fees. The last two columns in Table 4 present the sub-sample regression results for Big Four versus non-Big Four. In column (5), the coefficient of the interaction item ROACrisis is insignificant for firms audited by Big Four, but it generates a significantly negative coefficient for non-Big Four firms in column (6). An F test to compare the two interaction items is insignificant, which shows that the risk premium on audit fee is concentrated in firms audited by non-Big Four. The results for the control variables are unchanged. 6. Further analysis The above analysis shows that audit fees increase with firm risk under the crisis and it provides evidence for the risk premium of audit fee. A further interesting question is how the risk premium is generated. Our argument suggests that when firms face higher uncertainty, auditors might implement more audit procedures and request higher lawsuit compensation, leading to higher audit fees. Next, we examine how firm risk is associated with auditors’ time and attention to lawsuit risk. 6.1.

Audit time

We construct the following model to investigate whether auditors input additional effort when firm risk increases: Time ¼ b0 þ b1 ROA þ b2 Size þ b3 Lev þ b4 Liquidity þ b5 Diversify þ b6 Age X þ Industry þ e

ð2Þ

Time denotes audit time. As listed companies do not publicly disclose information about audit time, we use the period between the fiscal year end and the auditor’s report

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Zhang and Huang

Table 5.

Results of the audit time and opinion. Audit Time

Variables ROA Size Lev Liquidity Diversify Age Constant Industry Obs. Adj-R2

Audit Opinion

Coefficient (1)

P Value (2)

Coefficient (3)

P Value (4)

–21.1720⁄⁄⁄ –0.2043 2.9782 –0.1223 –0.1533 0.8607⁄⁄⁄ 93.9277⁄⁄⁄

(0.000) (0.683) (0.416) (0.810) (0.754) (0.000) (0.000)

–0.9034⁄⁄ –0.5784⁄⁄⁄ 2.0586⁄⁄⁄ –0.1647⁄⁄ 0.0093 0.0641⁄⁄⁄ 9.7444⁄⁄⁄

(0.024) (0.000) (0.000) (0.048) (0.880) (0.003) (0.000)

Yes 1172 0.08

Yes 1088 0.37

The dependent variable for the audit time regression is Time, the period between the fiscal year end and the auditor’s report date. The dependent variable for the audit opinion regression is Opinion, equal to one if a firm receives a modified audit report, and zero otherwise. ROA is defined as the ratio of net income to total assets. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Age is the number of years a firm has been listed. Industry represents the industry dummy variables. We winsorize the variables ROA, Lev and Liquidity at the top and bottom 0.5% to mitigate the effect of outliers. One, two, or three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

date as a proxy. The other variables are defined as before. The first two columns in Table 5 report the regression result for model (2), based on the 2008 data. We find that the coefficient of our risk measure, ROA, is –21.1720 and significant at the 1% level. The result indicates that accounting firms widen their audit scope and carry out additional processes for risky firms, which prolongs the auditing period. The regression also shows that accounting firms spend longer when auditing older firms. 6.2. Attention to lawsuit risk High risk may increase auditors’ concerns over potential lawsuits, thus generating higher audit fees. We employ the following logistic model to analyze this: Opinion ¼ c0 þ c1 ROA þ c2 Size þ c3 Lev þ c4 Liquidity þ c5 Diversify þ c6 Age X þ Industry þ e

ð3Þ

Opinion is the audit opinion variable, which equals one if a firm receives a modified audit report, and zero otherwise. The definitions of the other variables are the same as before. We expect that auditors are more likely to issue modified opinions to risky firms if high uncertainty increases auditors’ concerns over potential lawsuits. Columns (3) and (4) in Table 5 present the regression results for model (3) using the 2008 data. The coefficient of ROA is –0.9034 and significant at the 5% level, suggesting that risky firms are more likely to receive a modified audit opinion. The result confirms that high firm risk increases auditors’ concern to a potential lawsuit, resulting in a higher possibility of issuing a modified opinion. The results for the other variables show that modified audit opinion negatively correlates with firm size and current assets, but positively correlates with firm leverage and age. Finally, the sample size is slightly reduced because the probit regression deletes any observations with perfect prediction.

China Journal of Accounting Studies 7.

57

Robust analysis

7.1. The lagged effect of the crisis As there may be a lag between the happening of the financial crisis and its effects, we also employ 2009 as the crisis period. Specifically, we use 2007 and 2009 listed company data, and re-run the model (1) regression. The result is presented in the last two columns of Table 3. We find that the interaction item ROACrisis still generates a significantly negative coefficient. The result provides further evidence for the risk premium of audit fee, even when controlling for the lagged effect of the financial crisis. 7.2.

The increase in audit fees

We adjust audit fees by the annual inflation rate in the analysis, because audit fees might increase with time. To further rule out this time effect, we use the listed company data from 2004 to 2007 and re-run the model (1) regression for each successive two-year Table 6.

Results of the ‘other years’ analysis.

Variables ROA Nextyear ROANextyear Size Lev Liquidity Diversify Big4 Age Constant Industry Region Obs. Adj-R2

2006 and 2007 (1)

2005 and 2006 (2)

2004 and 2005 (3)

0.0154 (0.856) –0.0392⁄⁄ (0.033) 0.1267 (0.402) 0.2941⁄⁄⁄ (0.000) 0.1394⁄⁄⁄ (0.000) –0.0124⁄⁄⁄ (0.001) 0.0148⁄⁄ (0.040) 0.9218⁄⁄⁄ (0.000) 0.0156⁄⁄⁄ (0.000) 6.7153⁄⁄⁄ (0.000) Yes Yes 2100 0.62

–0.1781⁄⁄ (0.045) 0.0593⁄⁄⁄ (0.001) –0.0129 (0.907) 0.3260⁄⁄⁄ (0.000) 0.0835⁄⁄⁄ (0.000) –0.0117⁄⁄⁄ (0.002) 0.0056 (0.436) 0.7349⁄⁄⁄ (0.000) 0.0155⁄⁄⁄ (0.000) 6.0371⁄⁄⁄ (0.000) Yes Yes 2212 0.59

–0.2086⁄⁄ (0.038) 0.0365⁄⁄ (0.038) 0.0741 (0.529) 0.3165⁄⁄⁄ (0.000) 0.0796⁄⁄⁄ (0.000) –0.0120⁄⁄⁄ (0.001) 0.0065 (0.353) 0.5644⁄⁄⁄ (0.000) 0.0076⁄⁄ (0.030) 6.2358⁄⁄⁄ (0.000) Yes Yes 2183 0.53

This table presents the regression results for the successive two-year comparison between audit fees from 2004 to 2007. The dependent variable is the natural logarithm of audit fee. ROA is defined as the ratio of net income to total assets. Nextyear is a dummy variable, denoting the following year for each two-year period. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed. Industry represents the industry dummy variables. Region represents the region dummy variables. We winsorize the variables ROA, Lev and Liquidity at the top and bottom 0.5% to mitigate the effect of outliers. The change in the number of observations is due to differences in the number of listed companies and changes in companies’ disclosure of audit fees across years. P values are in parenthesis. One, two, or three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

58

Zhang and Huang

period. The regression result is reported in Table 6. Here, the variable Nextyear denotes the following year; for example, Nextyear equals one for 2005 when running the regression of the year 2004 and 2005. The coefficient of the interaction item ROANextyear is insignificant in all of the regressions, suggesting that our conclusion is free from the time-series increase in audit fees. 7.3.

Endogeneity

We use the 2008 financial crisis as an exogenous event to mitigate the endogeneity problem and analyze how audit fees relate to incremental firm risk when firms face a shock. To further resolve endogeneity in the analysis, we employ a Heckman two-stage regression (1979). First, we run a probit model on firm risk as follows: Risk ¼ d0 þ d1 Size þ d2 Lev þ d3 Diversify þ d4 Age þ

X

Industry þ

X

Year þ e ð4Þ

where Risk is an indicator variable, equal to one if the firm’s ROA is in the bottom quartile of sample firms in the same industry and year, and zero otherwise. The regression adds a year dummy to control for year effects. The definition of other variables is the same as before. Table 7.

Results of the Heckman regression. First-stage Coefficient (1)

ROA Crisis ROACrisis λ Size Lev Liquidity Diversify Big4 Age Constant Industry Year Region Obs. Adj-R2

Second-stage P Value (2)

–0.2128⁄⁄⁄ 1.1260⁄⁄⁄

(0.000) (0.000)

0.0545⁄⁄

(0.025)

0.0214⁄⁄ 3.1414⁄⁄⁄

(0.018) (0.000) Yes Yes No 2160 0.06

Coefficient (3)

P Value (4)

0.1645 0.0554⁄⁄⁄ –0.2602⁄ 1.5480⁄⁄⁄ 0.0076 1.2700⁄⁄⁄ –0.0320⁄⁄⁄ 0.0883⁄⁄⁄ 0.9943⁄⁄⁄ 0.0406⁄⁄⁄ 9.8736⁄⁄⁄

(0.145) (0.002) (0.070) (0.000) (0.821) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Yes No Yes 2160 0.62

The first-stage result is reported in columns (1) and (2). The dependent variable is the dummy variable Risk, equal to one if the firm’s ROA is in the bottom quartile of sample firms in the same industry and year, and zero otherwise. Columns (3) and (4) report the second-stage result. The dependent variable is the natural logarithm of audit fee. ROA is defined as the ratio of net income to total assets. Crisis is a dummy variable, equal to one if the fiscal year is 2008 and zero otherwise. λ is Mill’s Ratio, calculated in the first-stage regression. Size is the natural logarithm of total assets. Lev is the ratio of debts to total assets. Liquidity equals current assets divided by current liabilities. Diversify is the segment number. Big4 is a dummy variable, equal to one if a firm employs a Big Four auditor and zero otherwise. Age is the number of years a firm has been listed. Industry represents the industry dummy variables. Year represents the year dummy variables. Region represents the region dummy variables. We winsorize the variables ROA, Lev and Liquidity at the top and bottom 0.5% to mitigate the effect of outliers. The reduction in the sample size is because the probit regression deletes any observations with perfect prediction. One, two, or three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

China Journal of Accounting Studies

59

The regression results for model (4) are presented in columns (1) and (2) of Table 7. We find that the coefficient of Size is significantly negative, suggesting that large firms face lower uncertainty. Firm risk is higher for highly leveraged firms, and the coefficient of Diversify shows that firm risk also increases with operational complexity. Finally, older firms have higher risk. Based on the first-stage regression, we calculate a Mill’s Ratio (λ) and employ it in the second-stage regression. The result is reported in columns (3) and (4) of Table 7. The interaction item ROACrisis still generates a significantly negative coefficient. The regression provides further evidence that audit fees increase with firm risk under the crisis and there is a risk premium on audit fees. 7.4. Further analysis of auditor size The previous analysis shows that the risk premium of audit fee is insignificant for Big Four audited firms. However, this result might be caused by only a few observations in the Big Four regression because the Big Four have a low market share among Chinese listed companies. To rule out this alternative explanation, we classify our sample into ‘Big Ten’ and non-‘Big Ten’ audited firms according to accounting firms’ revenue, and re-run the regression. We find that the coefficient of the interaction item ROACrisis is insignificant for Big Ten audited firms, but attracts a significantly negative coefficient for non-Big Ten audited firms. We do not report the results to conserve space, but they are available upon request. The result further verifies that the risk premium on audit fee is concentrated in firms audited by small auditors. 7.5.

Alternative risk measure

As a robustness test we employ another performance variable, return on sales (ROS), as the measure of risk. The regression shows that the coefficient of the interaction item (ROSCrisis) is still significantly negative. Again, in the interest of space, we do not present the results, but they are available from the authors upon request. The analysis further supports our conclusion. That is, accounting firms charge higher audit fees to firms with higher risk under the crisis. 8.

Conclusion

We explore the relationship between firm risk and audit fees using the 2008 financial crisis as an exogenous event. We find that audit fees increase with firm risk under the crisis, suggesting that audit fees incur a risk premium. Further analysis reveals that the risk premium on audit fee is particularly high for export firms, which were seriously shocked by the crisis. The comparison shows that accounting firms do not charge more to SOEs with higher risk, due to the government’s implicit bailout guarantee, but the audit fees of private firms significantly increase with risk under the crisis. Finally, the risk premium of audit fee is only found for firms audited by smaller, non-Big Four auditors. By investigating firms’ audit fees under the crisis, this research improves our understanding of how firm risk is associated with audit pricing by controlling the endogeneity issue. Our analysis clarifies the controversy over the risk premium of audit fee in the extant literature. Further, our research shows that the 2008 financial crisis had a notable

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effect on firm auditing. For example, accounting firms expended more effort and were more concerned about potential lawsuits following the crisis. Finally, an implication of our research is that although most companies carried out cost-cutting projects under the crisis, accounting firms should respond by implementing stricter procedures and increasing the scope of their audits to avoid audit failure. Acknowledgements The authors acknowledge the financial support of the National Natural Science Foundation of China (71102136, 71272008, 71202044), the Key Research Institute Research Project of the Ministry of Education (11JJD790008), the Young Scholar Research Project of the Ministry of Education (11YJC790284, 10YJC790094), the Innovation Program of the Shanghai Municipal Education Commission and the 085 Project for Shanghai Institute of Foreign Trade. We thank anonymous referees and the editors for their insightful comments.

Notes 1.

2. 3.

4. 5. 6.

The risk of lawsuit is indeed low in China compared with other developed countries, but Chen, Li, Rui, and Xia (2009) point out that the litigation right of investors is gradually being recognized and protected as China’s legal system develops, which increases the risk of lawsuit for accounting firms. In November 2008, China’s export market experienced its first negative growth since entering the WTO. For example, the domestic accounting firms that are partners of the Big Four all had a government background when the Big Four entered the China audit market in the 1990s. The Chinese Institute of Certified Public Accountants (CICPA) has not undertaken an annual inspection of the Big Four for a long time. The mean of Crisis does not equal 0.5 because some companies do not disclose their audit fees and we require that firm observations should be at least two years after their IPOs. Because the analysis is based on two-year firm data, we admit that the test statistics could be overstated due to residual correlations, thus the significance levels should be interpreted with caution. We compare the deleted and undeleted firms on variables such as ROA, Size, Lev and Liquidity. The results show that the two groups of firms are quite similar, except in firm size: manufacturing firms are usually larger than non-manufacturing firms.

References Abbott, L., Parker, S., Paters, G., & Raghunandan, K. (2003). The association between audit committee characteristics and audit fees. Auditing: A Journal of Practice and Theory, 22, 17–32. Anderson, T., & Zeghal, D. (1994). The pricing of audit services: Further evidence from the Canadian market. Accounting and Business Research, 24, 195–207. Beatty, R. (1993). The economic determinants of auditor compensation in the initial public offerings market. Journal of Accounting Research, 31, 294–302. Chen, X., Li, M., Rui, M., & Xia, L. (2009). Judiciary independence and the enforcement of investor protection laws: Market responses to the ‘1/15’ notice of the supreme people’s court of China. China Economic Quarterly, 9, 1–28. Choi, J., Kim, J., Liu, X., & Simunic, D. A. (2008). Audit pricing, legal liability regimes, and Big 4 premiums: Theory and cross-country evidence. Contemporary Accounting Research, 25, 55–99. Chung, D., & Lindsay, W. D. (1988). The pricing of audit services: The Canadian perspective. Contemporary Accounting Research, 5, 19–46. DeAngelo, L. E. (1981). Auditor size and audit quality. Journal of Accounting and Economics, 3, 183–199. DeFond, M. L., Francis, F. R., & Wong, T. J. (2000). Auditor industry specialization and market segmentation: Evidence from Hong Kong. Auditing: A Journal of Practice and Theory, 19, 49–66.

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Journal of Risk Research Vol. 10, No. 1, 49–66, January 2007

ARTICLE

The UK’s Prudential Borrowing Framework: A Retrograde Step in Managing Risk? JOHN HOOD*, DARINKA ASENOVA*, STEPHEN BAILEY** & MELINA MANOCHIN{ *Decision Analysis and Risk and Centre for Risk and Governance, Glasgow Caledonian University, Glasgow, UK, **Globalisation and Public Policy, Glasgow Caledonian University, Glasgow, UK, {Finance, Accounting and Law Group, Aston Business School, Aston University, Birmingham, UK

ABSTRACT The contemporary understanding of public sector risk management entails a broadening of the traditional bureaucratic approach to risk beyond the boundaries of purely financial risks. However, evidence suggests that in reality public sector risk management does not always match the rhetoric. This paper focuses on the apparent inadequacy of any risk framework in the current Prudential Borrowing Framework (PBF) guidance in relation to that which was developed under Public Private Partnerships and Private Finance Initiative (PFI). Our analysis shows that the PBF and its associated indicators for local authorities adopt a narrow financial approach and fail to account for the full range of potential risks associated with capital projects. The PBF does not provide a framework for local authorities to consider long-term risk and fails to encourage understanding of the generic nature of risk. The introduction of the PBF appears to represent a retrograde step from PPP/PFI as regards risk and risk management. KEY WORDS: Prudential borrowing framework, prudential code, risk management, capital finance, public private partnerships

Introduction This paper focuses on the treatment of risk in the recently introduced Prudential Borrowing Framework (PBF) for UK local government. The PBF, Correspondence Address: Dr John Hood, Lecturer, Decision Analysis and Risk, Glasgow Caledonian University, Glasgow, G4 0BA, UK. Tel.: 0141 331 3154; Email: J.Hood@ gcal.ac.uk 1366-9877 Print/1466-4461 Online/07/010049–18 # 2007 Taylor & Francis DOI: 10.1080/13669870601054845

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and its associated Prudential Code, was introduced by the Local Government Act 2003 and the Local Government in Scotland Act 2003 for implementation on 1st April 2004. There is a parallel in the National Health Service (NHS) with the introduction of the Prudential Borrowing Code (PBC) for NHS Foundation Trust hospitals (Monitor, 2005). The focus of this paper, however, is on the PBF in local authorities. The paper considers PBF against the background of the current stage of development of risk management in public sector organisations. Theories of what constitutes risk in public sector organisations and the subsequent development of strategies and tactics to manage these risks have been widely discussed in recent years (see, for example, Dowlen, 1995; Vincent, 1996; NAO, 2000; Audit Commission, 2001; Hood and Allison, 2001; Hood, 2002; Hood and McGarvey, 2002; CIPFA, 2003b; Fone and Young, 2005). During this period, central government has published a number of guidance documents which focus on risk and risk management (HM Treasury, 2001, 2004; PFP, 1995). One key document is the HM Treasury Management of Risk: Principles and Concepts generally referred to as the ‘Orange Book’. This document defines risk as ‘‘the uncertainty of outcome, whether positive opportunity or negative threat, of actions and events’’ (HM Treasury Orange Book, 2004, p. 9), thus emphasising the possibility for both speculative gains and/or losses. This emphasis appears to bring public sector risk and its management out of the purely financial domain and into the conceptual framework of risk long understood by the private sector i.e. the holistic business environment. Despite, however, the increased attention to risk, the high profile of risk management and the increased volume of official standards and guidelines, this paper shows that its practical implementation remains problematic. Although financially measuring all private sector risks is not always straightforward, the primary focus, and the usual formalisation of risk, is based around uncertainty and its effect on financial performance. Whilst the public sector has become much more conscious of financial performance, fundamentally its risks are more often closely related to wider societal risks and to uncertainties in service delivery. Hood and Rothstein (in NAO, 2000) argue that business risk techniques could be integrated with managing public sector risks, but caution that (p. 30): Business risk management is emphatically not a panacea for solving all the intractable polyvalent policy problems faced by government…..Nor is it something that can effectively be done by numbers in an unreflective way…..Achievable successes are likely to be limited and in the middle range.

This key point aside, successful implementation of risk management in the public sector means a proactive rather than reactive approach and enabling management to take actions prior to the occurrence of risk. Therefore, a formalised approach for risk assessment and management can contribute

The UK’s Prudential Borrowing Framework 51 towards success (Ward and Chapman, 1995). One example of a formalised approach is the risk management framework used in PFIs. By contrast the newly introduced PBF and the related Prudential Code for its implementation produced by the Chartered Institute of Public Finance and Accounting (CIPFA) show an almost complete lack of appreciation of the generic nature of risk. This omission is out of step with recent government rhetoric and the imperatives of the ‘risk society’ and it therefore seems to represent a retrograde step in respect of public sector risk management. This paper analyses two key questions in order to establish whether the PBF and the Prudential Code are a retrograde step. First, is public sector risk management more rhetoric than reality? Second, how do the PBF and the corresponding Prudential Code account for risk? Risk Management in the Public Sector Over recent years public sector risk management has been a focus of attention for professional bodies such as the Association of Local Authority Risk Managers (ALARM), the Association of Insurance and Risk Managers in Industry and Commerce (AIRMIC) and the Institute of Risk Management (IRM) which, in 2002, produced jointly the first UK risk management standard. The standard recognised risk management as a central part of organisations’ strategic management and provided guidance to local authorities on the broad risk management issues (AIRMIC, ALARM&IRM, 2002). Another key document is the recent Orange Book (HM Treasury, 2004) which recommended that public sector bodies should take into consideration three main categories of risk: external risks, operational risk and risk associated with organisational change. The external risk category includes political, economic, socio-cultural, legal/ regulatory and environmental risks. Operational risks are associated with the delivery of services and/or products, as well as the availability of internal organisational capacity, including risk management expertise. Risk associated with organisational change refers to all activities and actions going beyond current organisational capabilities. Capital Project Risk Management Despite its general nature, the guidance on public sector risk management provides a foundation for understanding the risks faced by local authorities in capital finance projects. Clearly the risks in such projects are many, varied, dynamic and sometimes unpredictable which requires a framework for their systematic identification, assessment and management. Such a framework would be expected to contain two over-arching categorisations of risks; those which are financially calculable and those which may be difficult or impossible to accurately measure financially. Risks which fall into the former category include; default risk, liquidity risk, commercial risk, disposal of asset risk, external financing risk, project risk (design,

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construction, operation, residual value), commissioning and operating risk, demand, volume and usage risk – reflects the uncertainty that the project is economically viable and will attract sufficient number of customers/users. Those risks which are less financially quantifiable include service risk (quantity and quality), stakeholder risk, client/contractor risk, service user/ community risk, political risk, societal risk, reputation risk, technology/ obsolescence risk, regulation and legislative risk, public policy risk and force majeure risk. These categorisations are by no means exhaustive, but they reflect the very broad range of risks that local authorities should consider when entering into major capital projects. Not all of these would be a concern for private sector companies undertaking comparable capital projects (e.g. some of the stakeholder, societal and public policy risks). Nevertheless, such are the risk management systems of these companies that they utilise comprehensive and sophisticated systems to ensure that, as far as reasonably practical, they identify and assess all project risks (Chapman and Ward, 2002; Waring and Glendon, 1998). According to the head of the National Audit Office, the evidence from the public sector is less persuasive, (Bourne, 2002, p. 1) My message is that too often the public sector, or central government with which I am concerned, does not assess the risks and walks off the end of a cliff. It does not consider where it is going and when it marches forward it does not manage the risk. Too often there are no pilot projects, no training of those who have to deliver a new service and no contingency plans if it does not work.

Although Bourn’s criticism is substantially directed towards central government, there is evidence to suggest that the systems of managing risk in local authorities were also problematic prior to introduction of the PBF and the Prudential Code in 2004 (Hood, 2003; Fone and Young, 2005). Hood’s (2003) exploratory research suggested that, in relation to PPP/PFI projects, local authorities had weak systems for utilising the risk management expertise that existed centrally within the authority. In effect, most risk management responsibility was vested in service professionals within each specific department, rather than with risk professionals. Similarly, Asenova and Beck (2003) noted that in large scale PPP/PFI projects the utilisation of mechanistic approaches to risk management fails to produce the desired results and that a crucial question for local authorities relates to facilitation of the learning process. Hood (2003) also found that in PPP/PFI projects a large amount of risk management was outsourced, at considerable financial cost, to consultants. By adopting this approach, the local authority is failing to utilise a technique much used by the private sector – Integrated Risk Management (IRM). In terms of annual spend, local authorities are larger than most private sector companies and frequently face a more complex and diverse risk set. Why,

The UK’s Prudential Borrowing Framework 53 therefore, would evidence suggest that their use of IRM is relatively rare, whilst growing in the private sector? One of the greatest barriers to integrating risk management across a local authority should, in theory, be one of their strengths, namely their wide range of functions and services. In the main, however, they are managed by functional experts such as educationalists, road engineers, social welfare professionals etc. This ‘departmentalism’ is very much a feature of the traditional bureaucratic model and results in a narrow focus as to the meaning of risk but, more seriously, an antipathy or even hostility to anything which appears to be ‘corporate’. Almost by definition, IRM systems require a holistic, comprehensive and corporate approach; therefore successful implementation of IRM can be seen as problematic. Elsewhere in the public sector, there is some evidence for more homogenous and highly developed operationalising of risk than in local authorities. The ‘Controls Assurance’ system in the NHS attempts to create a framework which focuses on a wide range of risks, both clinical and nonclinical, which may impact upon the objectives of NHS organisations (NHS Executive, 2001, p. 3): CASU’s (Controls Assurance Support Group) central objective is to assist the NHS in improving risk management and the quality of its services through the provision of standards and through acting as a facilitator for identifying and sharing good practice on internal control and risk management activities. Specifically, the Unit is accountable to the NHS Executive’s Controls Assurance Team for: N N N N N N

Facilitating the maintenance of existing controls assurance standards Facilitating the development of new controls assurance standards Developing guidance Promoting benchmarking Assisting with identification of training needs for NHS organisations; and Evaluating the effect of the controls assurance programme

One highly publicised project which became a major embarrassment for the Scottish Executive was the Holyrood project for design and construction of the new Scottish parliament buildings. This project showed that the inability of some public sector bodies to foresee and manage project risks can result in a major fiasco. The Holyrood building was completed and officially opened in 2004, three years late and £380 million over the initial budget (MacDonnell, 2004). The subsequent inquiry uncovered numerous problems with the way the contract was awarded to the main construction company, as well as problems related to the type of contract used and its actual management (The Holyrood Inquiry, 2004). One of the key issues identified by the report was the lack of proper consideration and wrong political judgement in relation to the selection of finance and procurement options, whereby Ministers decided to use a ‘traditional’ construction management approach on the grounds that the PFI can lead to unacceptable delays (The

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Holyrood Inquiry, 2004, chapter 6). According to a letter by Lord Elder, one of Donald Dewar’s special advisors, quoted in the report ‘‘…as some would argue, this was the biggest single error, along with the decision on the fee structure…’’ (The Holyrood Inquiry, 2004, chapter 6). Furthermore, the report emphasised the lack of systematic assessment of the risks implicit in the selected procurement option as well as the complete failure to manage some well known project risks associated with it. Much of the same could be said of the Millennium Dome commissioned by the UK Westminster government. Clearly the management of risk in the public sector as a whole and in capital projects in particular displays considerable variation, being relatively well developed in the NHS, fragmented in local authorities and spectacularly incompetent on occasions within some areas of central government. In terms of answering the question posed as to whether public sector risk management is more rhetoric than reality, the examples and evidence cited provide no unequivocal answer. Although it has been much criticised, the centrality of risk and risk transfer to PFI and the risk frameworks provided by central government did, at least, identify a structure around which comprehensive risk management systems could be built. This structure is outlined in the following section. As our later analysis suggests, the PBF and its associated Code contains no such framework and so represents a return to a very limited vision regarding risk. Management of Risk in PPP/PFI Contracts There is a substantial body of literature relating to various aspects of PPP/ PFI. We intend only to focus, however, on the issues surrounding risk and in its management in such contracts. Before the introduction of PBF the PPP/ PFI was the preferred financing option for many local authority capital projects which sought to improve the quality of public services through involvement of private sector companies in their provision (TTF, 1997; HM Treasury, 1999; HM Treasury, 2000). Most of the PPP/PFIs have been characterised by features such as substantial running costs, complex contractual arrangements and extended contractual periods, all of which contribute to their heightened risk profile. In addition to the risk categories already identified above, PPP/PFI projects are exposed to financial risk which can arise from the inability of the project company to raise capital or can be related to other financial factors such as changes of interest rates (HM Treasury, 1997a). In a typical PPP/PFI contract, each of the risk categories is broken down into a number of individual risks which provides a very detailed risk map. For example, design and construction risk can also relate to site risk and industrial relations risk, as well as problems related to design and construction features (Hodge, 2004).

The UK’s Prudential Borrowing Framework 55 The PPP/PFI contractual framework goes beyond the risk identification to prescribe different treatment for different risk categories. Thus, most of the risks associated with design, construction, operation as well as project finance are supposedly transferred to the private sector. The demand risk and the technology risk are treated on a project-by-project basis (PFP, 1996), while the regulation/legislative risk are either partially shared or retained by the public sector. The fact that PPP/PFI contracts concern the provision of services over long period of time means that the residual value risk is generally not a major issue. Government guidelines place crucial importance on the proper identification and evaluation of the risks affecting particular projects, and these are reflected in the public sector financial model known as Public Sector Comparator (PSC). As previously indicated, however, rather than utilise in-house teams and systems local authorities often rely extensively on external consultants to provide expertise on technical, financial and operational issues as well as assurance that all project risks are thoroughly identified and evaluated for the entire duration of the contract. The risk evaluations are then incorporated in the PSC and during the negotiation process with the private sector bidders they are one important factor in the decision making process. The guidance goes even further to stipulate specific details on the accounting treatment of the PFI transactions (HM Treasury, 1997b). According to this guidance, the risks affecting particular projects should be treated as variations in the profits or losses which have to be allocated to the appropriate contractual partners. All significant risks should be assessed following a rigorous quantitative framework, which involves a number of stages (HM Treasury, 1997b, p. 16): N Identification of the main commercial risks borne by each party followed by detailed calculation of the potential profit variations, N Evaluation of the Net Present Value (NPV) of potential profit variations for the operator and the client, and N On the basis of the above calculation, to come up with a quantitative indicator of which party has an asset of the property.

Despite the availability of detailed guidance, in some early PFIs risk allocation issues were often misinterpreted in terms of the public sector client seeking to transfer all risks to the private sector partners, which had substantial cost implications. In each project there are certain categories of risk, over which the private sector has no control, and which therefore are better shared or handled by the public sector. Examples of such risks include volume or usage risk in prisons, court or road projects. Inevitably, the public sector also retains business risk associated with improper specifications, as well as any reputational risks from project failure (HM Treasury, 1995). In relation to the problems encountered in the project for National Insurance Recording System (NIRS), the NAO noted that it is duty of the government departments to understand fully the risks involved in such project and to provide contingency plans for those risks (NAO, 1998). Similarly, in

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educational projects, the local authority cannot transfer the ultimate responsibility for the provision of education. In some cases the private sector companies are willing to take particular risks, but this inevitably increases the project’s risk premium and affects the Value for Money (VFM) requirement. Therefore, as a general rule in PPP/PFI, the private partners should only be allocated such risks that they are capable of handling. Notwithstanding the detailed guidance and attention to risk, the experience with PPP/PFI projects has shown the intricacies and difficulties associated with the practical implementation of the risk management process. As noted earlier, criticisms of the treatment of risk and the level of actual risk transfer in PPP/PFI contracts has been levelled from a number of different perspectives. According to Lonsdale (2005) some contracts fail to account for key characteristics and relations associated with the main parties which potentially can have decisive impacts for the treatment of risk. Such key characteristics include an asymmetry of balance of power between the partners, superior commercial capabilities of the private sector and the political nature of public sector decision making. Such characteristic features and subsequent communication imbalances can explain the minimum level of risk transfer in some projects which favours the interests of private sector shareholders to those of the citizens (Duffy, 1999) and leads to excessive profits for the private sector. Some technical issues and methodologies have also been a source of concern. For example, in a recent UNISON report Pollock et al. (2005) questioned the evaluation of PPP/PFIs and specifically the methodological approach used in five government commissioned studies which compare the performance of PPP/PFI against traditional capital finance projects. The authors argued that while PPP/PFI projects are designed to include up to 24% ‘optimism bias’ risk adjustment, they fail to present sound data-based proof for the achievement of the key targets in terms of time and cost overruns. According to this analysis (Pollock et al., 2005: 3): Although 677 projects have been approved or completed since 1992, the Treasury has not fulfilled its objective of ‘‘sound evidence base’’ for rigorous investigation’’ of PFI. There is no evidence to support the Treasury’s chief justification for the policy, namely that PFI generates value for money savings by improving the efficiency of construction procurement.

Even if we are unwilling to accept that business models are often inaccurate, it is necessary to recognise that a true evaluation of the financial performance of most PFI projects will probably only be possible some years after it has entered the operational phase, perhaps even only after the full length of the contract. At more conceptual level, Hodge (2004) noted that the utilisation of PPP/PFI as a purchasing device can potentially jeopardise public interest and policy issues. According to the author while the risks in the commercial domain are given a lot of attention and are investigated in

The UK’s Prudential Borrowing Framework 57 detail, the risks in the governance domain are often neglected. One direct result of this neglect is questionable VFM for the taxpayer but more importantly loss of control and ‘‘shrinking stewardship responsibilities in governance’’ (Hodge, 2004, p. 46). Investigating the impact of long term contractual commitments typical of PFI contracts on the ability of the public sector to deliver services, Froud (2003) has questioned the compatibility of PFI with traditional assumptions about the appropriate role of the public sector. Froud’s analysis centres on the question as to how the contractual management of risks in PFI projects impacts on the ability of the state to respond flexibly to the needs of the public. According to Froud (2003, p. 587): The argument here is not that the private sector cannot provide a range of lower cost services…but that the tasks of planning and monitoring public services, of modifying policy, or working towards goals of equality and balanced interests are those which require the authority and mentality of well motivated state.

Other critics of the PPP/PFI risk allocation model have adopted a more polemic approach and focussed their analysis on issues such as the increased ‘marketisation’ of public services (Kerr, 1998) and the accuracy of the economic estimates (UNISON, 1999). While the PPP/PFI capital finance option has been rigorously pursued by the government, the criticism of PPP/PFI suggests that the risk allocation and transfer model has not been an unambiguous success and is probably not universally applicable. However, the existence of comprehensive risk model is a basic prerequisite for a more or less systematic approach to project risk and risk management. What is more, it corresponds to the current endorsement of risk management and the increased level of regulation and governance in local authorities. In the course of the PPP/PFI process some local authorities’ managers have become actively involved with variety of risk issues which are communicated with their advisors and negotiated with the commercial partners. They have accumulated risk awareness and management expertise which can be transferred to other projects or indeed to different tasks. The significance of the PPP/PFI risk framework is also highlighted by the fact that it has already been used as a benchmark for assessing the treatment of risk in other capital finance options (see for example The Holyrood Inquiry, 2004, chapter 6). The Prudential Borrowing Framework The strategic intention of the PBF is the provision of freedom and flexibility for local authorities to control their capital expenditure by removing the system of Credit Approvals. The PBF regime allows more freedom for authorities to ‘invest-to-save’ where expenditure will be repaid from future

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revenue savings. In some cases, savings may be sufficient to meet the full cost of servicing the new debt, meaning that no additional budget is required. In others, savings may only offer part of the financing costs, but nonetheless may represent good value for money. The PBF is not, however, an opportunity for unfettered borrowing and spending by local authorities and they are ‘‘bound by the finite resources available to the authority’’ (Welsh Assembly, 2004, p. 5). Authorities are required, under guidelines produced by the Chartered Institute of Public Finance and Accountancy (CIPFA, 2003a), to base their capital expenditure on a set of ‘prudential indicators’. In effect, they can borrow money to finance capital projects, provided they set safeguards on the affordability of their borrowing. The new system also provides the opportunity to look at alternative ways of financing office accommodation and buildings where they are currently under lease. In the previous system local authorities were restricted in their ability to enter into leases for assets which they could not afford to purchase outright due to limited capital resources. Only short term leases could be taken with the requirement to find new accommodation on their expiry (Finance Advisory Board, 2003). Theoretically a PBF capital finance option could present a challenge to the PPP/PFI orthodoxy, in that affordability could be demonstrated on the basis of the new PBF-related debt being serviced by savings, or additional revenue, generated by the capital investment funded by prudential borrowing. For example, if a new school estate was financed by PBF debt, as in the recently announced Glasgow City Council primary school renewal scheme (Brown, 2004) savings could be made on such aspects as school maintenance costs, the amalgamation of under-used schools, and improved energy efficiency, as well as the input of capital receipts from the disposal of assets. Savings could be further enhanced by the fact that the authority would be able to raise finance at lower interest rates than would be available to a private sector consortium, generally referred to as a Special Purpose Vehicle (SPV) under a PFI deal. The higher interest rates paid by SPVs have been one of the criticisms levelled by opponents of PFI contracts (see for example Froud and Shaoul, 1999; Gaffney et al., 1999 and Broadbent and Laughlin, 2005). Prior to the 2003 legislation, the highly centralised system of controlling local government capital expenditure which had been in place since the 1980s had resulted in very low levels of conventional capital spending by authorities. Since 1997, PPP/PFI has, rightly or wrongly, been increasingly seen as ‘the only game in town’. For local authorities, PPP/PFI was associated with the design, build, finance and operation of the contract. Under PBF, there is an opportunity to look at other value for money solutions to procure new services requiring capital investment. Savings may be generated from the new facilities and also from the reductions in procurement costs. The primary concept is that the private sector may continue to design build and operate such facilities, but with the public sector providing the finance. Savings can be achieved on the cost of

The UK’s Prudential Borrowing Framework 59 borrowing (interest rates and the internal rate of return on equity) between the public and private sectors and this may provide a value for money solution to be considered that will enable local authorities to procure integrated capital assets and services in future. The potential for savings should be considered against the possibility for price certainty in terms of both construction and operation which is embedded in a PPP/PFI option. As indicated by the Audit Commission (2004, para 35), however, local authorities proposing to use the PBF are required to prepare detailed plans to manage the financial aspects of any project: The clear implication of the prudential code is that an authority, after appraising different options, needs to have its capital strategy and asset management plan (AMP) in place at the same time that it sets its prudential indicators. The capital strategy should clearly link the authority’s capital investment to its service priorities and the AMP should identify the costs of maintaining existing assets. The CIPFA interim guidance suggests that authorities should use these documents to prepare a draft capital programme, revenue forecasts and treasury management strategy and approve or revise them in the light of prudential indicators so that they are affordable, prudent and sustainable.

A significant difference, however, between PPP/PFI and the PBF is on the issue of risk and how it should be managed. The legislation introducing the PBF contains little detail on the operationalisation of the policy framework. Any local authority considering use of the PBF would be expected to refer to the ‘Prudential Indicators’ (PIs) provided by CIPFA (2003a). These are summarised in Table 1. Beyond the PIs, there is no formal, universally expected framework for the use of the PBF. As can be seen from Table 1, the indicators do address a form of risk, but this risk is restricted to fiduciary prudence, i.e. a limited form of financial risk. In reality, the PIs appear little different to the capital finance criteria which local authorities would have utilised prior to the introduction of PFI, e.g. the Public Works Loan Board. Given the increased emphasis that has been placed on wider public sector risk in recent years, it could have been expected that the PBF and its associated codes would have had a much greater explicit appreciation of a comprehensive risk framework. The very narrow financial focus inherent in the PIs appears to be a reversion back to a form of departmentalism, on this occasion the predominance of the input of financial professionals at the expense of an input from those who could provide a more comprehensive risk overview. In effect, there is a total absence of IRM. An example of where this narrow focus could present long-run difficulties for local authorities is on the question of sustainability – which along with affordability and prudence is one of key factors of the PBF. Judging sustainability by simply comparing borrowing and debt with the

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John Hood et al. Table 1. Prudential indicators (based on Donaldsons/PMP, 2004)

Prudential indicators for affordability Estimates of the ratio of financing costs to net revenue stream (%) Estimates of the ratio of financing costs to net revenue stream (%) Estimates of the incremental impact of capital investment decisions on council tax Estimates of the incremental impact of capital investment decisions on housing rents Prudential indicators for prudence Net borrowing and the capital financing requirement Prudential indicators for Capital Expenditure Estimate of total capital expenditure (3 years) Actual capital expenditure for previous year Estimates of capital financing requirement (3 years) Actual capital financing requirement for previous year Prudential indicators for external debt Authorised limit for external debt (3 years) Operational boundary for external debt (3 years) Actual external debts as at 31st March of previous year Prudential indicators for Treasury Management Adoption of CIPFA Code of Practice for Treasury Management in the Public Services Upper limit on fixed interest rate exposures (3 years) Upper limit on variable interest rate exposures (3 years) Upper and lower limits for the maturity structure of borrowing Prudential limits for principal sums invested for longer than 364 days

capacity of the revenue budget to finance them is crude and is focused only on financial risk. This point is particularly relevant if local authorities finance capital projects, e.g. rebuilding a school estate, but still contract with private sector consortia to design, build and operate aspects of the service. Interestingly, during the development phases of the Code, the Prudential Code Steering Group (PCSG) which was closely involved in the drafting of the Prudential Code discussed more comprehensive interpretation of risk and uncertainty which was consequently discarded. Thus according to an Issues Paper (CIPFA, 2002, p. 1): In dealing with risk and uncertainty in capital developments, the local authority needs to manage all of the risks involved in large and small capital development works, including: cost over-runs, slippage of major projects, changing specifications. These risks have been well documented and studies in Audit Commission VFM studies. Local authorities have to manage these risks.

The UK’s Prudential Borrowing Framework 61 The paper also outlines other risks which can jeopardise the success of the PBF project including ‘‘risk in relation to the availability of capital financing from capital receipts, grants and external contributors’’. The more general risk associated with treasury management is also recognised (CIPFA, 2002, p. 2). As concerns the management of those risks the document emphasises that they have to be taken into account in the capital expenditure plans and indeed reflected in the PIs. Alternatively, authorities can utilise risk reserve in the form of ‘‘authorised borrowing limit’’ set at a level below the prudential borrowing limit. These recommendations were not reflected in the final draft of the Prudential Code. Similar concerns have been expressed by the Association of Chartered Certified Accountants (ACCA, 2003) which questioned different provisions including the long-term sustainability aspects of the Code, the reporting mechanisms for the PIs, its potential for maintaining appropriate investment levels, the overall context of the PIs and the distinction between indicators and targets. However, the main criticism levelled by ACCA focuses on sustainability. Despite the fact that the effect of capital investment and borrowing will be measured in decades, the Code does not make provisions for long-term planning. This analysis suggests that firstly, the main body of the Code makes insufficient reference to sustainability and secondly, that there is surprising lack of consideration for the potential long-term implications. Unlike PPP/ PFI projects where the project financing is virtually guaranteed by the government over 25–30 years, the PBF and the Code fail to commit the government to long-term support of the capital investment programme. Thus, while giving greater freedom to local authorities the new capital framework arguably reduces the overall responsibility of central government for local capital investment programmes and effectively transfers this responsibility with all associated risks to local government. One consequence of this transfer of responsibility is that local authorities will have higher level of responsibility for managing the risks associated with the application of the PBF than has been the case under PPP/PFI. The Lyons Inquiry into local government (Lyons, 2005, paragraph 2.97) noted that: while the new prudential borrowing powers are an important new flexibility, local authorities may be constrained in making full use of them by current constraints on revenue to finance additional borrowing.

Sir Michael Lyons suggested that those constraints could be removed by introduction of new taxes whose revenues are linked to success of the local area. Such taxes will be considered in the final report in late 2006. The ACCA also comments on the reporting structure necessary for the proper monitoring of the PIs and the need to consider the long-term implications of the proposed capital investment programme on the

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future level of local taxation. It specifically suggests (ACCA, 2003, paragraph 8): N explicit consideration of the effects on the Council Tax of differing levels of capital investment N a comparison of net external borrowing with the capital financing requirement of the Council’s investment programme, and N further comparative details of the Council’s total debt.

This information should be considered in conjunction with some historic figures on the relationship between the Council Tax and the level of investment over the past five years. In effect, the purpose of this exercise suggested by ACCA is to ascertain the risk of Council Tax increases introduced by the proposed borrowing programme. On the other hand, sustainability is hardly compatible with the lack of lower limits for capital investment which creates risks of underinvestment and/or inadequate maintenance programmes. A focus purely on financial risk ignores many of the most substantial risk elements of local authority service provision, e.g. ability to deliver a quality service to the community, long-term fitness for purpose of capital assets and the long term capacity of private sector partners to absorb risk and uncertainty. The Code is silent in relation to the maintenance of levels of quality in public services and does not draw a clear line between indicators and targets. In order to measure performance, local authorities should include in their capital expenditure plans a range of clear performance indicators which will be used as a performance measures (ACCA, 2003). This practice has already been utilised in PPP/PFI projects in which the quality of the services is assessed against a number of criteria known as key performance indicators and the failure of the contractor to meet the standard stipulated in these indicators results in payment deductions. Local authorities could argue that they assess these other risk elements quite separate from the PBF and PIs. Whilst this may be true in certain cases, the literature summarised above suggests that there are, in general, significant deficiencies in local authority risk management practices and the current articulation of the PBF fails to address this problem. It is plausible, therefore, that when using the PBF they would focus mainly on the PIs and ignore the systematic analysis of a wider risk framework. Our argument is that, as the PBF and its associated codes stand, the understanding of risk is too narrow and departmentalised where each local authority and indeed each government department using PBF can have their own rules and interpretations. In contrast, the overall control of PPP/PFI is highly centralised in the hands of HM Treasury, which significantly reduces the possibility for variations (Broadbent and Laughlin, 2002). Furthermore, the finance professionals appear not to have taken cognisance of the considerable body of both literature and practice (detailed above) that has grown up in recent years around the whole concept of ‘public’ risk and the developments in private sector risk management which are transferable to

The UK’s Prudential Borrowing Framework 63 the public sector. A consequence of this narrowness of approach is that local authorities will fail to make progress in incorporating risk management into their capital projects. This compares unfavourably with the risk frameworks utilised under PPP/PFI contracts (also detailed above). PPP/PFI has been subject to much criticism from a diverse range of commentators, and there is strong evidence that central government’s claims on its behalf have been based more in assertion than in proof. Despite its apparent flaws, it does, however, utilise a detailed and systematic risk identification and assessment framework as a preamble to risk transfer, which whilst not satisfactorily addressing all local authority project risks, represents a significant improvement on the PBFrelated system. The above analysis of risk management on PPP/PFI would suggest that many of the principles and practices of this policy would be appropriate to a formal and systematic risk framework for PBF projects. This conclusion addresses the second question posed above, namely how do the PBF and the Prudential Code account for risk compared with rest of the public sector. The answer is surely very badly indeed. Conclusion Both conceptually and in an operational context, public sector risk and its management has become increasingly high profile in recent years. However, despite exhortations from central government, QUANGOs and various audit bodies, there is evidence that the reality of risk management may not match the rhetoric. In effect, despite an attempt at homogeneity in the NHS, there is strong evidence that public sector risk management is highly variable, fragmented and characterised by a narrow focus on professional departmental disciplines. This failure to look at risk in its broadest sense, has, in local authorities, been exacerbated by the traditional bureaucratic model and the focus on professionalism and departmentalism. In addition, the almost total lack of explicit risk indicators has not provided them with particular incentives. Various initiatives to increase the focus towards corporatism, both generally and specific to risk management, have met with mixed, but limited, success. There is no question of the risk frameworks and systems found in PPP/ PFI contracts representing a panacea for risk management problems in local authority capital projects. Based on robust empirical evidence, there is a strong case for arguing that risk allocation in such contracts is not always equitable and that, at the very least, the public sector will remain the risk bearer of last resort irrespective of any risk transfer agreements in place. Notwithstanding these concerns, however, PPP/PFI does at least represent an opportunity for local authorities to use comprehensive risk identification and assessment techniques, which transcend the professional and departmental boundaries and account for some long term operational risks. In effect, it allows them the prospect of utilising more holistic, integrated risk management. Furthermore, the evolving nature of the policy has allowed

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local authorities to become more sophisticated in their risk-related dealings with private sector partners. The introduction of the PBF, although perhaps beneficial in other ways, appears to represent a retrograde step from PPP/PFI as regards risk, treatment of risk and risk allocation, with a lack of adequate treatment of risk under the current PBF and Code. If we accept the hypothesis that local authority risk management practices are underdeveloped, especially at the integrated level, it could be reasonably speculated that they will not go beyond the requirements of the guidance. Our analysis suggests that the PIs do not account for the full range of project risks and that there is no incentive for local authorities to consider the long term risk implications or to provide training for understanding risk and risk management. Over recent years there has been a trend for increased risk transfer from the public to the private sector through the use of, for example, PFI contacts for schools, and the PBF marks a reversal of this, with no evident framework for economic and equitable allocation of risk. There is also the possibility to use the PBF in a way which circumvents the current rigorous contractual and monitoring systems which exist for PPP/PFI. PPP/PFI has been, often rightly, criticised for its apparent pandering to the private sector. It did, however, offer local authorities at least the opportunity to consider risk in broad terms. We conclude that the PBF is not a panacea for constraints on capital spending or for dealing with many of the criticisms levelled at PPP/PFI. PBF as interpreted by the CIPFA Code is seriously flawed in its conception of risk, is highly conservative in its approach to identifying, evaluating and managing risk and to be an effective method of financing local authority capital projects will need to be expanded to account for exposure to the many and varied forms of risk. Acknowledgement The authors gratefully acknowledge the support of the British Accounting Association in funding this research. References ACCA (2003) Prudential Code for Capital Finance in Local Authorities, Comments from ACCA, Internet Publication. Available at www.accaglobal.com/technical (accessed 7 December 2005). AIRMIC, ALARM&IRM (2002) A Risk Management Standard, Internet Publication. Available at www.airmic.com/ (accessed 7 November 2005). Asenova, D. and Beck, M. (2003) Scottish Local Authorities and the Procurement of PFI Projects: A Pattern of Developing Risk Management Expertise? Public Works Management & Policy, 8(1), pp. 11–28. Audit Commission (2001) Worth the Risk: Improving Risk Management in Local Government (London: Audit Commission). Audit Commission (2004) Auditor Briefing 1/2004: Prudential Code (London: Audit Commission). Bourn, J. (2001) Reinforcing Positive Approaches to Risk Management in Government. Paper presented at The Institute of Risk Management, Third Annual Lecture, London, 19 June 2001.

The UK’s Prudential Borrowing Framework 65 Broadbent, J. and Laughlin, R. (2002) Accounting choices: Technical and political trade-offs and the UK’s private finance initiative, Accounting Auditing and Accountability Journal, 15(5), pp. 622–654. Broadbent, J. and Laughlin, R. (2005) The Role of PFI in the UK government’s modernisation agenda, Financial Accountability and Management, 21(1), pp. 75–97. Brown, L. (2004) Delivering regeneration for the city of Glasgow: Exercising freedom within boundaries, Spectrum, September 2004, p. 6. Available at www.cipfa.org.uk (accessed 12 January 2006). Chapman, C. and Ward, S. (2002) Managing Project Risk and Uncertainty, John Wiley and Sons: Chichester. CIPFA (2002) Issue Paper – Considerations of Risk and Uncertainty, Prudential Code Steering Group, Internet Publication. Available at www.cipfa.org.uk (accessed on 12 January 2006). CIPFA (2003a) The Prudential Code: Preliminary Guidance (London: The Chartered Institute of Public Finance and Accountancy). CIPFA (2003b) Guidance on Internal Control and Risk Management in Principal Local Authorities and Other Relevant Bodies to Support Compliance with the Accounts and Audit Regulations 2003 (London: The Chartered Institute of Public Finance and Accountancy). Donaldsons/PMP (2004) The Prudential Code for Capital Finance: A Practical Guide for Local Authority Managers (London: PMP). Dowlen, A. (1995) Learning to manage risk in public services, Executive Development, 8(2), pp. 19–24. Finance Advisory Network (2003) Responding to prudence – advisory report from the October 2003 series of FAN, on the final version of CIPFA’s prudential Code. Available at www.ipf.co.uk.fan Fone, M. and Young, P. (2005) Managing Risks in Public Organisations (Leicester: Perpetuity Press). Froud, J. and Shaoul, J. (1999) Appraising and evaluating PFI for NHS hospitals, Financial Accountability and Management, 17(3), pp. 247–270. Froud, J. (2003) The private finance initiative: risk, uncertainty and the state, Accounting Organisations and Society, 28, pp. 567–589. Gaffney, D., Pollock, A., Price, D. and Shaoul, J. (1999) NHS capital expenditure and the Private Finance Initiative – expansion or contraction, British Medical Journal, 319, pp. 48–51. HM Treasury (1997a) Partnership for Prosperity: The Private Finance Initiative (London: The Stationary Office). HM Treasury (1997b) Technical Note No 1 (Revised) How to Account for PFI Transactions (London: The Stationary Office). HM Treasury (2000) Public Private Partnerships, The Government’s Approach (London: The Stationary Office). HM Treasury (2001) Management of Risk, A Strategic Overview (London: HM Treasury). HM Treasury (2004) Management of Risk, Principles and Concepts (London: HM Treasury). Hodge, G. (2004) The risky business of public-private partnerships, Australian Journal of Public Administration, 63(4), pp. 7–49. Hood, C. (2002) The risk game and the blame game, Government and Opposition, 37(1), pp. 15–37. Hood, J. and Allison, J. (2001) Local authority corporate risk management: A social work case study, Local Governance, 27(1), pp. 3–18. Hood, J. and McGarvey, N. (2002) Managing the Risks if Public-Private Partnerships in Scottish Local Government, Policy Studies, 23(1), pp. 21–35. Hood, J. (2003) ‘Minimising risk: The role of the local authority risk manager in PPP/PFI contracts’, Public Policy and Administration, 18(2), pp. 57–70. Kerr, D. (1998) The PFI miracle, Capital and Class, 64(Spring), pp. 17–28. Lonsdale, C. (2005) Risk transfer and the UK private finance initiative, Policy and Politics, 33(2), pp. 231–249. Lyons, M. (2005) Lyons Inquiry into Local Government: Consultation Paper and Interim Report (London: HMSO). Available at www.lyonsinquiry.org.uk/

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Macdonell, H. (2004) £4.31m Court Battle Looms, Building Firm that Missed out Says Rules were Broken, Scotsman, Internet Publication, available at www.scotsman.com (accessed 16 November 2005). Monitor (2005) Prudential Borrowing Code (PBC) for NHS Foundation Trusts (Monitor: Independent Regulator of NHS Foundation Trusts), Report IRG 03/05. NAO (1998) The Contract to Develop and Upgrade the Replacement National Insurance Recording System, Ordered by the House of Commons, Prepared by the Controller and the Auditor General, National Audit Office (London: Stationary Office). NAO (2000) Supporting Innovation: Managing Risk in Government Departments (London: National Audit Office). NHS Executive (2001) Governance in the new NHS: Controls assurance statements 2000/2001 and establishment of the Controls assurance support unit, HSC 2001/005. Pollock, A., Price, D. and Player, S. (2005) The Private Finance Initiative: A Policy on Sand, An Examination of the Treasury’s Evidence Base for Cost and Time Overrun Data in Value for Money Policy and Appraisal (London: UNISON). Private Finance Panel (1995) Private Opportunity, Public Benefit: Progressing the private finance initiative (London: HMSO). Private Finance Panel (1996) Risk and Reward in PFI Contracts: Practical guidance on the sharing of risk and structuring of PFI contracts (London: HMSO). The Holyrood Inquiry (2004) A Report by the Rt Hon Lord Fraser of Carmylle QC on his Inquiry into the About Holyrood Project, Internet Publication. Available at www.scottish.parliament.uk/vli/ holyrood (accessed 16 November 2005). TTF (1997) Treasury Taskforce Guidance, Partnerships for Prosperity, The Private Finance Initiative (London: HMSO). Vincent, J. (1996) Managing risk in public services: A review of the international literature, International Journal of Public Sector Management, 9(2), pp. 57–64. Ward, S. C. and Chapman, C. (1995) Risk-management perspectives on the project lifecycle, International Journal of Project Management, 13(3), pp. 145–149. Waring, A. and Glendon, I. (1998) Managing Risk (London: Thomson Learning). Welsh Assembly (2004) Guide to the Prudential Framework for Capital Finance for Local Authorities in Wales, July 2004.

Journal of Sustainable Finance & Investment, 2014 Vol. 4, No. 2, 147–160, http://dx.doi.org/10.1080/20430795.2013.837810

Towards a new framework to account for environmental risk in sovereign credit risk analysis Margot Hill Clarvisa,b*, Martin Hallec, Ivo Mulderb and Masaru Yarimed a

Institute of Environmental Sciences, University of Geneva, Geneva, Switzerland; bUNEP-FI, Geneva, Switzerland; cGlobal Footprint Network, Geneva, Switzerland; dGraduate School of Public Policy, University of Tokyo, Tokyo, Japan (Received 14 August 2013; accepted 21 August 2013) Despite the growing body of evidence on ecosystem degradation and on-going development in measuring its economic implications, there remains a lack of understanding and integration of environmental risks into investment decision. There is, therefore, currently a weak financial rationale and a limited choice of tools to assess the materiality of environmental risk for the sovereign bond market. Improving investor understanding of the materiality of environmental risks is likely to be crucial to limiting risk exposure of important investments and to encouraging the transition to a greener more sustainable economy. This article presents the development and initial application of a framework that aims to improve the financial rationale for assessing the materiality of environmental risk in the sovereign bond market. It is the result of a collaborative and inter-disciplinary project of researchers and practitioners from a group of financial institutions, the United Nations Environment Programme Finance Initiative, and Global Footprint Network. Results not only show the long- and short-term implications of environmental risk for a wide variety of resource profiles, but also how these risks relate to macroeconomic factors that are already recognised as relevant to sovereign credit risk. This, therefore, presents a more accurate reflection of how these factors might influence the risk or return situation for an investor. More collaborative and innovative research between scientists and practitioners could improve both knowledge and methods to effectively account for the financial materiality of natural resource risks for a country’s economy. Keywords: sustainability; resource risks; sovereign credit worthiness analysis

1.

Introduction

Evidence of ecosystem degradation (Bascompte et al. 2012; Rockström et al. 2009a, 2009b; UNEP 2012) and its significant economic effects (Grantham 2011) continues to mount. While there has been a growing body of evidence on the economic value of ecosystem services (Costanza et al. 1997; TEEB 2010), less headway has been made in communicating the risks and rewards of functioning ecosystems to investors in a way that informs their investment decisions (Urwin 2012). Achieving a better understanding and integration of environmental risks into investment decision-making is likely to not only be crucial to limiting risk exposure of important investments, but also to encouraging the transition to a greener more sustainable economy (UNEP 2011).

*Corresponding author. Email: [email protected] © 2013 Taylor & Francis

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Sovereign and government bonds (S&P 2012) are one of the most important asset classes for investors, yet integration of environmental risks has to date been particularly limited in sovereign credit ratings (S&P 2012). What little analysis exists tends to present stand-alone sustainability ratings of countries without specifying or quantifying the link to economically material risks (Oekom 2012; Sarasin 2011). Many academics and investors are advising growing caution against expecting past risks and performance to play out in the same way in the future (Grantham 2011; Rockstrom et al. 2009a, 2009b; Urwin 2012). This is all the more relevant, given the significant challenges that a number of ‘wealthy’ countries have faced in relation to their sovereign debt over the past two years. For example, in the USA, as well as Spain, Greece, Portugal and many other nations primarily in the Eurozone, were downgraded in the period 2011–2012. Sovereign bonds have generally been considered safe securities, especially of most OECD countries, but that picture is now quickly changing. The research presented here was, therefore, driven by a growing recognition of the need to improve the currently weak financial rationale and the limited choice of tools to assess the materiality of environmental risk for the sovereign bond market. This article presents the results of a collaborative research project between a group of financial institutions, the United Nations Environment Programme Finance Initiative, and Global Footprint Network to investigate the potential for environmental risk integration in sovereign credit analysis (E-RISC) and assess its financial materiality. After reviewing the challenges to integrating economic risks of environmental degradation into financial practice, the article presents the E-RISC framework in order to discuss its implications for addressing current gaps and weaknesses. The article therefore aims to practicably contribute to both theory and practice, presenting the development and application of a more finance relevant framework. In doing so, it discussed how such a framework could address the current challenges of integrating and accounting for environmental risks in not only risk frameworks addressing sovereign bonds, but potentially also other crucial areas of finance. 2.

Background Sovereign bonds and credit ratings

2.1

Sovereign bonds are issued by a central government to raise money on capital markets. They are a debt security issued by a national government within a given country and denominated in either the country’s own currency (government bond) or a foreign currency (sovereign bond). At over USD 40 trillion, they represent over 40% of the global bond market and are therefore one of the most important asset classes held by investors around the world The level of outstanding sovereign debt amounts to 69% of global GDP in 2010, up from 46% in 2000, with significant jumps in 2009 and 2010, due to the mix of stimulus packages, lost tax revenues and weak growth after the financial crash of 2008 (McKinsey Global Institute 2011). Sovereign credit default risk assessment seeks to measure the ability and willingness of a country to pay back its debt in time and in full. Factors conventionally used to measure a country’s sovereign credit worthiness (Moody’s 2008; S&P 2012) are: .

.

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Economic development: Economic structure and growth prospects (GDP, GDP per capita, inflation); economic diversity and volatility, income levels. Government debt burden: Sustainability of a sovereign’s deficits, its debt burden, debt structure, and funding access (total debt as a per cent of GDP, interest payments, average debt maturity). Budgetary performance: Fiscal performance and fiscal flexibility, long-term fiscal trends and vulnerabilities, and potential risks arising from contingent liabilities (budget deficit as per cent GDP).

Journal of Sustainable Finance & Investment .

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Foreign liquidity and balance of trade: External liquidity and international investment position, status of a sovereign’s currency in international transactions, the sovereign’s external liquidity, and its external indebtedness, which shows residents’ assets and liabilities relative to the rest of the world (foreign debt as per cent of GDP, foreign currency and reserves, trade deficit/surplus, dependence on a single commodity). Monetary flexibility: A monetary authority’s ability to use monetary policy to address domestic economic stresses, particularly through its control of money supply and domestic liquidity conditions; the credibility of monetary policy, as measured by inflation trends; and the effectiveness of mechanisms for transmitting the impact of monetary policy decisions to the real economy, largely a function of the depth and diversification of the domestic financial system and capital markets. Institutional and political factors: Institutional effectiveness and political risks; delivering sustainable public finances; promoting balanced economic growth; responding to economic or political shocks. It also reflects the view of the transparency and reliability of data and institutions, as well as potential geopolitical risks.

Sovereign credit ratings provided by rating agencies play a role in determining interest rates and hence the borrowing costs of a debtor nation. These institutions therefore play a significant role in bond markets and have a responsibility to adjust and realign their risk models to the latest trends and new forms of emerging risks (van Duyn and Milne 2011). Debt repayment requires sustainable revenue for governments through taxes, royalties and other types of income, which in turn require stable and sustainable economic activities (Sarasin 2010). Given that many sectors are directly or indirectly dependent on natural resources such as forest products, energy, agriculture, pharmaceutical and chemical industries, there is a clear need to better understand and value the underlying natural capital that can be the foundation of a nation’s economy. 2.2

Current frameworks

Credit rating Agencies (CRAs), investors, banks and information providers all utilise different sovereign credit risk assessment methodologies, internal processes and rating methodologies, which range in their adequacy for understanding of environmental or natural resource-related risks. The three major CRAs (Moody’s, Fitch and Standard and Poor’s), as well as those with lower market shares, tend to factor governance issues (G) in their sovereign risk analysis through political risk scores. However, there is very little integration of social (S) and environmental (E) criteria in their sovereign ratings (PRI 2012). Some risks relating to risk exposure from recurrent natural hazards (i.e. hurricanes, earthquakes, etc.) and economic reliance on single commodities are included in credit rating assessments of a country’s institutional coping capacity (S&P 2012). However, not only does there remain a lack of public information on the extent to which such environmental risks are factored into the actual rating process, but also a gap remains in the analysis of other forms of environmental risk. Banks, institutional investors and investment managers are starting to factor in environmental criteria in their sovereign credit risk models, often at an early ‘contextualisation’ phase and disconnected from the mainstream financial analysis. For example, some use resource-based metrics such as the Ecological Footprint as an indicator amongst others for assessing country level sustainability performance (Sarasin 2010, 2011). Other institutions factor in environmental issues in the pre-screening process, for example, filtering out countries that do not meet certain international environmental obligations or treaties (UNEP-FI and Global Footprint Network 2012). A growing number of banks and investors are buying ratings, indices or additional data and analysis from environmental, social and governance (ESG) information providers to supplement

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their own sovereign credit risk analysis (MSCI 2013). Many ESG specialists compare sovereign ESG performance with sovereign credit ratings (Oekom 2012) in order to show or assess correlations. These forms of analysis have added a valuable new layer of information to traditional analysis, but there has been less research directly linking environmental (as well as other social and governance) factors to the economic, fiscal and political factors that make up a sovereign’s credit rating. While correlations provide valuable information on comparative performance of sovereigns across a range of ESG issues, there remains a gap of qualified or quantified connections between the ESG risk criteria and core economic or financial indicators. Social and governance factors such as property rights and education levels may contribute to a stronger work force and innovation, which in return can stimulate economic growth. It would, therefore, not be unrealistic to expect a positive correlation between sovereign credit ratings and these ‘S’ and ‘G’ factors. Similarly, some ‘E’ factors can be shown to be linked with economic performance and bond risk, but some are not. A nation’s CO2 emissions or the ratification of environmental treaties may be useful for screening purposes from a values perspective (Reutimann and Steinemann 2012), but are perhaps less likely to render a negative correlation to risk or GDP. Other ‘E’ factors, such as those investigated here, may be much more likely to have material impact on economic performance. 2.3

Challenges in the financial application of current approaches for valuing ecosystems

At present, environmental risks are therefore rarely considered by financial institutions when analysing risks at the country level (UNPRI and UNEP-FI 2011). This is mostly due to the widespread perception that these risks are not material to the country’s economic and financial performance in the timeframe that they typically consider, which rarely exceeds two to three years (S&P 2012). Where environmental factors are considered, they are mostly seen as ‘extra financial’ and, as such, often grouped with social and governance factors. This means that environmental factors are not considered because of a proven and recognised impact on financial performance but rather, they are considered either for ethical reasons or on the unproven assumption that these factors may prove material over time (Hill et al. 2011). In fact, the financial performance of investments that include ESG factors compared to those that do not is hotly debated (Renneboog, Ter Horst, and Zhang 2011). Demonstrating the materiality of environmental risks has proved difficult and much of the recent research into the economic implications of environmental pressures has fallen short of providing financial analysts with both a clear case for integrating environmental risks in their analysis and the tools to do so. All economic activity ultimately depends on the availability of ecosystem services and so there has been a growing body of evidence on ecosystem degradation (Bascompte et al. 2012; Rockström et al. 2009a, 2009b; UNEP 2012), its significant economic effects (Grantham 2011) and thus its effect on economic prospects (Butchart et al. 2010; Costanza and Daly 1992). This type of research is essential to the development of a better understanding of the linkages between environmental health and economic performance but it lacks the coverage and periodicity to be used consistently in financial risk analysis. The little academic and trade research on environmental impacts on credit risk has tended to firstly focus on corporate bonds (Graham and Maher 2006; Schneider 2011), or correlations between environmental risk and bond pricing and ratings (Drut 2010; Dwyer 2012; Scholtens 2010). More effort needs to be made on assessing the specific implications of environmental and natural resource risks to macroeconomic indicators already used in investment analysis. Another strand of research that has attracted a lot of attention in the past years is the valuation of ecosystem services. Major initiatives such as The Economics of Ecosystems and Biodiversity, the UN System of Environmental-Economic Accounting, and the World Bank-led programme on

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Wealth Accounting and Valuation of Ecosystem Services (Bartelmus 2009; Christie et al. 2006; Common and Perrings 1992) have focused on quantifying the values of currently intangible natural assets or services that have been taken for granted, with the goal of ‘making nature’s values visible’ (Costanza et al. 1997; Kareiva et al. 2011; MEA 2005). While these initiatives have shed light on the potential value of ecosystem services at different scales, they are of limited applicability to financial analysts focusing on country-level risks who must predict actual costs and revenues rather than potential values. It, therefore, appears that a scientifically grounded method that investors might use to demonstrate the materiality of environmental risks for a country’s economic performance remains absent. Such a method would need to clearly identify environmental risks that can affect the key levers of macroeconomic performance. It would also need to quantify these impacts on indicators that financial institutions are familiar with in order to facilitate integration. These objectives are at the heart of the E-RISC project, which aims to develop an investor-focused methodology to identify and quantify how natural resource-related risk might impact prospects for macroeconomic performance and sovereign credit risk.

3. Towards an analytical framework 3.1 The Ecological Footprint The aim of the E-RISC methodology is to identify and quantify the economic risks that a country is exposed to as a result of its pattern of natural resource consumption. The principal starting point for this analysis is Ecological Footprint accounting (Borucke et al. 2013), a comprehensive resource accounting tool that compares demand for renewable natural resources and services with the ability of the biosphere to generate them. The Ecological Footprint is a measure of consumption that incorporates information on domestic production as well as international trade (see Figure 1). Comparing total demand (Ecological Footprint of consumption) with demand met from domestic production (Ecological Footprint of production) and the regenerative capacity of the domestic biosphere (biocapacity) allows a first classification and assessment of risk in a country. The difference between the Footprints of consumption and production represents a country’s

Figure 1. The Ecological Footprint and its components.

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net trade in renewable natural resources and services; this component of demand is, therefore, subject to risks linked with availability and prices of resources on international markets. The difference between the Ecological Footprint of production and biocapacity, on the other hand, represents either an overuse of domestic bioproductive land and marine areas or a reliance on global commons to absorb waste in the form of carbon dioxide emissions. Both of these carry their own distinct set of economic risks for the country’s economy (UNEP-FI & Global Footprint Network 2012).

3.2 The framework The E-RISC framework (Figure 2) incorporates 20 qualitative and quantitative indicators (see Table 1) across four dimensions (resource balance, trade-related and degradation related risk, financial resilience) to answer one fundamental question: How financially-material are natural resource risks for a country’s economy?

Figure 2. Conceptual overview of the E-RISC framework.

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Description

Resource balance Footprint/ Ratio of Ecological Footprint over biocapacity biocapacity ratio Trade-related risk (short-term risk factors) Exposure to price Change in trade balance from a 10% change in volatility resource prices, expressed as a percentage of GDP Footprint trade Ratio of the country’s Ecological Footprint of ratio consumption over its Ecological Footprint of production Fuels trade ratio Ratio of the country’s fossil fuel consumption over its fossil fuel production Footprint trade Average yearly growth rate over the last 10 ratio trend years of the net trade component of the country’s Footprint (EFc-EFp) Fuels trade ratio Average yearly growth rate over the last 10 trend years of the net trade component of the country’s fuel consumption (fuel consumption – fuel production) Dependency Ratio of the country’s total resource demands, in biocapacity terms, which is met through domestic production Risk of supply Number of countries, out of the country’s top disruption five resource suppliers in biocapacity terms, who are themselves have a biocapacity deficit Natural resource Exports minus imports of natural resources, in trade balance share of GDP Degradation-related risks (medium term risk factors) Resource overuse Ratio of the country’s Ecological Footprint of ratio production over its biocapacity Agricultural output Share of Agriculture in the country’s total value added, as percentage of GDP Agricultural Percentage of the country’s total employment employment accounted for by agriculture Agricultural Percentage of the country’s total merchandise exports accounted for by food and exports agricultural raw materials Agricultural Percentage of the country’s total merchandise imports imports accounted for by food and agricultural raw materials Exposure to Change in the country’s trade balance as a result degradation of a 10% fall in production of renewable natural resources. Expressed as a share of GDP Exposure to Average yearly growth rate over the last 10 degradation years of the resource overuse component of trend the country (Ecological Footprint of production minus biocapacity) Financial resilience Debt Country’s general government gross debt, as share of GDP

Source Global Footprint Network, National Footprint Accounts Calculated from UNCTAD trade data Global Footprint Network, National Footprint Accounts US Energy Information Administration Global Footprint Network, National Footprint Accounts US Energy Information Administration Calculated from Global Footprint Network, National Footprint Accounts Calculated from Global Footprint Network, National Footprint Accounts UNCTAD data Global Footprint Network, National Footprint Accounts World Bank, World Development Indicators CIA World Factbook UNCTAD data UNCTAD data Calculated from Global Footprint Network, National Footprint Accounts Calculated from Global Footprint Network, National Footprint Accounts IMF World Economic Outlook Database (Continued)

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Table 1. Continued. Indicator

Description

Source

Government budget balance Trade balance

Country’s general government net lending/ borrowing, as a share of GDP Country’s total merchandise exports minus total merchandise imports Yearly average consumer inflation over the last five years

IMF World Economic Outlook Database UNCTAD data

Inflation

World Bank, World Development Indicators

Note: The four dimensions are: (1) resource balance, (2) trade risk, (3) degradation risk, and (4) financial resilience.

To answer this question, the E-RISC framework is divided into three components. (1) Natural resource risks: The first component examines the patterns of production and consumption of natural resources in order to identify resource-related risks. (2) Economic significance of resource risks: The second component examines the structure of the economy, including trade, to assess the expected magnitude of impact of risks identified in the first component. (3) Country’s economic resilience: The third and final component considers the financial situation of the economy and its resilience to the shocks that may emerge as identified in the first two components.

3.2.1 Natural resource-related risks Natural resource-related risks are divided into three categories according to their principal risk drivers. The first category concerns the risks related to shocks in the country’s natural resource trade. The drivers of these shocks are short-term in nature, mostly tied to movements in international prices for soft commodities and fossil fuels and mineral resources or to trade policy decisions. The analysis considers the current state and historical time trends of the country’s trade in natural resources. These include renewable natural resources and services (obtained as the difference between the Ecological Footprint of consumption and of production) but also fossil fuels as well as ores and minerals in order to provide a complete overview of the country’s natural resource situation. The second category assesses risks tied to environmental degradation occurring as a result of resource overuse. Specifically, it examines the pattern of renewable resource production in the country in the context of the limits set by that country’s biocapacity. Overharvesting of renewable resources will over time lead to declines in the productive capacity of land and marine areas. The third category of risk is related to the build-up of anthropogenic carbon emissions in the atmosphere and is assessed through the carbon component of the country’s Ecological Footprint. 3.2.2

Economic significance of risk

Not all countries are equally exposed to resource risks. The economic and social structure of countries plays a large part in determining the magnitude of the impact of risks that have been identified in the first component. In particular, this component explores the trade structure of the economy to assess the economic significance of trade-related risks (exposure to resource price volatility and to supply disruption). It also examines the country’s economy in terms of production. The relative importance of agriculture in terms of value added and employment is, for

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example, an indication of how exposed the economy might be to falling productivity, resulting from overharvesting of resources. The resource-intensiveness of industry is also included in this evaluation. On the carbon front, the methodology seeks to evaluate the possible impacts of carbon pricing on the economy.

3.2.3

Financial resilience to risk

The last component of the analysis is similar to traditional credit risk analysis as undertaken by credit rating agencies and other financial institutions. Essentially, the aim is to assess through a few key macroeconomic indicators how resilient the economy of the country may be in the face of a resource shock as described in the first two components. A country already experiencing economic difficulties resulting in high debt levels or imbalances in trade and government budgets will have a less margin of manoeuvre to respond to external shocks. Inflation is also considered as it may be directly influenced by the movement of commodity prices, especially if the country’s natural resource trade is heavily imbalanced.

4.

Applying the framework

The results presented in Figure 3 allow individual users to select or weight these indicators in order to arrive at a rating allowing them to incorporate this information in their risk analysis. To date, the framework has been applied in full to five countries: Brazil, France, India, Japan, and Turkey. These countries were chosen based on consensus of the participating banks and investors with the aim to provide a varied and representative panel of environmental and financial risk profiles. While it is not the aim of this article to focus on the country results, but rather the development and significance of the framework, its initial application does reveal an interesting set of contrasting resource risks (UNEP-FI & Global Footprint Network 2012) as shall briefly be elucidated through the cases of Turkey and India. Turkey demands nearly twice as much biocapacity as it has available (UNEP-FI & Global Footprint Network 2012). Its increasing dependence on imported resources to meet this growing gap has increased its vulnerability to trade-related risks. Since agriculture remains economically important in terms of output, exports and employment, Turkey is highly exposed to risks

Figure 3. Rating the four dimensions of risk for the five case study countries. The horizontal axis shows the grading of the four resource risk dimensions, based on a total set of 20 indicators (see Table 1) whereby each indicator receives a score of between −2 (more exposure to risk) and +2 (less exposure to risk) for each country (UNEP-FI & Global Footprint Network 2012).

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linked to its worsening levels of environmental degradation (UNEP-FI & Global Footprint Network 2012). A manageable sovereign debt level coupled with a balanced government budget gives the country considerable resilience to external shocks, which, however, is threatened by an important trade deficit and high inflation. While India’s per capita Footprint remains relatively low, its demand for renewable natural resources and services is 1.8 times greater than its ability to provide them. This ecological deficit has mainly been maintained by an overuse of its domestic ecological assets resulting in widespread degradation (deforestation, and soil erosion are prime examples) (UNEP-FI & Global Footprint Network 2012). This puts agricultural production at risk, a big problem as the sector employs over half of the population and accounts for a quarter of output. India’s growing reliance on imported fossil fuels also makes it vulnerable to price movements, a vulnerability that is compounded by a large fuel subsidy programme. In macroeconomic terms, sovereign debt levels are still moderate but high budget and trade deficits combined with persistent inflation will make managing resource shocks a challenge (UNEP-FI & Global Footprint Network 2012). As can be seen from these two countries, the type and magnitude of resource-related risks varies considerably from country to country. Nonetheless, even though the results are only preliminary, the examination of the five countries listed above has highlighted a number of notable findings: (1) The framework captures the wide variety of resource profiles that countries possess. Countries are not equal in their resource endowments and neither are they in resource management or in the resource-intensity of consumption. Some countries, such as Brazil, still consume less natural resources and services than it can generate. Others such as India or Japan use more relying either on overharvesting of resources (in India’s case) or on imports (in Japan’s) to make up the difference. The framework captures these differences and their consequences in terms of risk exposure. (2) The framework demonstrates the materiality, or significance, of natural resource related risks for countries’ macroeconomic performance. It notably establishes how the growing reliance on imports to meet demand for natural resources and services can translate into exposure to commodity price volatility with potentially severe economic consequences. One of the metrics generated by the framework is a comparative measure of the exposure of countries to resource price volatility. A simulation of a 10% change in commodity prices shows changes in countries’ trade balance equivalent to between 0.2% and 0.5% of GDP. This is far from negligible, especially given the large and sudden price movements observed on commodity markets in the last decade. Potential effects are even larger when risks linked to environmental degradation resulting from overharvesting are taken into consideration. Looking again at effects on the trade balance of countries, the framework simulates a 10% fall in productive capacity of land and marine areas. Provided that the consumption level of countries is maintained and that the shortfall in production is compensated through imports, the impact on the trade balance is equivalent to between 1% and 4% of GDP. These magnitudes of environmental risks are potentially significant enough to affect the sovereign credit risk. (3) Contrary to the popularly held belief in much of the financial sector, the results of the framework show that environmental risks are not only a long-term concern. The risks on the trade side can materialise very suddenly as a result of price movements or policy decisions such as export restrictions. Even the risks that are linked with more cumulative drivers can, however, also materialise very suddenly. Ecosystem collapse or extreme climatic events aggravated by climate change may indeed result from a long-term

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overburdening of ecosystems’ capacity to provide resources or services, yet, these phenomena are subject to important threshold effects and other nonlinear reactions that make them highly unpredictable (Liu et al. 2007). By using a probabilistic model based on past trends as well as current situations, the E-RISC framework generates results that are relevant even in short time-frames of a few years.

5. Implications The E-RISC framework directly identifies and quantifies linkages between environmental risks (rather than values of intangible services) and macroeconomic factors that are already recognised as relevant to sovereign credit risk (rather than stand-alone sustainability factors). To date, environmental impacts on macroeconomic performance have largely been ignored in credit risk analysis or they have been analysed in a very limited scope (e.g. taking carbon emissions as a proxy for the full suite of environmental risk) (AXA Investment Managers 2013). By analysing macroeconomic indicators through the lens of environmental risk, the E-RISC framework could allow different market participants (CRAs, investors, bond issuers) to better link these environmental risk indicators to the metrics that they already use in their credit risk or country risk assessments. Expressing environmental risks in terms of potential impacts on trade balance, for example, makes the results more intelligible to country risk analysts and facilitates integration of the information. The incentive to integrate such risks into mainstream assessment is given more urgency by worsening global resource scarcity. Humanity already consumes over one and a half times more natural resources and services than the planet is able to regenerate (Borucke et al. 2013; WWF and Global Footprint Network 2012), the risks to economic performance linked with poor natural resource management will increase along with rising global consumption and population levels. By embedding environmental risks in sovereign credit ratings and in the selection and weighting of sovereign bonds, governments that issue debt are given a signal to better account for the economic and financial importance of its natural capital. The protection of a country’s natural capital has historically been the domain of the Ministry of Environment. However, if the Ministry of Finance or Economy would more precisely understand how natural capital underpins the growth and security of sectors that are relevant to economic growth, it could lead to a stronger argument for the protection of the environment for social and financial reasons in addition to ecological ones. The need for improved assessment of sovereign credit risk is also increasing due to recent evolutions in the financial regulations on capital adequacy requirements for banks (Basel III, Basel Committee on Banking Supervision) and insurers (Solvency II, Directive 2009/138/EC). New regulations on capital adequacy require banks to strengthen their Tier 1 capital ratios and minimise over-leverage through risky assets. According to Basel III, holding triple-A rated sovereign debt would not require an adjustment in a bank’s core capital ratio to make up for extra risk, and it would, therefore, be seen as adequately capitalised. Thus, by categorising triple-A rated sovereign debt as ‘risk-free’, Basel III is encouraging or even requiring banks to hold an increased level of these securities as part of the holdings in their investment portfolio. This has led many investors to worry, in light of the recent downgrades and potential defaults in the Eurozone crisis (van Duyn and Milne 2011; Kowsmann 2012) that new regulations are in fact leading to the next potential asset bubble. It is, therefore, becoming increasingly important for investors to understand how and in what sense triple-A rated countries may differ, with natural resource risks being an increasingly important factor in developing and deepening that understanding.

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Highly-rated sovereign bonds have traditionally been considered safe securities and a reliable and, in most cases, risk-free investment. However, as the ramifications of recent sovereign downgrades reverberate, greater attention should be paid to adequacy and comprehensiveness of information used to assess sovereign credit risk. Even though the current turmoil in the sovereign bond market did not originate from environmental constraints, the potential materiality of a number of growing trends currently not taken into account in sovereign credit risk assessment: rising commodity prices, increasing pressure on natural resources, growing populations, increasing consumption and climate change impacts. Fixed income in general, and sovereign bond credit analysis in particular, must catch up with other investment areas (e.g. equities market) to develop the requisite metrics and frameworks to account for these forms of emerging and intensifying risks. However, it should be added that greater progress in accounting for natural resource and environmental risks is required across the finance sector. Rating agencies use many factors to determine a nation’s creditworthiness. However, tentative results from the proposed framework in this article suggest that these criteria alone are insufficient. Current risk frameworks used to assess the exposure of financial products to local, national and global risks must, therefore, better reflect both the resource constraints and the interconnected global economy, characteristic of the twenty-first century. The framework presented here provides an initial insight into how factors such as resource prices, ecosystem degradation, and future climate change policies can impact national economies. It also provides a sense of how these criteria can be factored into sovereign credit risk models and hence in the selection and weighting of sovereign bonds and sovereign credit ratings. The framework presented is only a start and significant work remains in the development of the methods, metrics and application to a greater range of countries. However, it should serve as an important example of how research on environmental risks and natural capital could be framed to more adequately reflect the needs of investors. This is vital to not only develop more systemic approaches to risk assessment in the finance sector, but by doing so, also better support the drive towards more sustainable economies. In this regard, deeper engagement between environmental science faculties and business schools or finance institutes could complement practitioner-led approaches such as E-RISC. Furthermore, lessons learnt and results from such a framework extend beyond sovereign credit analysis for fixed-income investments. It also has relevance to other investment categories such as project finance, trade finance, and development finance, as well as insurance and re-insurance. An improved sense of short-, medium- and long-term country risk can contextualise a wide variety of investments and lending portfolios across different nations.

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Accounting and Business Research, 2014 Vol. 44, No. 3, 260 –279, http://dx.doi.org/10.1080/00014788.2014.883062

Transfer pricing as a tax compliance risk SVEN P. JOSTa,b, MICHAEL PFAFFERMAYRa,c and HANNES WINNERc,d∗ a

Department of Economics and Statistics, University of Innsbruck, Universitaetsstrasse 15, A-6020 Innsbruck, Austria; bInternational Tax Services – Transfer Pricing, Ernst & Young LLP, 8484 Westpark Drive, McLean, VA 22102, USA; cAustrian Institute of Economic Research (WIFO), Universitaetsstrasse 15, A-6020 Innsbruck, Austria; dDepartment of Economics and Social Sciences, University of Salzburg, Residenzplatz 9, A-5010 Salzburg, Austria This paper studies the role of transfer pricing as a critical compliance issue. Specifically, we analyse whether and to what extent the perceived risk associated with transfer pricing responds to country-, industry- and firm-specific characteristics. Empirically, transfer pricing risk awareness is measured as a professional assessment reported by the person with ultimate responsibility for transfer pricing in their company. Based on a unique global survey conducted by a Big 4 accounting firm in 2007 and 2008, we estimate the number of firms reporting transfer pricing being the largest risk issue with regard to subsequent tax payments. We find that transfer pricing risk awareness depends on variables accounting for general tax and transfer pricing specific strategies, the types and characteristics of intercompany transactions the multinational firms are involved in, their individual transfer pricing compliance efforts and resources dedicated to transfer pricing matters. Keywords: international taxation; multinational firms; transfer pricing

1. Introduction This paper deals with compliance issues raised by the transfer pricing behaviour of multinational entities (MNEs). In contrast to the majority of (analytical) tax research focusing on the effects of tax rate differentials and tax competition on transfer pricing within MNEs (i.e. shifting income from high-tax countries to affiliates located in low-tax countries), we analyse whether and to what extent the perceived risk associated with such activities (i.e. the transfer pricing risk awareness) is affected by common tax considerations (e.g. experience from previous tax audits) and transfer pricing specific determinants. Taking a stronger compliance perspective on transfer pricing has not only been emphasised by tax authorities, tax practitioners and public administrations (see, e.g. OECD 2013a,b) but also by the management accounting literature (Cools and Emmanuel 2007 provide a comprehensive survey).1 ∗

Corresponding author. Email: [email protected]

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To quantify the role of country-, industry-, and firm-specific characteristics on transfer pricing risk awareness of MNEs, we rely on a unique cross-sectional survey of more than 350 MNEs around the world conducted by a Big 4 accounting firm in 2007 –2008, in which the person with ultimate responsibility for transfer pricing matters was interviewed, i.e. in most cases the Chief Financial Officer (CFO) or tax director (details on the survey are provided in Section 3.1). Our dependent variable – the number of MNEs in a given country and industry that report transfer pricing to be a substantive risk issue – is derived from a categorical survey variable varying from the ‘largest risk issue’ over ‘a risk issue but not the largest’ to ‘not a risk issue at all’. This variable measures how the presence of transfer pricing opportunities and/or the actual transfer pricing choices an MNE makes may affect the assessment of the likelihood of being subject to tax audits in any given country. Our covariates include variables accounting for (i) the general tax and transfer pricing specific strategies, (ii) the types and characteristics of intercompany transactions the MNEs are involved in, and (iii) the individual transfer pricing compliance efforts and resources dedicated to transfer pricing matters within the MNEs. This information, together with the professional assessment of risks associated with transfer pricing, enables us to provide new insights into the transfer pricing compliance practice of MNEs. Our empirical findings suggest that the number of MNEs in a country and industry considering transfer pricing the largest risk issue is positively related to specific determinants of transfer pricing, such as the materiality of intangible goods transactions and the compliance approach. Furthermore, we observe that the set of variables used to account for common tax considerations, and in particular previous tax audit experience, also affects significantly the perception of transfer pricing as a corporate risk issue. This, in turn, suggests that compliance considerations – apart from tax optimisation strategies – should be accounted for not only in empirical work but also in theoretical transfer pricing models. The paper is organised in the following way. In Section 2, we briefly review the related literature. Section 3 explains the technique for estimating the expected number of MNEs reporting a high transfer pricing risk awareness in an industry in a given country. It further summarises the data and discusses the variables used in the empirical model. There, we also present and discuss the empirical findings. Section 4 concludes. 2. Related previous research Transfer prices are charged for transactions between firms that are part of an affiliated group of legal entities. They are typically utilised to approach optimal decision-making in decentralised firms, which often implies to maximise overall profits (Eccles 1985, Borkowski 1990, 1996). However, transfer pricing is not only intended to provide performance incentives for managers and divisions, but also opens a wide range of opportunities for re-allocating profits between legal entities, thereby reducing tax payments. Especially, MNEs are inclined to transfer pricing manipulation as they are able to exploit international tax rate differentials. The regulatory framework to determine transfer prices within an MNE is based on international tax law. In particular, Article 9 of the Organisation for Economic Cooperation and Development (OECD) Model Tax Convention states that intercompany transactions should be charged with a price equal to the one that would has been charged between unrelated firms (arm’s length principle).2 How to calculate such prices is stipulated in the OECD Transfer Pricing Guidelines (OECD 2010), and the choice of the corresponding method depends on the circumstances and characteristics of the underlying transaction. From a firm perspective, there is in fact some leeway for setting transfer prices but also a substantial risk of paying penalties and losing reputation in case of non-compliance (see, e.g. Wright 2007). It is therefore less surprising that MNEs

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typically pay strong attention to tax compliance issues today when setting transfer prices (see, e.g. Cools and Emmanuel 2007). Our study investigates whether and to what extent transfer pricing risk awareness is influenced by country-, industry-, and firm-specific characteristics. Basically, it belongs to three different strands of research, which will be discussed briefly now. First, we relate to the management accounting and control literature, analysing tools and techniques of motivating selfish managers to contribute to an organisation’s strategies and goals (Anthony 1988, Anthony and Govindarajan 2007). In this regard, recent research emphasises the decisive role of transfer pricing tax compliance on the design and use of management control systems within MNEs. Cools et al. (2008) were the first to investigate this issue. They found that the gains from tax compliant transfer pricing (i.e. using the same values for managerial and tax purposes3) have to be balanced with the disadvantages of losing managerial flexibility and motivation (see also Cools and Slagmulder 2009, focusing on responsibility accounting).4 In a similar vein, Rossing and Rohde (2010) observed that MNEs adjust their (overhead) cost allocation systems after implementing a transfer pricing tax compliance strategy. Both papers are based on detailed case studies within one MNE, and they provide valuable insights into management and cost accounting practices under different situational contexts. Perhaps most important from our perspective, their findings suggest that MNEs change their compliance strategy in response to the risk of being audited.5 Managers seem to recognise that transfer pricing decisions are inherently associated with considerable risks, which in turn affects their transfer pricing strategies. Our study, though not directly related to the management accounting and control literature, is based on this observation and tries to complement the existing case study evidence with a thorough empirical analysis on the key drivers of transfer pricing risk awareness using a global survey of large MNEs (Ernst & Young 2008). Second, our paper might be of interest from a risk management perspective discussing how executives might deal with different types of risks (Mikes 2009, 2011). Recently, Kaplan and Mikes (2012) proposed a distinction between risks that are (i) preventable (risks arise from within the firm and are controllable or ought to be eliminated), (ii) strategy (risks that are accepted to gain a strategic advantage), or (iii) external (risks arise from outside the firm and are not controllable). According to the authors, only preventable risks can be efficiently addressed by a ruleor compliance-based approach (see also Simons 1999, for an earlier contribution). However, whether risks associated with transfer pricing belong to the class of preventable or even strategy risks remains an open question.6 Third, we draw on the tax accounting and public economics literature which, one the one hand, deals with the government response to transfer pricing and the corresponding welfare effects (Kant 1990, Haufler and Schjelderup 2000, or Raimondos-Møller and Scharf 2002). On the other hand, much of the empirical work has been devoted to optimal firm strategies and the question whether evidence could be found that support income shifting behaviour from high- to low-tax countries as suggested by the game theory (Hines 1997, Devereux 2007 provide comprehensive surveys). The findings from these studies suggest that MNEs use transfer prices as a means of profit maximisation. Unfortunately, transfer pricing is non-transparent by nature,7 so that it can be only observed indirectly assessing whether firms in low-tax countries are more profitable than ones in high-tax economies or whether economic activity varies across locations. Most frequently used proxies are reported profits of MNEs in high- and lowtax countries (Grubert and Mutti 1991), tax liability as a fraction of sales or assets in a hightax country (Harris et al. 1993), or foreign direct investment in high- and low-tax countries (Hines and Rice 1994). Notice that most of the existing research restricts the analysis of transfer pricing matters to North America (Clausing 2009 provides an overview). Only few studies focus on European data, such as Oyelere and Emmanuel (1998), Huizinga and Laeven (2008), Egger

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et al. (2010), or Mura et al. (2013). In contrast to these studies, Clausing (2003) and Bernard et al. (2006) rely on country-level intercompany prices from US trade data. However, using such aggregate data makes it impossible to estimate firm-specific factors that determine the transfer pricing behaviour of individual MNEs (e.g. income under-reporting, the transfer pricing method selection or transfer pricing risk management). Furthermore, it is also ignored that, according to the OECD Transfer Pricing Guidelines (see OECD 2010), transfer prices cannot be set equal to the marginal cost of production, but must be set according to the functions performed, risks borne and assets employed by the transaction parties. This paper aims to provide new insights to transfer pricing as a tax compliance issue looking at the risk sensitivity of different industries across a large number of developed countries. Thereby, we rely on a unique survey which contains in-depth data on numerous transfer pricing aspects obtained from MNEs around the world. Furthermore, given that our results are based on a professional judgement by a key tax person in each MNE, we address both tax authorities and MNEs. Tax professionals as well as tax authorities observed that the complexity of new regulations is increasing steadily and, likewise, are the intercompany transactions taking place within an MNE. This being said, there needs to be a mutual understanding what the key drivers are behind transfer pricing (risk) management to efficiently manage the transfer pricing compliance burden.

3.

Empirical analysis

3.1. Data description and choice of the dependent variable We rely on a data set of more than 350 MNEs (parent companies) in 24 markets (countries) across 12 different industries that has been surveyed in Ernst & Young’s Global Transfer Pricing Survey 2007 – 2008 (henceforth ‘the Survey’). The design of the Survey has been developed by Ernst & Young’s transfer pricing professionals. It has been conducted by telephone interview. Interviews were carried out with the person with ultimate responsibility for tax policy and strategy in each MNE (Ernst & Young 2008).8 To preserve confidentiality, the firm-level data were aggregated, i.e. records were summed for each country and industry combination in which MNEs operate. Overall, 12 industry classifications were incorporated. Hence, the observational unit is the country and industry dimension. The Survey has been made available exclusively to us in this aggregated dimension and it forms the raw data of our sample.9 The subsequent analysis is based on own calculations using these data, and it deviates strongly from the report of Ernst & Young (2008) in that it provides a more disaggregated and thorough analysis of one particular item of the questionnaire. In particular, as our aim is to estimate the main factors behind the perceived risk of transfer pricing as a compliance issue of MNEs, we refer to a question measuring how the presence of transfer pricing opportunities and/or the actual transfer pricing choices an MNE makes may affect its assessment of the likelihood of being subject to tax audits in any given country. Let us discuss the nature of the response variable used in the empirical analysis in a first step, which is derived from a categorical variable restricted to three levels. The person with ultimate responsibility for transfer pricing in an MNE was asked to evaluate the following question: To what extent do you consider transfer pricing a risk issue with regard to severe subsequent tax payments and penalties?

Respondents could choose from (i) the largest risk issue,

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S.P. Jost et al. (ii) a risk issue but not the largest, and (iii) not a risk issue.

For the purpose of this analysis and to limit the loss of valuable information, the dependent variable is defined as the number of MNEs in a specific country and industry that reported transfer pricing to be the largest risk issue in the aforementioned survey question. 3.2. Estimation approach Given the nature and distribution of the dependent variable, we use a count data model to estimate the relationship between the response variable and the set of explanatory variables. For variables with low expected counts (in absolute terms), as in our case, this is regularly more suitable than using a standard regression model. As we do not find strong evidence for overdispersion (see summary statistics in Table 3 and the test statistics reported in Table 6) and the Vuong test rejects the existence of zero inflation, we fit a non-inflated Poisson count data model. Hence, let the number of tax directors considering transfer pricing the largest risk issue within their group be represented by mij for each observed combination of country i and industry j which is generated under a Poisson regression model. Thereby, each observation, i.e. the MNEs in each country and industry, is allowed to have a different value of m. Our basic specification reads as y

e−mij mijij Pr (yij |m) = , yij !

for

y = 0, 1, 2, . . . ,

(1)

where mij = E(yij |x) = exp (xi b1 + xij b2 ). yij defines the outcome variable; xi is a column vector of country-specific explanatory variables; xij is a vector of country-/industry-specific regressors such as transfer pricing specific determinants obtained from the survey. b1 and b2 represent vectors of structural parameters. In our specification, we include country- and industry-specific control variables that are mainly motivated by the existing literature on transfer pricing (see Tables 3 and 4 for summary statistics and the definition of all variables, respectively). As mentioned above, it is often believed that MNEs manipulate transfer prices such that income is taxed in low-tax countries.10 Therefore, we include the statutory corporate tax rate of the parent country. Furthermore, we include additional information on the existence of country-specific statutory transfer pricing regulations along with penalty regimes and the time of introduction of domestic transfer pricing regulations. We will take these two factors into consideration as we believe that an MNE will most likely align its transfer pricing practice to the nature of domestic regulations and the behaviour of domestic tax authorities. The remaining set of explanatory variables is derived from the survey of MNEs around the world (also see Section 3.3 for more information) and can be broadly divided into three sets of regressors. The first set controls for drivers of general tax considerations of MNEs. These strategic components are likely to provide general insights into a company’s tax focus areas given its previous experience and future expectations. The second set concerns decisive information about the materiality and nature of intercompany transactions. More specifically, intangible transactions are relatively complex in terms of transfer pricing. Therefore, traditional transfer pricing methods (i.e. the comparable uncontrolled price method, the resale price method, and the cost plus method) can usually not be applied and tax authorities are particularly sceptical about whether and to what extent intangible transactions comply with the arm’s length principle. The third set of independent variables controls for transfer pricing compliance practices and resource allocation.

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Descriptive statistics

Table 1 provides the basic descriptives of the data set. Overall, our sample originally included 368 firms that reported their individual transfer pricing risk awareness. However, to preserve confidentiality, we had to aggregate firm-level information such that the final data set was based on a country and industry dimension. Therefore, the data set could have potentially had 288 observations (i.e. 24 countries and 12 industries). Due to missing observations in the explanatory variables and the fact that respondents were not operating in all possible countries and industries, we use 137 observations in our basic model specification. With regard to the distribution of firms, almost all industries are represented once in the USA. By contrast, firms from only one industry are observed in China and New Zealand. Consumer product firms are the most important group in our data set, asset management companies are only present in three countries. In our empirical model, we account for this pattern of unbalanced distribution (see below). Figure 1 illustrates the transfer pricing risk awareness by parent country. As can be seen, the difference in reported risk perception could hardly be starker. Interestingly, it is the Anglo-Saxon countries, such as Australia, New Zealand, the UK, and the USA that are particularly aware of transfer pricing as a risky compliance issue; almost half of all parent companies located in the UK consider transfer pricing to be the largest risk issue within their group. From a macro-level perspective, a potential reason might be the relatively early incorporation of transfer pricing regulations into law in these countries; Australia and the USA were the two first countries that issued detailed transfer pricing legislation. Consequently, one might expect that MNEs located in these countries as well as tax authorities are relatively experienced with transfer pricing issues (Table 2). Tax aspects, albeit less dominant from a compliance perspective, might also play a role as it is particularly high-tax countries that are seated in the top group (e.g. Germany and the USA). At the other end of the table, non-OECD countries such as Brazil prevail, with no firms perceiving transfer pricing as the largest corporate risk issue. Similarly, albeit not very well represented, China is also located at the lower end of the table. One of the pioneer countries, China incorporated transfer pricing regulations into law in 1991. However, as a developing country, the transfer pricing system in China is still in an elementary stage and has many problems, such as the lack of well-trained transfer pricing expertise and the shortage of experience in handling sophisticated transfer pricing investigations. This particularly applies to Chinese tax authorities who audit only a small proportion of MNEs per year. Likewise, the transfer pricing risk awareness differs significantly among industries (Figure 2). MNEs operating in the pharmaceutical and telecommunications industries seem particularly exposed to and aware of transfer pricing risks. One reason might lie in the unprecedented GlaxoSmithKline case, in which the Internal Revenue Service (IRS) in the USA imposed a penalty of Table 1. Selected sample characteristics. Total number of firms surveyed Total number of countries Total number of industries Total number of observations Average number of industries per country Average number of MNEs per industry Most diverse country Least diverse country Most represented industry Least represented industry

368 24 12 137 6 31 USA (10) China (1), New Zealand (1) Consumer products (23) Asset management (3)

Notes: The table provides selected sample characteristics of the data set. Reference is also made to Table 2 which shows the 12 subordinate parent industries that were collapsed to 5 industry classifications.

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Figure 1. Perceived risk awareness by parent country.

US$3.2 billion to the pharmaceutical giant due to allegedly wilful manipulation of transfer prices (Sikka and Wilmott 2010). In complying with transfer pricing regulations, MNEs operating in the telecommunication business face the problem of documenting intercompany service transactions, which are particularly difficult to document as the costs and benefits have to be adequately allocated between a service provider and a service receiver within the group. Furthermore, the allocation of research and development expenses (also relevant to pharmaceutical companies) also poses a great challenge to these companies. At the other end of the table, one can find financial services companies (although this classification does not cover banks). Having just become ‘popular’ to tax authorities, financial services transfer pricing requires an in-depth knowledge of transfer pricing techniques. Let us briefly turn our attention to the summary statistics reported in Table 3. On average, we observe about one MNE per country and industry that considers transfer pricing the largest risk issue in our sample, corresponding to slightly more than 21%.11 This share increases to 22.1% considering the 51 observations that are located in pioneer countries (these values are not reported in Table 3). On the contrary, the average transfer pricing risk awareness decreases to 20.6%

Table 2. Location and industry classification of MNEs. Commodities

Finance

Resource

Pharma

Telco/media

Year Banking of TP and reg. Consumer Real Asset capital Oil and Media and intro. Automotive products estate Utilities management markets Insurance gas Biotechnology Pharmaceuticals entertainment Telecommunication Total 1999 1995

0 0

7 4

0 4

0 1

1 0

2 3

0 0

2 5

1 0

2 1

0 2

0 1

15 21

1999 1997 1999 1998 2003 2007 1996 2003 2001 2000 2001 1996

0 3 2 0 1 0 3 5 2 0 2 6 6

2 3 3 0 5 8 7 10 1 5 7 2 8

0 0 0 0 0 0 0 0 0 1 0 0 0

1 0 0 0 0 1 4 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

3 1 0 0 3 1 2 0 0 0 2 0 3

0 1 1 0 0 0 1 0 1 0 1 1 0

0 3 4 1 1 0 0 0 1 0 0 3 2

0 0 1 0 2 1 0 0 0 0 0 0 0

1 0 0 0 2 0 0 1 0 0 3 2 0

0 0 1 0 1 1 1 0 0 0 0 0 1

2 0 0 0 1 0 0 0 1 1 0 1 1

9 11 12 1 16 12 18 16 6 7 15 15 21

1996 2001

0 1

9 7

1 2

1 1

1 0

3 2

1 1

0 1

1 0

0 0

1 0

1 0

19 15

1996

0

7

0

0

0

0

0

0

0

0

0

0

7

2007 2006 2006

1 2 1 0 1 5 41

2 3 4 6 12 17 139

1 0 1 0 0 0 10

0 3 0 1 0 3 16

1 0 0 0 0 0 3

0 1 4 2 5 5 42

0 0 0 0 2 2 12

4 1 1 0 2 5 36

0 0 0 0 0 5 11

1 0 0 3 1 4 21

1 0 2 0 6 3 20

1 0 1 0 2 4 17

12 10 14 12 31 53 368

1999 1994

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Notes: Missing entries in the first column indicate that this country does not have incorporated statutory transfer pricing regulations into law. [H] indicates a high-risk country, [P] indicates a pioneer country, and ‘TP’ stands for transfer pricing.

Accounting and Business Research

Argentina [H] Australia [H] [P] Belgium Brazil [P] Canada [H] China [H] [P] Denmark [H] Finland France [P] Germany [H] India [H] Ireland Italy [P] Japan Rep. of Korea [H] [P] Mexico [H] [P] The Netherlands New Zealand [P] Norway [H] Spain [H] Sweden Switzerland UK[H] USA [H] [P] Total

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Figure 2. Perceived risk awareness by the parent industry classification.

considering the firms located in countries that are not among the first ones incorporating transfer pricing regulations into law.12 We can see that the average statutory corporate tax rate in our sample is 31.3%, with 18.3% of the respondents located in a low-tax country and 21.9% located in a high-tax country.13 It is also worth mentioning that approximately 60% of all Table 3. Descriptive statistics. Variable

Mean

SD

Min.

Max.

Dependent variables TP risk awareness (rel.) TP risk awareness (abs.)

0.212 0.628

0.34 1.163

0 0

1 8

Country parameters Pioneer country [D] Statutory tax rate

0.372 0.313

0.485 0.056

0 0.125

1 0.407

Survey parameters Tax audit performed Management responsibility T/N subject to customs Reliance on audit firm Resources increased Intangible T/N Documentation prepared IC agreements in place

0.434 0.458 0.469 0.928 0.655 0.758 0.909 0.692

0.426 0.436 0.423 0.193 0.386 0.227 0.244 0.374

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

Observations

137

Notes: The table provides summary statistics of all relevant variables used in the final model. All survey parameters are expressed as shares. Variables are defined as in Table 4. [D] indicates a dummy variable, ‘IC’ means intercompany, ‘TP’ stands for transfer pricing, and ‘T/N’ stands for transaction.

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Table 4. Definition of variables. Country parameters Statutory tax rate High-risk country [D]

Pioneer country [D] Survey variables Tax audit performed Management responsibility T/N subject to customs Reliance on audit firm Resources increased Intangible T/N Documentation prepared IC agreements in place Number of firms (log)

Definition Statutory corporate tax rate in the parent country of the MNE in 2007. Source: KPMG (2008) Dummy variable indicating that a country has statutory transfer pricing regulations (e.g. legal documentation requirements, guidance on the application of transfer pricing methods, etc.) and penalty regimes in case of non-compliance [1 ¼ yes, 0 ¼ no]. Source: Ernst & Young’s audit risk tool, interviews with Ernst & Young transfer pricing professionals Dummy variable indicating that a country was among the top 10 countries that introduced statutory transfer pricing regulations [1 ¼ yes, 0 ¼ no] Share of MNEs in a CIC that have undergone a transfer pricing audit since 2003 Share of MNEs in a CIC reporting that CFO is ultimately responsible for transfer pricing Share of MNEs in a CIC reporting that IC transactions are subject to customs or other import duties Share of MNEs in a CIC reporting increased reliance on audit firm with regard to transfer pricing advice Share of MNEs in a CIC reporting that need for transfer pricing resourced has increased in the last three years Share of MNEs in a CIC reporting that intangible transactions are ‘material’ or ‘significant’ Share of MNEs in a CIC that have prepared a transfer pricing documentation in the past Share of MNEs in a CIC reporting that IC transactions are adequately covered by IC agreements Natural logarithm of the number of firms in a observed country and industry

Notes: [D] indicates a dummy variable, ‘CIC’ means country and industry classification, ‘IC’ stands for intercompany, and ‘T/N’ stands for transaction.

MNEs are located in high-risk countries, which are characterised by statutory transfer pricing documentation requirements and stringent penalty regimes in the case of non-compliance with domestic transfer pricing rules. With respect to the set of explanatory variables derived from the Survey of MNEs, we find that, on average, 43.4% of MNEs have undergone a transfer pricing audit in the last years. The share of MNEs preparing a transfer pricing documentation is fairly high (i.e. above 90%); however, this number includes MNEs that prepared a documentation on an as-necessary basis with limited or no coordination between affiliated companies. At the same time, a relatively high share of MNEs supplements their transfer pricing documentation with legally binding intercompany agreements that rule the terms and conditions of intercompany transactions. It is also striking that almost half of all observations report that the CFO or the audit committee is ultimately responsible for transfer pricing within their group, which underlines the practical importance of this tax issue. It is rather not surprising that about two-thirds of MNEs have increased resources devoted to transfer pricing over the last years. Reference is made to Table 4 which provides the definition of all variables. Table 5 reports the correlation matrix. 3.4.

Empirical findings

This section connects the empirical findings of the existing literature to common transfer pricing practices of MNEs around the world using the aforementioned characteristics and determinants obtained from the Survey. Table 6 presents the empirical results of our count data model. Our

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Table 5.

Correlation matrix. (2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

1.000 0.050 0.156 0.153 0.305 20.029 0.272 20.281 0.166 0.065 0.004 20.044 0.543

1.000 0.096 0.153 20.113 20.059 20.004 0.038 0.115 0.231 0.019 0.010 0.033

1.000 0.228 20.020 20.198 20.049 20.040 20.152 20.016 20.104 20.035 0.200

1.000 0.057 20.205 0.151 20.051 20.056 20.123 20.076 20.150 0.203

1.000 20.132 0.181 20.234 0.261 0.236 0.074 0.219 0.245

1.000 0.078 0.154 0.042 20.125 20.150 0.041 20.086

1.000 20.032 0.029 0.049 20.324 0.145 0.220

1.000 20.030 0.096 0.104 0.072 20.106

1.000 0.119 0.037 0.107 0.032

1.000 0.122 0.173 0.058

1.000 0.088 20.045

1.000 20.063

1.000

Notes: [D] indicates a dummy variable.

S.P. Jost et al.

(1) TP risk awareness (2) High-risk country [D] (3) Pioneer country [D] (4) Statutory tax rate (5) Tax audit performed (6) Management responsibility (7) T/N subject to customs (8) Reliance on audit firm (9) Resources increased (10) Documentation prepared (11) Intangible T/N (12) IC agreements in place (13) Number of firms (log)

(1)

Table 6.

Estimation results.

Country parameters Pioneer country [D] Statutory tax rate [1]

(2)

(3)

0.456 (0.244)∗ [0.170]∗∗∗ 21.294 (1.702) [1.235]

0.547 (0.27)∗∗ [0.221]∗∗

0.383 (0.215)∗ [0.150]∗∗ 2.041 (3.659) [3.651]

0.930 (0.316)∗∗∗ [0.353]∗∗∗ 0.302 (0.219) [0.207]

3.125 (1.665)∗ [1.976] 1.003 (0.510)∗∗ [0.572]∗ 25.216 (4.498) [5.051] 21.031 (0.583)∗ [0.699] 1.098 (0.381)∗∗∗ [0.340]∗∗∗ 20.949 (0.420)∗∗ [0.368]∗∗∗ 1.005 (0.395)∗∗ [0.382]∗∗∗ 1.562 (0.763)∗∗ [0.805]∗

1.320 (0.484)∗∗∗ [0.586]∗∗ 0.972 (0.501)∗ [0.519]∗

1.035 (0.375)∗∗∗ [0.351]∗∗∗ 20.996 (0.414)∗∗ [0.368]∗∗∗ 1.038 (0.403)∗∗∗ [0.378]∗∗∗ 1.252 (0.746)∗ [0.804] 20.587 (0.494) [0.640] 20.794 (0.380)∗∗ [0.341]∗∗ 1.104 (0.157)∗∗∗ [0.137]∗∗∗ 137 15.154∗∗∗ 0.82 0.398 229.4 276.1

20.804 (0.369)∗∗ [0.336]∗∗ 1.072 (0.147)∗∗∗ [0.134]∗∗∗ 137 30.283∗∗∗ 0.39 0.404 231.7 284.3

20.889 (0.580) [0.626] 1.067 (0.378)∗∗∗ [0.351]∗∗∗ 21.116 (0.439)∗∗ [0.379]∗∗∗ 0.919 (0.392)∗∗ [0.387]∗∗ 1.244 (0.684)∗ [0.787] 20.539 (0.468) [0.568] 20.740 (0.325)∗∗ [0.311]∗∗ 1.040 (0.148)∗∗∗ [0.135]∗∗∗ 137 15.041∗∗∗ 0.74 0.400 227.0 270.8

Accounting and Business Research

Survey parameters Tax audit performed [2] Management responsibility [3] [1] × [2] [2] × [3] T/N subject to customs Reliance on audit form Resources increased Intangible T/N Documentation prepared IC agreements in place Number of firms (log) Observations Industry effects (F-statistic) Vuong test (Z-statistic) McFadden’s R2 Akaike information criterion Bayesian information criterion

(1)

Notes: The table provides the results from the Poisson regression model. All variables defined as in Table 4. [D] indicates a dummy variable. ‘IC’ stands for intercompany, ‘TP’ stands for transfer pricing, and ‘T/N’ means transaction. White (1980) robust standard errors are in parentheses and country clustered standard errors in brackets. ∗ Significance at 10% level. ∗∗ Significance at 5% level. ∗∗∗ Significance at 1% level.

271

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Figure 3. Observed and predicted counts (firms in an industry/country cell that consider transfer pricing all the largest risk issue).

dependent variable is defined as the number of MNEs in a given country and industry cell reporting transfer pricing as the largest risk issue. All 3 models include 11 industry dummies based on the parent industry classification. The first column presents the basic model specification which contains two country-specific parameters as well as a set of transfer pricing specific determinants obtained from the Survey. It also includes the statutory corporate tax rate to account for larger tax compliance risks in high-tax countries. In the second specification, we add two interaction effects between the share of MNEs with previous tax audit experience and (i) the statutory corporate tax rate and (ii) management responsibility for transfer pricing matters. This specification tests whether the impact of previous audit experiences depends on the tax levels and the management responsibilities in the respective country. The difference between columns two and three lies in the exclusion of all non-significant explanatory variables to obtain a parsimonious model (with a significance level of less than 10%). In all specifications, we report both the results of robust (White 1980) and clustered standard errors. This allows us to investigate whether the level of aggregation gives rise to the presence of country-wise clustered errors, which are typically present in studies in which observations are randomly sampled but the explanatory variables are measured at a different aggregate level than the dependent variable. More specifically, our data may induce clustering of errors at the country level; that is, the aggregated firms in our industries are correlated in some unknown way within countries, but different countries do not have correlated errors. From Table 1, we know that the respondents/firms are not equally distributed across countries and industries. We accounted for this difference in representativeness by including the number of firms in a country and industry cell in our model. This ensures that our results and, hence, the coefficients of the explanatory variables are not driven solely by countries with a high number of observations (e.g. the USA). In general, the model seems well specified. The control variables have the expected sign, the industry effects are significant, and the pseudo-R2 is relatively high.14 As mentioned earlier, the summary statistics of the left-hand side does not show any signs of overdispersion or excess zeros.

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This is also confirmed by the insignificant Vuong (1989) test statistic as well as a graphical comparison between observed and predicted counts as illustrated in Figure 3. Within the realm of transfer pricing, there are almost exclusively structural reasons for firms to assess their transfer pricing practice as highly risky, and there is no reasonable indication that reporting transfer pricing as a high risk is a matter of chance. Hence, there is no basis for a zero-inflated model. We find that the time of introduction of transfer pricing regulations matters, i.e. the pioneer country variable is weakly significant at the 10% level. As mentioned before, this result is quite intuitive and less surprising from a practitioner’s point of view: relatively speaking, both MNEs and tax authorities in pioneer countries tend to be relatively experienced in transfer pricing matters and statutory transfer pricing documentation requirements have developed into detailed guidelines for MNEs in these countries (the first country that established statutory transfer pricing regulations was the USA in 1994; see also Table 3 for the time of introduction of transfer pricing regulation per country). The above-mentioned management accounting literature has drawn a similar conclusion. In particular, Cools et al. (2008, p. 614), in their case study of transfer pricing tax compliance within a large worldwide operating MNE, cite a corporate tax director who stated that at the time where the corporate tax department recognised the strengthened US initiatives on transfer pricing: . . . we did not actively oversee the transfer pricing policy in the product divisions, nor did we have a written version of their policy. Until 1993 there were a number of general transfer pricing principles, but the parties involved had some freedom to negotiate transfer prices . . . From 1993 on, the fulfilment of the compliance requirements became the primary goal for Corporate Tax.

Furthermore, transfer pricing audits might have also become more sophisticated and MNEs have become relatively experienced in preparing their annual transfer pricing compliance work. The significance of industry fixed effects tells us that the awareness of transfer pricing, as the largest corporate risk issue differs across industries. For pharmaceutical companies this might be quite obvious as GlaxoSmithKline was involved in an unprecedented tax dispute with the IRS, which might have sustainably affected the tax authorities’ attention towards intercompany transactions involving research and development activities and intangibles as such. Our results also indicate a non-significant coefficient for the statutory corporate tax rate. Against the background of the existing research on transfer pricing, this result may seem intriguing at first sight. However, there is a reasonable explanation of this finding: The left-hand side is a subjective variable highly correlated with perceptions. As such, it primarily reflects the CFO’s previous experiences as the ultimate person responsible for transfer pricing issues as well as their expectation in the future. Hence, resources and compliance efforts will most likely be allocated to countries with high-risk exposure, irrelevant of the corporate tax rate (i.e. tax minimisation aspects are less important from a compliance perspective). Our evidence is also consistent with the above-mentioned case studies from the management control literature suggesting that MNEs increased efforts to implement fiscally acceptable transfer prices as a response to tighter transfer pricing regulations in the early 1990s (Cools et al. 2008, p. 614). With respect to our variables of interest, we find evidence that most of the transfer pricing specific determinants derived from the Survey significantly affect the risk awareness of MNEs. First and foremost, we find that previous transfer pricing audit experience increases the number of MNEs perceiving transfer pricing as the largest risk issue. MNEs seem to learn from their previous experience with competent authorities (and vice versa). This results may also be considered in line with the existing literature on audit selection; see, for example, Erard and Feinstein (1994). Relative to the other coefficients, this effect also proves to be among the strongest (see Table 7 for a discussion of marginal effects). We also observe that

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Table 7. Marginal effects. Selected parameters

(1)

(2)

(3)

Pioneer country [D]

0.124∗ 0.110∗ 0.237∗∗∗ 0.177∗∗∗ 0.077 0.058 0.263∗∗∗ 0.197∗∗∗ 20.253∗∗ 20.190∗∗ 0.264∗∗ 0.198∗∗ 0.319 0.239 20.202∗∗ 20.151∗∗

0.146∗ 0.176∗ 0.768∗∗ 0.754∗ 0.246∗∗ 0.242∗ 0.270∗∗∗ 0.265∗∗∗ 20.233∗∗ 20.229∗∗ 0.247∗∗ 0.242∗∗ 0.384∗∗ 0.377∗ 20.198∗∗ 20.194∗∗∗

0.104∗ 0.116 0.339∗∗∗ 0.330∗∗ 0.250∗∗ 0.243∗ 0.274∗∗∗ 0.266∗∗∗ 20.287∗∗∗ 20.279∗∗ 0.236∗∗ 0.230∗ 0.320∗ 0.311 20.190∗∗ 20.185∗∗

Tax audit performed Management responsibility T/N subject to customs Reliance on audit firm Resources increased Intangible T/N IC agreements in place

Notes: The table reports the marginal effect of a selection of relevant variables. Marginal effects presented in the first (second) row are evaluated at the mean (median) of all other explanatory variables. A discrete change from 0 to 1 is assumed for dummy variables, which is indicated by a [D]. All variables are defined as in Table 4. ‘IC’ stands for intercompany, and ‘T/N’ is the abbreviation for transaction. ∗ Significance at 10% level. ∗∗ Significance at 5% level. ∗∗∗ Significance at 1% level.

the share of MNEs with management responsibility for transfer pricing matters positively affects the transfer pricing risk awareness in a given country and industry.15 The two interaction terms between the tax audit experience and the statutory corporate tax rate and management responsibility show negative coefficients. That is, if the share of MNEs with previous tax audit experience increases, it has a strong positive effect on the number of firms in a country and industry perceiving transfer pricing as the largest risk issue, which is, however, reduced with an increasing number of firms with management responsibility for transfer pricing issues (or with an increasing statutory corporate tax rate). One potential explanation is that firms with previous tax audit experiences and management responsibility have already reacted to previous findings for the CFO has taken over responsibility, thereby reducing their risk exposure. From another perspective, in the case of an increasing share of firms with management responsibility, the positive effect of previous tax audit experience is almost cancelled out by the interaction term. That is, if management has taken over the ultimate responsibility for transfer pricing, a previous tax audit experience does not seem to play a role in the awareness of transfer pricing as the largest risk issue. With only one exception for the second interaction effect, however, the coefficients are insignificant. Interestingly, the reliance on their auditor for transfer pricing advice negatively affects the transfer pricing risk sensitivity. This might have two explanations: first, a Channel 1 firm (i.e. an MNE that is both audited and provided tax advice by the same auditing company) will most likely not receive different opinions on the appropriateness of their internal pricing system from their audit team and their counterparts at the consulting arm of an auditor. Hence, if the same company audits a company’s books that has provided advice in the management of transfer prices, the risk awareness is reported significantly lower. Second, a Channel 2 firm (i.e. an MNE that is provided advice in other fields than transfer pricing by an auditing company, but not audited by this company) might be trusting in a long-term relationship with

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its auditor, who may have been providing audit-related consulting services in the past. It is also less surprising to see the significant positive effect of an increase in resources devoted to transfer pricing matters affect the risk awareness. This might indicate that the compliance efforts dedicated to transfer pricing matters in the past proved to be insufficient. As might have been expected, we further find that the share of MNEs reporting material intangible goods transactions significantly increases the number of MNEs considering transfer pricing the largest risk issue in a given country and industry. Intercompany transactions involving intangible goods, such as royalties for the licensing of brands or service payments for the provision of cash pooling services, are particularly difficult to price (i.e. the application of standard transfer pricing methods is hardly possible) and regularly trigger a tax authority’s scrutiny. The last two variables of interest are the share of MNEs preparing a transfer pricing documentation and the share of MNEs that cover their intercompany transactions with intercompany agreements. Both variables concern a company’s attempt to comply with local transfer pricing regulations and we expect at least one of them to significantly affect the number of MNEs considering transfer pricing a high risk. The former variable is insignificant throughout all specifications. The latter, however, enters our model significantly at a level of 5%. This might suggest that the implementation of intercompany agreements (i.e. a legally binding document) is more relevant than the mere preparation of a transfer pricing documentation. By nature, a transfer pricing documentation is prepared ex post and, hence, does relatively little in explaining a company’s effort to comply with the arm’s length principle – compared to a binding group-wide transfer pricing guideline or the conclusion of intercompany agreements that both rule the terms and conditions applied to intercompany transactions ex ante (see Cools et al. 2008, for a related case study evidence). Nevertheless, the significance of the variable capturing the share of MNEs that have implemented intercompany agreements suggests that such compliance efforts taken by an MNE significantly reduces the number of MNEs considering transfer pricing the largest risk issue. Finally, Table 7 reports the corresponding marginal effects of the tax considerations and the other survey parameters for the three models presented in Table 6. Except for the two variables controlling for previous tax audit experience and management responsibility the size of the marginal effects are relatively stable throughout the different model specifications.16 For instance, taking the specification in column three, the marginal effect of previous tax audit experience evaluated at the mean of all other variables is around 0.339, and about 0.330 for a firm in a given country and industry with median transfer pricing characteristics – besides the existence of intangible property transactions the strongest effect. Accordingly, a change in the share of MNEs with previous tax audit experience of 10 percentage points in a given country and industry is associated with an increase in the number of MNEs considering transfer pricing the largest risk issue by about 3.4%. Given the seemingly ever-increasing role of transfer pricing as a tax (compliance) issue, we believe that the share of MNEs that will have experienced a tax audit with focus on transfer pricing will also increase significantly over the next years. Assuming that this share will increase to 75% (90%) in the years ahead, while holding all other variables at their mean, the likelihood of at least one firm considering transfer pricing to be the largest risk issue in a given country and industry increases by 9.6 (15.2) percentage points to 32.3% (37.8%). 4.

Conclusions

Motivated by the existing literature on transfer pricing, we ask whether and to what extent the risk sensitivity of MNEs for transfer pricing issues is influenced by firm-specific determinants and the corresponding tax environment. For this purpose, we use unique data obtained from a global survey on the transfer pricing risk perception of large MNEs compiled by a Big 4 accounting firm. More specifically, we use the professional judgement by the key person with ultimate

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responsibility for transfer pricing in an MNE (in most cases, the CFO or tax director) as our dependent variable to analyse the awareness of transfer pricing as a corporate risk issue. The estimation results provide strong evidence that the number of MNEs considering transfer pricing their largest risk issue is systematically affected by their previous tax audit experiences. In addition, we observe significant effects of typical transfer pricing-related determinants, such as the nature of intercompany transactions or the use of intercompany agreements to appropriate cover intercompany transactions. Our estimation results also suggest that corporate statutory tax rates are not the principal determinant in explaining the risk perception of transfer pricing. Apparently, the procedural risk of tax compliance as perceived by the responding MNEs seems to play a more crucial role than the level of statutory corporate tax rates. Our findings may not only be interesting from a tax policy perspective (e.g. harsher and more frequent transfer pricing audits would increase tax compliance), but also for the (further) development of theories in management accounting and public economics. First, our results seem to be in line with the management accounting literature which has shown that managers are aware of the risks associated with non-compliant transfer pricing. This, in turn, affected a company’s compliance strategies such that in many cases unified transfer prices for tax and (management) incentive purposes were implemented as a result of such risks. However, the management control literature has demonstrated that this is not always beneficial for the overall performance of or within a firm. Furthermore, recent research has shown that rule-based compliance strategies are not necessarily warranted from a risk management perspective (Kaplan and Mikes 2012). This is especially the case if risks are almost uncontrollable by a firm’s management. Whether risks from transfer pricing belongs to these types of risks is not analysed so far, and might be answered in future research. From a public economics perspective, there is no theoretical contribution on transfer pricing that explicitly accounts for the compliance risks of such activities. Linking the existing theory on transfer pricing closer to the (traditional) tax compliance literature might improve our understanding of profit shifting activities of MNEs.

Acknowledgements We are grateful to the editor (Vivien Beattie), an anonymous referee and Simon Loretz, Harald Oberhofer as well as seminar participants at the University of Innsbruck for helpful comments and discussions. Furthermore, we would like to thank Ernst & Young for making the raw data of its transfer pricing survey available to us. We declare that there is no conflict of interest regarding the material discussed in the manuscript.

Funding Financial support from the Austrian Fonds zur Fo¨rderung der wissenschaftlichen Forschung (FWF, grant no. P 17028-G05) and the Jubilaeumsfonds der Oesterreichischen National Bank (OeNB, project no. 12459) is gratefully acknowledged.

Notes 1.

Among others, reference is made to the increasing number of (public) hearings by tax authorities around the world, such as those involving Her Majesty’s Revenue and Customs (HMRC) in the UK or the IRS in the USA on transfer pricing practices of MNEs. Much of this recent attention on transfer pricing compliance resulted from the UK Public Accounts Committee’s (PAC’s) annual review of HMRC’s accounts, which led to the public questioning of representatives from Starbucks, Amazon, and Google. The PAC’s strongly worded report, released in early December 2012, described the current situation MNEs proclaim to comply with local regulations as ‘outrageous’. It also called for a change in mindset at HMRC, which must be ‘more aggressive in policing and prosecuting companies that paid too little tax’ and ‘be seen to challenge practices to prevent the abuse of transfer pricing, royalty payments, intellectual property pricing and interest payments’ (see PAC 2012).

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The OECD (2010) encourages all member countries to follow its transfer pricing guidelines in their domestic tax practices. Taxpayers are encouraged to follow these guidelines in evaluating for tax purposes whether their transfer pricing complies with the arm’s length principle. Since the first published version of 1995, a vast majority of all member states has incorporated into law the fundamental aspects of the OECD Transfer Pricing Guidelines. Over the last decade or so, over 60 countries have adopted transfer pricing rules based on the framework published by the OECD. While the tax accounting and economics literature discussed below mainly focuses on a unified set of transfer prices, recent analytic research in accounting analysed distinctive transfer prices for alternative purposes (Smith 2002, Baldenius et al. 2004, or Hyde and Choe 2005). In practice, MNEs usually rely on non-differentiated transfer prices, mainly for reasons of simplicity and in order to avoid any disputes with tax authorities (Ernst & Young 2008, Cools and Slagmulder 2009). In practice, smaller companies tend to have one set of books that serves both management accounting and tax accounting purposes to keep administrative efforts at a minimum. Bigger MNEs with substantial headcount in corporate finance and tax departments may prefer to keep separate books if a local manager’s key performance measures are inherently linked to transfer pricing. Rossing and Rohde (2010, p. 213), for instance, cite a corporate tax manager who stated that . . . [t]his is not about control – it is about minimizing the risk from audit adjustments to transfer prices . . . we basically sat down and asked ourselves: where are the biggest risks and what are the vulnerable areas that tax authorities will go for.

6.

7.

8.

9. 10.

11.

From a methodological point of view, transfer pricing compliance itself might be viewed as an endogenous rather than an exogenous variable (Cools and Emmanuel 2007, Jegers 2010). Strictly speaking, there is no such thing as the ‘right’ transfer price. In practice and in particular upon audit, the facts and circumstances surrounding an intercompany transaction affecting the determination of a reasonable transfer price as well as the underlying method are regularly discussed in length and in detail with an uncertain outcome. See, e.g. OECD (2012). There is generally no reliable source of information about transfer prices on a firm level and, more obviously, taxpayers are generally very reluctant to publish sensitive data about their transfer pricing behaviour as tax authorities have become more aggressive and public scepticism about transfer pricing as a means of income tax manipulation is prevalent. The far-reaching international tax proposals unveiled by the Obama administration on 4 May 2009 carry broad implications for MNEs, not only in the significant limits they would place on income deferral but also in major changes to the foreign tax credit and new restrictions on the use of disregarded entities, practitioners told the Bureau of National Affairs. More specifically, the 4 May document reads that they will ‘eliminate loopholes for disappearing offshore subsidiaries and crack down on foreign tax havens’. Generally, the Survey included a country’s largest MNEs and relied on the following selection criteria: Firms with a global ultimate owner in the USA or Canada (Asian countries) needed a revenue of at least US$500 million (US$250 million) and had to held subsidiaries in at least two continents outside North America (Asia). If the list of companies fulfilling these criteria was short, it was completed with the next largest firms within those countries. Firms with global ultimate owners in Europe (Latin America, Australia, or New Zealand) should have subsidiaries in at least five other countries worldwide (two other continents), i.e. there were no revenue criteria here. Again, if the list of companies felt short, it was replenished with companies holding subsidiaries in at least three other countries (only one other continent). Ernst & Young does not make the data available publicly, but gratefully gave us a special data access for research purposes. Furthermore, it should be noticed that all of our empirical results shown in the tables and figures represent original work and are not published elsewhere. There is no reliable source of firm-specific data on actual transfer prices and transfer pricing behaviour such as the number of transfer pricing adjustments per year. The data and model specifications used in the existing literature on transfer pricing thus only allow for implicit conclusions on the link between income shifting behaviour and transfer prices. Harris et al. (1993), for example, find that US tax liability is related to the location of foreign subsidiaries, suggesting that income is shifted by means of manipulation of transfer prices. Similar implicit conclusions are drawn by Grubert and Mutti (1991) and Huizinga and Laeven (2008). The difference between the mean and the variance of our dependent variable is relatively low; the variance (1.35) is just twice as large as the mean. The overall distribution of our dependent variable is not displaying signs of over-dispersion, that is, we do not observe a significantly greater variance than

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12.

13. 14. 15.

16.

S.P. Jost et al. might be expected from a Poisson distribution (Figure 3). Furthermore, we also observe very large standard errors on the coefficients in the inflation equation which further implies a definite lack of fit in a zero-inflated model. ‘Pioneer’ countries are among the top 10 countries worldwide introducing statutory transfer pricing regulations. Compared to all other countries, pioneer countries are believed to be relatively experienced in transfer pricing matters and their tax authorities are believed to follow a rather sophisticated approach in challenging a taxpayer’s transfer pricing system. We defined low-tax countries as countries with a corporate tax rate lower than the first quartile of tax rates of the full sample. Similarly, high-tax countries are located within the upper quartile of the tax rate distribution. Several pseudo-R2 for count models have been defined by analogy to the R2 in the linear regression model. We only report McFadden’s pseudo-R2 (see, e.g. Long 1997). This finding seems to be in line with practitioners’ experience that the level of ‘C-suite support/awareness’ (as it is often referred to in practice) for transfer pricing has become increasingly important. It has been observed that companies tend to hire senior tax directors with multiple years of international tax (compliance) experience and able enough to stir management through the forests of worldwide tax regulations. At the same time, the C-suite is increasingly involved in tax risk management decisions as failure to comply with regulations can have a significant detrimental effect on the image of a company. Hence, transfer pricing risk management is one of the aspects that continues to be on the rise on managements’ agenda. See Groysberg et al. (2011). The increase in the size of these two variables is predominantly driven by the inclusion of interaction effects in the second model specification.

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The European Journal of Finance, 2014 Vol. 20, No. 1, 80–103, http://dx.doi.org/10.1080/1351847X.2012.681791

Transparency, idiosyncratic risk, and convertible bonds Yi-Mien Lina∗ , Chin-Fang Chaob,c and Chih-Liang Liud a Department of Accounting, National Chung Hsing University, Taichung, Taiwan; b Department of Finance, Ling Tung University, Taichung, Taiwan; c Feng Chia University, Taichung, Taiwan; d Department of Industrial and Business Management, Chang Gung University, Taoyuan, Taiwan

(Received 23 March 2011; final version received 29 March 2012) We first investigate the relationship among a company’s information transparency, idiosyncratic risk, and return of its convertible bonds. The effects of a company’s idiosyncratic risk on its equity’s value volatility and its credit risk are also examined. The findings indicate that when a company discloses a significant amount of information, it is likely to have a higher idiosyncratic risk and a lower credit risk, with no impact on returns on convertible bonds. The volatility of stock returns is positively related to returns on convertible bonds, and it is found that diversified strategies and returns on a company’s equity help to improve its credit rating and that a better credit rating triggers an increase in returns on convertible bonds and idiosyncratic risk, indicating that evaluations of the value of convertible bonds must take pure bonds and equity (option) values into account. After excluding conversion values and estimating the idiosyncratic risk on daily, weekly, and monthly bases, this study suggests that there is a positive relation between returns on convertible bonds and information transparency when estimating idiosyncratic risk on a monthly basis and that a positive association also exists between credit rating, idiosyncratic risk, and returns on bonds.

Keywords: convertible bonds; information transparency; idiosyncratic risk; credit risk

1. Introduction Recent studies have suggested that idiosyncratic risk is based on a firm’s operation dimension, meaning that managers can manipulate it, which suggests that a company capable of controlling its idiosyncratic risk can increase its influence on stock returns. This article, therefore, contends that idiosyncratic risk may increase or decrease in the period following the issuance of convertible debt. The relationship between the issuance of convertible bonds and the issuer’s risk has not been well established in the literature. This paper contributes to the literature by examining how idiosyncratic risk affects returns on convertible bonds and the credit rating of the issuer and how transparency is associated with idiosyncratic risk and returns on convertible bonds. Furthermore, we also isolate the bond price whilst excluding its conversion value to obtain the estimated bond return with the conversion feature stripped out. Taiwan’s bond market has adopted the dual systems of exchange and over-the-counter (OTC) trading whereby convertible bonds are listed and traded in the exchange trading system, and government bonds and corporate bonds are listed and traded in the OTC trading system. Government bonds are quoted and offset in the Electronic Bond Trading System, which was developed by the GreTai Securities Market (GTSM) in 2001. In 2003, the GTSM established the Fixed

∗ Corresponding

author. Email: [email protected]

© 2012 Taylor & Francis

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Income Securities Trading System, which provides auction and bargaining mechanisms for corporate bonds and financial debentures.1 Convertible bonds issued by listed and OTC companies are required to be sold by underwriters using only a firm-commitment underwriting method, and companies whose stocks are emerging, unlisted or not traded in brokerage houses are not allowed to issue convertible bonds to the public. Moreover, the issuance of convertible bonds does not require that the bonds have a credit rating in Taiwan; thus, investors themselves must evaluate the issuer’s credit risk.2 In recent years, the Financial Supervisory Commission has actively pushed government bonds to join global bond indices in Taiwan. In 2005 and 2006, the government successfully introduced Taiwan’s government bond index to Citibank and Lehman Brothers and attracted international bond businesses and pension funds to invest in Taiwan’s bond market. The trading volume of foreign capital on bonds increased in Taiwan such that Taiwan’s bond market became more internationalised with improved international visibility. These changes promoted Taiwan’s domestic bond market system and information transparency. Healy, Hutton, and Palepu (1999) show that companies with higher information transparency (HT) have higher rates of return on stocks than do companies with lower information transparency (LT), even after controlling for earning performance, risk, growth factors, and company size. Companies with HT receive more attention from institutional investors and analysts than do companies with LT and experience less volatility of stock returns from investment uncertainties. Alexander, Edwards, and Ferri (2000) find that high-yield corporate bonds with HT also have high liquidity, while Edwards, Harris, and Piwowar (2007) find that companies that publicise more of their information have lower transaction costs, as do companies with HT on their bond prices. Using the information transparency perspective, we investigate whether the performance of Taiwan’s bond market is positively correlated with the information transparency of a company, as its stock market is. Morck, Yeung, and Yu (2000) find that idiosyncratic variance in stock returns is lower in emerging markets but higher for individual companies in the developed market. In explaining horizontal variances in stock returns, Malkiel and Xu (2003) suggest that a company’s idiosyncratic volatility is stronger than beta risk or company size.3 Bello (2005) uses idiosyncratic risk to observe returns on mutual funds and finds that a non-linear correlation exists; after controlling for the beta of the portfolio, returns on mutual funds have a positive linear correlation with fund size and a negative linear correlation with the P/B ratio. Clearly, much of the literature has addressed the relationship between idiosyncratic risk and financial instruments of equity securities; however, correlations between convertible bonds and idiosyncratic risk have not yet been thoroughly investigated. This paper attempts to examine the impact of idiosyncratic risk on the rate of return on bonds. Because of the effect of endogenous variables on information transparency, credit risk and idiosyncratic risk (i.e. the interrelationship among these variables), we use simultaneous equation models to examine their relationships. Our samples are selected from listed and OTC companies that issued convertible bonds during the period of 2002–2007 and whose bonds were listed on the Information Disclosure and Transparency Ranking System of the Securities and Futures Commission (SFC). Based on criteria from the SFC’s Information Disclosure and Transparency Ranking System announced in 2008, we adopt manually collected data to develop scores for the level of information transparency to evaluate the selected sample companies. The findings indicate that information transparency is positively related to idiosyncratic risk and to a company’s credit rating but is unrelated to returns on convertible bonds. When a company discloses a significant amount of information, its idiosyncratic risk is higher, whereas less disclosure results in less idiosyncratic risk. Idiosyncratic risk significantly impacts returns on

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convertible bonds. The volatility of stock returns is positively related to returns on convertible bonds, indicating that when evaluating convertible bonds, both pure bonds and option (conversion) value must be considered. We also find that a company’s credit rating (as a negative indicator of credit risk) is positively correlated with idiosyncratic risk, which means that when a better credit rating signals less credit risk, more idiosyncratic variance in the company exits. A diversified strategy and returns on equity also positively affect the company’s credit rating. Credit risk is positively related to returns on convertible bonds and the volatility of stock returns. Accordingly, when a company’s credit rating degrades, its credit risk explicitly becomes higher, and returns on convertible bonds decline. We also measure the returns on convertible bonds with the estimated pure bond return, excluding its conversion value. The evidence indicates that returns on convertible bonds without conversion value are positively related to information transparency under monthly-based idiosyncratic risk but are unrelated to transparency under weekly- and daily-based idiosyncratic risk. A company’s idiosyncratic risk is positively associated with information transparency and returns on convertible bonds without conversion value, and returns on convertible bonds without a conversion feature are positively related to the volatility of the company’s stock returns and its credit risk. A negative relationship exists between idiosyncratic risk and credit risk. Theoretically, the presence of a call option in the convertible bond should have an impact on the company’s share price. We further analyse how the beta and the returns correspondingly change before and after the issuance of convertible bonds. The findings show that stock returns significantly increase before and after the issuance of convertible bonds, implying that the issuance of convertible bonds is positive for investors. Moreover, the beta increases before the issuance of convertible bonds and significantly decreases after their issuance; thus, issuance of convertible bonds implies that the company has better prospects for investors such that systematic risk may decrease. The remainder of this paper is organised as follows. Section 2 reviews the related literature and establishes the research hypotheses. Section 3 describes the research design. Section 4 provides empirical results. Conclusions and suggestions are discussed in Section 5. 2. Literature review and hypotheses For companies with growth opportunities, convertible bonds are a costless financial instrument with which to raise funds (Mayers 1998) and reduce the problems of over-investment and asset replacement.4 Green (1984) shows that companies can reduce agency costs generated between creditors and shareholders by issuing convertible bonds. When the equity market is hot, companies are inclined to raise capital by issuing convertible bonds (Alexander, Stover, and Kuhnau 1979) because their convertibility to common shares increases investors’ interest (Billingsley and Smith 1996; Mann, Moore, and Ramanlal 1999). Brennan and Schwartz (1988) observe that compared to newly issued stock, convertible bonds are less influenced by company risk: when company risk increases, default risk will increase, and bond prices will usually dip (Merton 1974), but the value of convertible options will also increase because of asset volatility (Ingersoll 1977a, 1977b). These two counteracting forces result in convertible bonds’ value being less influenced by company risk. Another perspective comes from Stein’s (1992) Backdoor Equity theory, which suggests that companies can use convertible bonds as indirect financial instruments to prevent the problem of reversal options when they are raising capital by issuing equity. When designing its convertible bond, the company normally requires that it be redeemed before the next

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investing opportunity is offered, so that the company can recover its debt capacity when the convertible bonds are finally converted into common shares and can then re-finance its next investment project (Korkeamaki and Moore 2004). This argument is consistent with Mayers’ (1998) sequential-financing theory.5 According to financial theories, issuing a convertible bond can reduce interest costs and prevent the dilution of equity. When a company announces an issue of convertible bonds, the company’s future is viewed positively, and a signalling effect should occur. Stein (1992) finds that the stock prices for companies that announce issuances of convertible bonds are better than those of companies that announce issuances of common shares. However, some researchers suggest that a convertible bond announcement will have a negative effect, such as a drop in stock prices (Mikkelson and Partch 1986; Lee and Loughran 1988). Lee and Loughran suggest that this negative effect occurs because of the perception that convertible bonds are issued when managers believe that the firm’s stock prices are overestimated, thus causing them to issue convertible bonds to maximise current shareholders’ wealth. Karpoff and Lee (1991) and Kahle (2000) expressed a similar point of view in observing that insiders in companies that issue convertible bonds dispose their shareholdings before the issue because they think their companies’ stock prices are overestimated. However, the decision to issue convertible bonds is not only made by issuers; the opinions of potential investors on convertible bonds are also significant. As a result, what determines the decision to invest in convertible bonds is a significant issue for both buyers and sellers. Part of the value of a convertible bond is its conversion value because when stock prices inflate, the bond’s conversion value inflates as well. HT helps the value of potential stocks, as does the quality of convertibility. Preventing asymmetric information between shareholders and bondholders can help lower agency costs and makes investors more willing to buy the company’s bonds. Healy, Hutton, and Palepu (1999) find that companies with HT had higher stock prices than did companies with LT, even after controlling for the HT companies’ earning performance, risk, growth, and size. The companies with HT were better at attracting institutional traders’ and analysts’ attention and at reducing the uncertainty of investment. Alexander, Edwards, and Ferri (2000) find that high-income bonds with HT have better liquidity, while Bessembinder, Maxwell, and Venkataraman (2006) suggest that insurance companies’ transaction costs in the bond market have dropped since the introduction of the TRACE system in 2002. Edwards, Harris, and Piwowar (2007) contend that the transaction costs of bonds from companies with HT are lower and that costs decline as the transparency of bond prices increases. Goldstein, Hotchkiss, and Sirri (2007) also find that a company’s transaction costs decline with the level of information transparency because investors have more bargaining power when they know more about bond prices. Jin and Myers (2005) indicate that the lower the information transparency, the lower the relative idiosyncratic volatility and the higher the stock returns will be. However, we suggest that if a company has HT, transaction costs will decrease, stock liquidity will increase, and the capital cost of a company may decline. If the company discloses even more information, the degree of information asymmetry and agency costs will be further reduced, and investors will be even more willing to hold their stock. Based on most articles’ evidence that companies’ stock returns are positively related to information transparency, we extend this view to the returns from convertible bonds and investigate whether the Taiwan bond market has a similar positive correlation with information transparency. We hypothesise the following: H1: A company’s level of information transparency is positively correlated with the return on convertible bonds

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Portfolio risk can be divided into systematic risk and non-systematic risk. Systematic risk cannot be avoided by changing the portfolio, while non-systematic risk can be reduced by diversifying the portfolio. Non-systematic risk is also called diversifiable risk or idiosyncratic risk. Theoretically, the value of convertible bonds consists both the value of the pure bond and the value of equity. Idiosyncratic risk can observe the changes in a company’s equity values. Campbell et al. (2001) use idiosyncratic risk analysis to calculate the average volatility of companies from 1962 to 1997 and find that the return volatility of certain companies was greater than the average market volatility and that stock’s interaction and explanatory ability of the market model gradually declines. To reduce risk, investors must diversify by holding more kinds of stocks.6 Malkiel and Xu (2003) indicate that idiosyncratic volatility is stronger than beta risk or company size in explaining the cross-sectional variances of returns on stocks. Goyal and Santa-Clara (2003) indicate that the volatility of lagged market returns is not related to expected market returns. Without considering the positive correlation of the equal-weight average volatility of market prices and the value-weight portfolio returns on portfolio, one may assume that equal-weight average volatility has better predictability in measuring idiosyncratic risk. Malkiel and Xu (2003) indicate that the volatility of individual stocks appears to have increased over time because the number of stocks traded in the market has increased. The level of individual volatility is also related to the company’s size.7 However, there is no consensus on whether idiosyncratic risk can predict a company’s equity value. Bali et al. (2005) contradict Goyal and Santa-Clara’s research, suggesting that their results are influenced by small-cap stocks traded on the NASDAQ. Because Goyal and Santa-Clara’s results are not applicable to stocks traded on the New York Stock Exchange or American Stock Exchange, they cannot prove the relationships between a portfolio’s value-added returns and the median and weighted-average volatility of stock prices. Information transparency and idiosyncratic risk may be correlated to some degree. Durnev, Morck, and Yueng (2004) state that high-idiosyncratic variance is correlated with effective asset allocation because stock prices with high-idiosyncratic variance also contain information about future earnings. Roll’s (1988) research on idiosyncratic risk finds that the relationship between idiosyncratic risk and stock prices is affected by private information but not by public information. Jin and Myers (2005) also indicate that less transparency of accounting information results in lower idiosyncratic variance. In fact, convertible bonds have the characteristics of both securities and bonds; thus, we expect that idiosyncratic risk is affected by both information transparency and returns on convertible bonds, leading to the following hypothesis: H2: The information transparency of a company is positively correlated with its idiosyncratic risk

Operational risk is the risk specific to a company, that is, idiosyncratic risk. The main difference between idiosyncratic (or non-systematic) risk and market (or systematic) risk is that managers can control idiosyncratic risk. Higher operational risk often accompanies higher bankruptcy risk. Therefore, larger-scale companies reduce bankruptcy risk by controlling operational risk to protect the interests of blockholders and retail investors. Malkiel and Xu (2003) examine cross-sectional differences in stock returns and find that the ability of idiosyncratic risk to explain stock returns is greater than the beta risk for firm size. Hence, idiosyncratic risk is a main determinant of stock returns, and if a company can control idiosyncratic risk, its managers have a greater influence on stock returns that form the counter scale effect, which leads to a positive relationship between firm size and stock returns. Morck, Yeung, and Yu (2000) find that idiosyncratic variance is lower in the emerging markets but higher in the developed market. Bello (2005) uses idiosyncratic risk to observe returns on

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mutual funds and finds that these two variables have a non-linear correlation; after controlling for the beta of the portfolio, Bello finds that returns on mutual funds have a positive linear correlation with fund size and a negative linear correlation with the P/B ratio. Ang et al. (2006) indicate that idiosyncratic volatility induces a reverse effect on stock returns, that is, higher current idiosyncratic volatility will be accompanied by higher current abnormal returns, followed by lower abnormal returns in the next period. This reverse effect of stock returns almost always focuses on small companies. Fu (2009) finds that current stock returns are positively related to firm size after controlling for idiosyncratic risk. In contrast, Lewis, Rogalski, and Seward (2002) suggest that, all else being equal, an unexpected decline in the convertible bond issuer’s systematic risk is consistent with a positive share-price reaction to the offer announcement. However, unsystematic risk significantly increases in the period following a convertible bond offering. This change is at least partially attributable to trends in the idiosyncratic risk conditions in the issuer’s industry, suggesting that increases in idiosyncratic risk more than offset decreases in systematic equity risk. While many articles have focused on idiosyncratic risk and equity securities, few have examined the relationships between returns on convertible bonds and idiosyncratic risk. Because the value of convertible bonds is positively derived from conversion rights, which are highly influenced by stock returns, we extend the positive relationship between stock returns and idiosyncratic risk to convertible bonds: H3: A company’s idiosyncratic risk is positively correlated with the returns on its convertible bonds

Part of the value of convertible bonds is attached to their conversion value, and conversion value is related to the volatility of the company’s value. When the company’s volatility increases, the impact on conversion value also increases. Ingersoll (1976) and Smith (1984) use the volatility of asset value as a significant parameter in establishing a pricing model for convertible bonds. However, the volatility of a company value is difficult to determine; thus, in both practical and academic communities, only equity values are adopted to estimate the company’s value. There are currently no consistent views about the impact of issuing bonds on equity securities. Ammann, Fehr, and Seiz (2006) and Dutordoir and Van de Gucht (2004) find that stock prices respond negatively to announcements of convertible bond issues, while Smith (1986) and Magennis, Watts, and Wright (1988) believe that the declaration of debt has no relationship with variations in stock prices and that only equity securities issues are related to a decline in stock prices. As for the timing of an issue, Billingsley, Lamy, and Smith (1990) indicate that the stock market has a clearly negative response on the date of a bond issue, but Kang and Stulz (1996) find that the Japanese market responds positively. We adopt the volatility of stock prices as a proxy for the volatility of company value and expect that the conversion value will increase with the volatility of stock prices. Expecting a positive influence from these factors on the returns of convertible bonds, we form Hypothesis 4. H4: The volatility of a company’s value is positively correlated with returns on its convertible bonds

Credit risk is the risk of potential losses resulting from a company default or changes in the credit rating (Bank for International Settlements).8 Past research has generally argued that bond prices are related to credit ratings. Jewell and Livingston (1997) state that bond ratings have a long-range, deep impact on issuers’ long-term stock returns. Mikkelson and Partch (1986) find that when companies with class-A bond ratings announce a convertible bond issue, a

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negative market reaction occurs; however, if companies with class-B or lower ratings announce a convertible bond issue, there is no response. Brennan and Schwartz (1988) suggest that companies whose investment risks are difficult to estimate are more inclined to issue convertible bonds because of the insensitivity of the prices of their convertible bonds to the risk of investment. Erb, Harvey, andViskanta (1999) find evidence of a significant negative relationship between the country risk rating and the emerging market bond spread; they also find that idiosyncratic country risk may be a determinant of emerging equity or bond market returns (Erb, Harvey, and Viskanta 1996; Erb, Harvey, and Viskanta 1999; Erb, Harvey, and Viskanta 2000). Under the assumption of a non-zero collection ratio, some studies, such as Davis and Lishka (1999), Takahashi, Kobayashi, and Nakagawa (2001) and Ayache, Forsyth, and Vetzal (2003), use credit risk to investigate the value of convertible bonds. Johnston, Markov, and Ramnath (2009) document that frequent debt reports around credit-rating downgrades also impact equity price.9 In Taiwan, it is not easy to acquire accurate bond rating information because Taiwan’s government does not require companies to reveal bond-rating information before issuing convertible bonds. We use companies’ credit rating information as a proxy for bond ratings and adopt other financial indices, such as financial leverage and return on equity (ROE), to measure credit risk. When a company’s credit rating drops and its credit risk is higher, its convertible bonds become less attractive to investors. Credit risk is also an idiosyncratic risk, and both credit risk and idiosyncratic risk may have endogenous variable problems. Thus, we consider the impact of credit risk on the returns on convertible bonds. H5: A company’s idiosyncratic risk is negatively associated with its credit risk H6: A company’s credit risk is negatively associated with returns on convertible bonds

3. Research design 3.1

Data

We collected data from Taiwanese bond markets for the period 2004–2010. Because the volatility of return rates on securities and bonds requires calculating 3 years’ standard deviation (the occurring year and its previous 2 years), the actual research period is 2002–2010. Information transparency is built in accordance with the criteria of the SFC’s Information Disclosure and Transparency Ranking System, announced in 2008 (the fifth period), which includes 5 categories and 100 items. We use hand-collected data to evaluate the sample companies in detail and distinguish their information transparency. Information for other variables was acquired from the Taiwan Economic Journal (TEJ) Co., Ltd database. Companies that lack sufficient information or that are in the insurance and financial industries are excluded from the sample, as are companies not found on the SFC’s bulletin board. 3.2

Empirical model

This paper explores the impacts of information transparency, idiosyncratic risk, volatility of stock returns, and credit risk on convertible bond returns. The endogenous variables’ problem of information transparency, credit risk, and idiosyncratic risk are also considered. To test the interactive relationships of information transparency, credit risk, and idiosyncratic risk with convertible bond

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returns, we adopt the simultaneous equations model. The empirical models are shown below. Cratingit = a0 + a1 Transpait + a2 Svolatilityit + a3 IdioRisk it + a4 Breturnit0 + a5 Maturityit + a6 Betait + a7 Couponit + a8 Treasuryit + a9 Roeit + a10 Levit + a11 Ageit s + a12 Diver it + a13 Mbit + eit ,

(1)

IdioRisk it = a0 + a1 Transpait + a2 Svolatilityit + a3 Cratingit + a4 Breturnit + a5 Betait + a6 Roeit + a7 Levit + a8 Ageit + a9 Diver it + a10 Mbit + et ,

(2)

Breturnit = a0 + a1 Transpait + a2 Svolatilityit + a3 IdioRisk it + a4 Cratingit + a5 Maturityit + a6 Betait + a7 Couponit + a8 Treasuryit + a9 Roeit + a10 Levit + a11 Ageit + a12 Diver it + a13 Mbit + et ,

(3)

where Breturnit denotes the convertible bond returns for company i at time t; Transpait denotes the transparency of financial information for company i at time t; Maturityit denotes the bond maturity for company i at time t; IdioRisk it denotes the idiosyncratic risk for company i at time t; Cratingit denotes the bond’s credit rating for company i at time t, which is used to measure the default risk; Svolatilityit denotes the volatility of stock returns for company i at time t, which is measured with the criterion of the standard deviation of the 3-year return rate; Couponit denotes the interest rate of the treasury bill contracted 3 months for company i at time t; Treasuryit denotes the spread of 2- and 10-year treasury bonds at time t; Betait denotes the beta risk for company i at time t; ROEit denotes the return on equity for company i at time t; Levit denotes the debt-equity ratio for company i at time t; Diver it denotes the diversification degree for company i at time t, which is a dummy variable; Ageit denotes the number of years that company i existed in the capital market before 2010; and Mbit denotes the ratio of market price to a stock’s book value. 3.3

Measurement of variables

Included below are the definitions and measurements of variables used in the empirical model (Table 1). 3.3.1 Bond return (Breturn) The annual return on a firm’s convertible bond i in year t is calculated as: [(Purchase price of bond − Selling price of bond) +Interest income for period of holding] Breturnit = × 100% Purchase price of bond

(4)

3.3.2 Information transparency (Transpa) In 2004, the Taiwan Securities and Futures Institute began using a firm’s information transparency as a rating criterion. To increase the validity of the rating criterion for information transparency, we use hand-collected data based on the 2008 criteria of the Information Disclosure and Transparency Rankings System of the Securities and Futures Institute to measure the transparency of a company. This rating system divides the rating indices of corporate governance factors into the following five categories: (a) compliance with the regulations and laws concerning information disclosure;

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Table 1. Definition of variables. Variables

Definition

Breturn

Return on convertible bond Breturnit = [(purchase price of bond − selling price of bond) + interest revenue from holding] ÷ purchase price of bond × 100%. Returns on convertible bonds excluding conversion value. The conversion value of convertible bonds is first calculated by the conversion premium percentage less 100% and then by returns on convertible bonds less returns of conversion right. We then further calculate the annual bond price and returns based on the formula above. Measured based on the Information Transparency and Disclosure Ranking System in Taiwan. Taiwan Corporate Credit Rating Index. Using the variance of 12-month residuals between actual and expected returns of a firm as annual idiosyncratic risk. The remaining life of bonds. The volatility of stock returns of company i, which is measured according to the criterion of standard deviation of the 3-year return rate. Interest rate of 3-month treasury bill. Interest rate spread of 2- and 10-year treasury bonds. Beta risk. Net income divided by average shareholder’s equity. Liabilities divided by shareholder’s equities. A dummy variable that equals 1 if a company has a diversified strategy and 0 otherwise. Age of a company. Ratio of stock market price to stock book value as a proxy for growth opportunity.

Breturn1

Transpa Crating Idiorisk Maturity Svolatility Coupon Treasury Beta ROE Lev Diver Age Mb

Source TEJ TEJ

Collected by hand TEJ TEJ TEJ TEJ TEJ TEJ TEJ TEJ TEJ TEJ TEJ TEJ

(b) timeliness in disclosing information; (c) disclosure of predictive financial information; (d) disclosure on annual reporting, including transparency of financial and operational information, composition of board of directors, and ownership; and (e) disclosure on enterprise website. The index is composed of 100 items. The highest point of each item is 1 point, meaning that the highest possible total point score is 100. We classify our sample firms into 1 of 10 grades in which every grade spans 10 points. Hence, the higher the grade, the better the information transparency of the firm will be. 3.3.3 Idiosyncratic risk (IdioRisk) We adopt Fama and French’s three-factor model to estimate idiosyncratic risk. Fama and French (1993) calculate abnormal returns in 12 months after the event date, i.e. Rspt − Rft = a + b1 (Rmt − Rft ) + b2 SMBt + b3 HMLt + ept , where Rspt denotes the average return (raw return), Rft denotes the return on 1-month treasury bills, Rmt denotes the return on Center for Research in Security Prices value-weighted index that represents the market index return, SMBt is returns on portfolio of large-cap stocks less returns on portfolio of small-cap stocks, and HMLt is the return on a portfolio of high book-to-market ratio stocks less the return on a portfolio of low book-to-market ratio stocks. First, the data of each company for the past 36 months are used to estimate regression coefficients, which are substituted into a three-factor model to calculate residual error between

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actual monthly firm’s return and monthly market return. Next, the monthly residual error is used to calculate the standard deviation of annual residual errors, i.e. idiosyncratic risk (or idiosyncratic volatility).10 In addition to the above calculation of monthly-based idiosyncratic risk, we also estimate (1) residual variances (idiosyncratic risk) on a daily basis for a span of 250 trading days before the convertible bonds’ issue date and (2) residual variances (idiosyncratic risk) on a weekly basis with a span of 52 weeks before the convertible bonds’ issue date. 3.3.4 Volatility of company value (Svolatility) The value of the conversion right on a convertible bond is related to the volatility of the issuing company’s value. The higher the volatility of company value, the more positive the relationship between company value and the value of the conversion right will be.11 Because the volatility of company value cannot be observed empirically, most studies have used the volatility of stock prices as a proxy for the volatility of company value. Similarly, we use the standard deviation of stock returns within 4 years (the current year and the previous 3 years) to estimate the volatility of company value. We compare the average annual returns of convertible bonds and stocks in Table 2. To observe the trend more clearly, we present the results in Figures 1 and 2. Although their graphical trends are similar, we find that the volatility of stock prices is much larger than the volatility of convertible bond returns. The possible reason is that convertible bonds are less affected by announcements of significant information. Table 2. Summary of convertible bonds return, stock return, and stock return volatility. Year Bond return Stock return Stock return volatility IdioRisk− (monthly) IdioRisk (weekly) IdioRisk (daily)

2004

2005

2006

2007

2008

2009

2010

0.121 −0.145 0.452 0.068 0.041 0.017

0.129 0.161 0.434 0.055 0.022 0.012

0.086 0.295 0.450 0.046 0.022 0.007

0.018 0.029 0.517 0.070 0.031 0.012

0.119 −0.540 0.618 0.106 0.050 0.019

0.027 1.561 1.132 0.098 0.030 0.019

0.059 0.100 1.215 0.062 0.028 0.011

2.000 1.500 1.000 Stock Return

0.500 Bond Return

0.000 2004

2005

2006

2007

2008

-0.500 -1.000

Figure 1. Return on convertible bonds vs. stock returns.

2009

2010

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0.800 Volatility of Stock Return

0.600 0.400 0.200 0.000 2004

2005

2006

2007

2008

2009

2010

Figure 2. Return on convertible bonds vs. volatility of stock returns.

3.3.5 Credit risk The measurement of credit risk includes suitable yields and bond credit ratings. For the yields, we use the interest rate of a treasury bill due within 3 months as the risk-free interest rate and beta risk as the sensitivity of firms to the market, maturity, and interest spread between 2- and 10-year treasury bonds. As for the bond credit rating, issuing convertible bonds in Taiwan does not require the company to first acquire the bond’s credit rating, so we adopt a company’s credit rating (Crating) and other financial indicators, including financial leverage (Lev) and ROE, as proxies for credit risk. The credit rating of a company comes from the Taiwan Corporate Credit Risk Index (TCRI) of the TEJ database. Based on the TCRI rating, scores are divided into 10 levels under which higher credit ratings are included in smaller numbered levels; the 10th level means the company is in financial distress. Scores are put in descending order before being placed into the model. If a company is on the best credit rating level, it has a score of 10, and if a company is on the worst credit rating level, it has a score of 1. 3.3.6 Control variables The age of a company will affect its reputation and, thus, its bond rating. Younger companies have insufficient records to refer to, so investors are less interested in their bonds. Companies that have significant growth opportunities or that are highly diversified require more debt financing, and it is likely that these companies will raise capital by issuing convertible bonds. Therefore, company’s age (Age), company’s growth opportunity (Mb) and company’s product diversification (Diver) are used as the control variables in the regression model. 4. Empirical results 4.1

Descriptive statistics

Table 3 shows the descriptive statistics of all variables for a total of 1637 firm years. The average credit rating is approximately 5.45. Thus, companies that issued convertible bonds generally had mediocre credit ratings. As for the transparency of disclosure, the mean is approximately 0.29,

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Table 3. Descriptive statistics. Variables Crating Transpa Svolatility IdioRisk (monthly) IdioRisk (weekly) IdioRisk (daily) Breturn Maturity Beta Coupon Treasury ROE Lev Age Mb Diver

Minimum

Median

Maximum

Mean

Standard deviation

1.000 0.000 0.006 0.020 0.013 0.003 −0.181 1.000 −0.215 0.530 0.005 −2.528 0.042 1.000 0.270 0.000

5.000 0.000 0.458 0.063 0.030 0.013 0.091 5.000 0.949 1.175 0.012 0.100 0.927 19.000 1.490 1.000

10.000 1.000 4.198 1.745 0.537 0.134 0.758 10.000 2.041 2.160 0.022 0.903 45.623 48.000 9.280 1.000

5.445 0.290 0.622 0.071 0.032 0.014 0.097 4.839 0.925 1.324 0.013 0.090 1.097 20.132 1.735 0.672

1.511 0.454 0.548 0.049 0.018 0.006 0.183 0.693 0.347 0.488 0.005 0.230 1.327 6.594 1.206 0.470

Notes: Total number of samples is 1637. Variable definitions: Breturn denotes return on convertible bonds; Transpa denotes the level of a company’s information transparency; Maturity is the remaining life of a bond; IdioRisk denotes idiosyncratic risk; Crating denotes a proxy for bond’s credit ranking; Svolatility denotes the volatility of stock return; Coupon denotes the treasury bill’s interest; Treasury denotes interest spread between short- and long-term treasury bonds; Beta denotes Beta risk; ROE is a ratio of net income to equity; Lev denotes leverage; Age denotes the seniority of a company; Mb is a ratio as a proxy for a company’s growth opportunity; Diver is a dummy variable for whether a company adopted a diversified strategy. We also estimate residual variances (idiosyncratic risk) on a daily basis over 250 trading days prior to the convertible bond’s issue date, and we estimate residual variances (idiosyncratic risk) on a weekly basis over 52 weeks prior to the convertible bond’s issue date.

which shows that issuers of convertible bonds disclose some transparent information. There is also a large variance in stock returns (Svolatility) in the sample, based on a standard deviation of 0.548, and the average return for the period is approximately 0.622. The averages for idiosyncratic risk (IdioRisk) on monthly, weekly, and daily bases are approximately 0.071, 0.032, and 0.014, respectively. The minimum and maximum returns on convertible bonds (Breturn) are −18.1% and 75.8%, respectively. There is a large difference in returns on convertible bonds, and the average return is approximately 9.7%. On average, the convertible bonds have 4.839 years to maturity, and the age of the issuers is over 20 years. With respect to the risk-free interest rate (Coupon) and interest spread between short- and long-term bonds (Treasury), they have lower volatility based on their minimum and maximum values. 4.2 Analysis of correlation Table 4 shows the Pearson correlation coefficients of selected variables, which indicate that idiosyncratic risk is positively associated with transparency (0.11) and is consistent with Hypothesis 2. Idiosyncratic risk also has positive associations with volatility of stock returns (0.22) and company credit rating (0.26) and is positively associated with beta risk (0.02). Idiosyncratic risk and returns on convertible bonds have a positive relationship. Based on ROE, better performance contributes to a company’s credit rating (0.49), and reactions to good news result in higher volatility of stock returns. Moreover, the firm’s age is negatively related to returns on convertible bonds and idiosyncratic risk but is positively related to transparency.

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(1) (1) Breturn (2) Transpa (3) Svolatility (4) IdioRisk (5) Crating (6) Maturity (7) Beta (8) Coupon (9) Treasury (10) ROE (11) Lev (12) Age (13) Mb (14) Diver

(2)

(3)

1.00 −0.05∗∗ 1.00 0.08∗∗∗ −0.02 1.00 0.07∗∗∗ 0.11∗∗∗ 0.22∗∗∗ 0.05∗∗ −0.23∗∗∗ 0.10∗∗∗ 0.04∗ 0.05∗∗ −0.16∗∗∗ −0.02 0.07∗∗∗ 0.28∗∗∗ 0.01 −0.05∗∗ −0.37∗∗∗ −0.07∗∗∗ 0.06∗∗ 0.33∗∗∗ 0.03 0.07∗∗∗ 0.07∗∗∗ 0.06∗∗ 0.00 −0.03 −0.20∗∗∗ 0.15∗∗∗ 0.02 0.00 0.04 0.22∗∗∗ −0.04 0.10∗∗∗ 0.02

(4)

(5)

1.00 0.26∗∗∗ −0.09∗∗∗ 0.02 −0.42∗∗∗ 0.30∗∗∗ −0.19∗∗∗ 0.02 −0.09∗∗∗ −0.07∗∗∗ −0.11∗∗∗

1.00 −0.14∗∗∗ −0.15∗∗∗ −0.06∗∗∗ 0.03 0.49∗∗∗ 0.30∗∗∗ −0.28∗∗∗ −0.19∗∗∗ −0.22∗∗∗

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13) (14)

1.00 −0.04∗ 1.00 0.15∗∗∗ −0.08∗∗∗ 1.00 −0.11∗∗∗ 0.01 −0.76∗∗∗ 1.00 0.01 0.06∗∗ 0.10∗∗∗ −0.04∗ 1.00 −0.06∗∗∗ −0.08∗∗∗ −0.02 0.00 −0.27∗∗∗ 1.00 −0.05∗∗ 0.13∗∗∗ −0.13∗∗∗ 0.06∗∗ −0.03 0.06∗∗ 1.00 −0.01 0.09 0.05∗ 0.08∗∗∗ 0.37∗∗∗ −0.12∗∗∗ −0.12∗∗∗ 1.00 0.06∗∗∗ 0.26∗∗∗ 0.01 −0.01 0.03 0.01 0.26∗∗∗ 0.01 1.00

Notes: Total number of samples is 1637. Variable definitions: Breturn denotes return on convertible bonds; Transpa denotes the level of a company’s information transparency; Maturity is the remaining life of a bond; IdioRisk denotes idiosyncratic risk; Crating denotes a proxy for bond’s credit ranking; Svolatility denotes the volatility of stock return; Coupon denotes treasury bill’s interest; Treasury denotes interest spread between short- and long-term treasury bonds; Beta denotes Beta risk; ROE is a ratio of net income to equity; Lev denotes leverage; Age denotes the seniority of a company; Mb is a ratio as a proxy for a company’s growth opportunity; Diver is a dummy variable for whether a company adopted a diversified strategy. *Significant at 10% level. **Significant at 5% level. ***Significant at 1% level.

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Table 4. Pearson correlations between variables.

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Transparency, idiosyncratic risk, and stock return volatility

The results of using a simultaneous equations model to address the problem of endogenous variables are presented in Table 5. The results of model (3) show that information transparency is unrelated to returns on convertible bonds; thus, Hypothesis 1 is not supported, regardless of whether idiosyncratic risk is measured on monthly, weekly, or daily bases. Apparently, although the nature of convertible bonds theoretically comprises the nature of pure bonds and the features of conversion options, the impact of information transparency on the returns of convertible bonds shows insignificant results, unlike the impact of information transparency on the stock returns for the same company. Because the essence of convertible bonds is still the debt contract, the fluctuation of convertible bonds is primarily driven by yield or market interest rates. Therefore, the returns on convertible bonds are less influenced by a company’s information transparency. Regarding the effect of idiosyncratic risk on returns on convertible bonds, column 3 of Table 5 shows a positive relationship. Therefore, Hypothesis 3 is supported in Taiwan’s bond market. However, according to the Block–Scholes pricing option model, the volatility of stock prices was deemed to be the most significant parameter. Ingersoll (1976) and Smith (1984) also establish their pricing model of convertible bonds with volatility of stock prices as a significant parameter. Column 3 of Table 5 shows similar results that the volatility of stock returns has a significant positive effect on the returns of convertible bonds. This finding may be attributed to the fact that the volatility of stock returns influences the conversion of convertible bonds and, in turn, influences the valuation of convertible bonds. Therefore, Hypothesis 4 is supported with the data collected from Taiwan’s bond market. Based on model (2), column 2 of Table 5 shows that companies with more information transparency have higher idiosyncratic risk, regardless of whether idiosyncratic risk is calculated on monthly, weekly, or daily bases. These findings are consistent with Jin and Myers’s (2005) results. Thus, Hypothesis 2 is verified by the above outcomes. Moreover, columns 1 and 2 of Table 5 show that using a company’s credit rating as a negative indicator for credit risk results in a positive correlation with idiosyncratic risk on monthly, weekly, and daily bases. This finding implies that when a better company’s credit rating signals less credit risk, the company has more variance in idiosyncratic risk, thereby supporting Hypothesis 5. The results for the influence of credit risk on returns on convertible bonds are presented in columns 1 and 3 of Table 5. The credit rating is positively related to returns of convertible bonds and volatility of stock returns. Thus, when a company’s credit rating degrades, its credit risk rises, and its returns on convertible bonds fall. Hence, our result supports Hypothesis 6. Both diversified strategy and company age negatively affect a company’s credit rating, but financial leverage positively affects its credit rating, suggesting that a company with a high-debt ratio or high-operating performance has low credit risk. In addition, we find that beta risk is negatively associated with the credit rating but that it has no relationship to returns on convertible bonds, possibly because beta risk is applied to the equity market and because the impact on bonds is indirect. Next, consistent with the bond pricing theory that discounting by a high yield to maturity results in a low value of bonds, both the riskfree interest rate and the spread of treasury bonds have a negative association with the returns on convertible bonds. Regarding profitability, ROE positively contributes to a company’s credit rating and negatively contributes to idiosyncratic variance but does not significantly affect returns on convertible bonds. Both returns on convertible bonds and idiosyncratic risk are negatively associated with company age but are positively related to the market-to-book ratio, suggesting that investors evaluate the bonds’ value based not only on its history but also on its future growth opportunities, and the reason is that an older company can be observed with more credit record

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Table 5. Determinants of credit rating, idiosyncratic risk, and return on convertible bonds. Crating

IdioRisk− monthly

Breturn

Model (1)

Model (2)

Model (3)

Independent variables

Coefficient (t-value)

Coefficient (t-value)

Coefficient (t-value)

Panel A Transpa Svolatility IdioRisk− monthly Breturn Crating Maturity Beta Coupon Treasury ROE/100 Lev Age Diver Mb R2 Adjusted R2 F-value

0.337∗∗∗ (5.39) 0.215∗∗∗ (3.07) 13.347∗∗∗ (14.60) 0.690∗ (1.92) – −0.302∗∗∗ (−5.88) −0.673∗∗∗ (−7.19) 0.007 (0.18) −0.156 (−1.41) 2.731∗∗∗ (15.51) 0.409∗∗∗ (8.46) −0.053∗∗∗ (−10.65) −0.225∗∗∗ (−3.48) −0.177∗∗∗ (−5.58) 0.540 0.535 119.79∗∗∗

0.003∗ (1.93) 0.022∗∗∗ (13.31) – 0.034∗∗∗ (3.43) 0.010∗∗∗ (14.74) – 0.025∗∗∗ (9.97) – – −0.019∗∗∗ (−3.55) −0.002 (−1.48) −0.000∗∗ (−2.35) −0.002 (−1.18) 0.006∗∗∗ (7.44) 0.443 0.438 105.61∗∗∗

0.002 (0.48) 0.019∗∗∗ (3.47) 0.311∗∗∗ (4.16) – 0.004∗ (1.92) 0.005 (1.38) −0.008 (−1.09) −0.040∗∗∗ (−5.85) −0.062∗∗∗ (−7.47) −0.005 (−0.36) 0.010∗∗∗ (2.67) −0.001∗ (−1.90) 0.005 (0.97) 0.007∗∗∗ (3.04) 0.092 0.084 10.42∗∗∗

Panel B Transpa Svolatility IdioRisk− weekly Breturn Crating Maturity Beta Coupon Treasury ROE/100 Lev Age Diver Mb R2 Adjusted R2 F-value

0.285∗∗∗ (4.97) 0.315∗∗∗ (5.20) 25.225∗∗∗ (13.22) 0.871∗∗∗ (2.66) – −0.293∗∗∗ (−6.23) −0.653∗∗∗ (−7.50) 0.127 (1.51) 0.079 (0.80) 2.508∗∗∗ (14.62) 0.345∗∗∗ (7.56) −0.054∗∗∗ (−11.70) −0.242∗∗∗ (−4.21) −0.197∗∗∗ (−6.58) 0.511 0.507 124.66∗∗∗

0.002∗∗∗ (3.02) 0.004∗∗∗ (5.96) – 0.016∗∗∗ (3.97) 0.004∗∗∗ (13.38) – 0.016∗∗∗ (15.32) – – −0.020∗∗∗ (−9.05) −0.002∗∗∗ (−2.74) −0.000∗∗∗ (−4.39) −0.000 (−0.51) 0.003∗∗∗ (8.20) 0.409 0.405 107.45∗∗∗

0.004 (0.96) 0.011∗∗ (2.36) 0.537∗∗∗ (3.46) – 0.005∗∗∗ (3.66) 0.006 (1.52) −0.006 (−0.90) −0.038∗∗∗ (−5.90) −0.050∗∗∗ (−6.68) 0.013 (0.89) 0.007∗∗ (1.96) −0.001∗∗∗ (−3.15) 0.006 (1.34) 0.008∗∗∗ (3.23) 0.078 0.070 10.03∗∗∗

0.001∗∗∗ 0.001∗∗∗ – 0.004∗∗∗ 0.001∗∗∗ – 0.007∗∗∗ – – −0.006∗∗∗

0.003 (0.72) 0.010∗∗ (2.22) 1.587∗∗∗ (3.23) – 0.005∗∗ (2.32) 0.006∗ (1.66) −0.005 (−0.61) −0.038∗∗∗ (−5.86) −0.051∗∗∗ (−6.71) 0.016 (1.15)

Dependent variables

Panel C Transpa Svolatility IdioRisk− daily Breturn Crating Maturity Beta Coupon Treasury ROE/100

0.294∗∗∗ (5.09) 0.289∗∗∗ (4.77) 84.617∗∗∗ (14.18) 0.777∗∗ (2.32) – −0.283∗∗∗ (−5.93) −0.738∗∗∗ (−7.82) 0.086 (1.02) 0.034 (0.34) 2.403∗∗∗ (12.81)

(2.66) (5.36) (3.27) (14.29) (19.31) (−8.68)

(Continued)

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Table 5. Continued

Dependent variables Independent variables Lev Age Diver Mb R2 Adjusted R2 F-value

Crating

IdioRisk− monthly

Breturn

Model (1)

Model (2)

Model (3)

Coefficient (t-value)

Coefficient (t-value)

Coefficient (t-value)

0.326∗∗∗ −0.050∗∗∗ −0.247∗∗∗ −0.179∗∗∗ 0.508 0.504 117.75∗∗∗

(7.04) (−10.74) (−4.21) (−5.95)

−0.001∗∗∗ (−2.70) −0.000∗∗∗ (−5.52) −0.000 (−0.28) 0.001∗∗∗ (6.77) 0.465 0.461 128.90∗∗∗

0.008∗∗ (2.19) −0.001∗∗∗ (−2.87) 0.004 (0.93) 0.007∗∗∗ (3.16) 0.073 0.065 9.01∗∗∗

Notes: t-Values are given within parentheses. Total number of samples is 1637. Variable definitions: Breturn denotes return on convertible bonds; Transpa denotes the level of a company’s information transparency; Maturity is the remaining life of a bond; IdioRisk denotes idiosyncratic risk; Crating denotes a proxy for bond’s credit ranking; Svolatility denotes the volatility of stock return; Coupon denotes treasury bill’s interest; Treasury denotes interest spread between short- and long-term treasury bonds; Beta denotes Beta risk; ROE is a ratio of net income to equity; Lev denotes leverage; Diver is a dummy variable for whether a company adopted a diversified strategy; Age denotes the seniority of a company; Mb is a ratio as a proxy for a company’s growth opportunity. Idiosyncratic risk IdioRisk is estimated using the Fama–French three-factor model. We adopt monthly, weekly, and daily bases in Panels A, B, and C, respectively. ∗ Significant at 10% level. ∗∗ Significant at 5% level. ∗∗∗ Significant at 1% level.

so as to be assessed with appropriate yield and better growth opportunities, which contributes to the rising value of its conversion value. 4.4

Returns on convertible bonds excluding conversion value

We also measure the returns on convertible bonds, excluding the conversion value. The conversion value of convertible bonds is first calculated by the conversion premium percentage less 100% as the returns of conversion right and then returns on convertible bonds less returns of conversion right. Next, we further calculate the annual bond price and returns based on the formula found on page 15 and then re-run Equations (1)–(3), replacing the convertible bond returns with the estimated pure bond return, excluding its conversion value. From Table 6, the evidence indicates that returns on convertible bonds without the conversion value are positively related to information transparency for monthly-based idiosyncratic risk but unrelated to transparency for weekly- and daily-based idiosyncratic risk. Hence, the findings partially support Hypothesis 1. Moreover, we find that the idiosyncratic risk of a company is positively associated with information transparency and returns on convertible bonds without a conversion value. Thus, the evidence also supports Hypotheses 2 and 3. The evidence also demonstrates that returns on convertible bonds without a conversion feature are positively related to the volatility of stock returns and the credit risk of a company. These results support Hypotheses 4 and 6. Finally, our results find that there is a negative relationship between the company’s idiosyncratic risk and credit risk and therefore support Hypothesis 5. 4.5

Sensitivity analysis

We divide the sample into high and low groups based on diversification, growth potential and information transparency to analyse the properties of idiosyncratic risk and returns on convertible

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Table 6. Determinants of credit rating, idiosyncratic risk, and return on convertible bonds excluding its conversion value.

Dependent variables Independent variables

Crating

IdioRisk− monthly

Breturn1

Model (1)

Model (2)

Model (3)

Coefficient (t-value)

Coefficient (t-value)

Coefficient (t-value)

Panel A Transpa Svolatility IdioRisk− monthly Breturn1 Crating Maturity Beta Coupon Treasury ROE/100 Lev Age Diver Mb R2 Adjusted R2 F-value

0.345∗∗∗ (5.54) 0.217∗∗∗ (3.10) 13.290∗∗∗ (14.59) 0.788∗∗∗ (2.71) – −0.298∗∗∗ (−5.81) −0.666∗∗∗ (−7.11) −0.012 (−0.14) −0.148 (−1.35) 2.728∗∗∗ (15.51) 0.411∗∗∗ (8.53) −0.051∗∗∗ (−10.29) −0.226∗∗∗ (−3.49) −0.172∗∗∗ (−5.46) 0.541 0.537 120.40∗∗∗

0.004∗∗ (2.10) 0.022∗∗∗ (13.23) – 0.021∗∗∗ (2.62) 0.010∗∗∗ (14.71) – 0.026∗∗∗ (10.03) – – −0.019∗∗∗ (−3.62) −0.002 (−1.32) −0.000∗∗ (−2.19) −0.002 (−1.17) 0.007∗∗∗ (7.67) 0.441 0.436 104.75∗∗∗

0.010∗ (1.69) 0.019∗∗∗ (2.87) 0.297∗∗∗ (3.23) – 0.007∗∗∗ (2.71) 0.001 (0.12) −0.015 (−1.63) −0.028∗∗∗ (−3.37) −0.064∗∗∗ (−6.25) 0.001 (0.05) 0.005 (1.12) −0.002∗∗∗ (−4.65) 0.005 (0.90) 0.001 (0.45) 0.100 0.091 11.37∗∗∗

Panel B Transpa Svolatility IdioRisk− weekly Breturn1 Crating Maturity Beta Coupon Treasury ROE/100 Lev Age Diver Mb R2 Adjusted R2 F-value

0.294∗∗∗ (5.14) 0.317∗∗∗ (5.27) 24.838∗∗∗ (13.09) 1.341∗∗∗ (4.72) – −0.286∗∗∗ (−6.13) −0.634∗∗∗ (−7.31) 0.133 (1.60) 0.104 (1.06) 2.503∗∗∗ (14.65) 0.344∗∗∗ (7.61) −0.051∗∗∗ (−11.01) −0.245∗∗∗ (−4.28) −0.193∗∗∗ (−6.49) 0.516 0.512 127.05∗∗∗

0.002∗∗∗ (3.20) 0.004∗∗∗ (5.91) – 0.011∗∗∗ (3.03) 0.004∗∗∗ (13.19) – 0.016∗∗∗ (15.41) – – −0.020∗∗∗ (−9.09) −0.002∗∗∗ (−2.61) −0.000∗∗∗ (−4.23) −0.000 (−0.50) 0.003∗∗∗ (8.42) 0.407 0.403 106.35∗∗∗

0.006 (1.09) 0.011∗∗ (2.06) 0.452∗∗ (2.55) – 0.011∗∗∗ (4.72) 0.001 (0.23) −0.013∗ (−1.64) −0.030∗∗∗ (−4.09) −0.051∗∗∗ (−5.94) 0.022 (1.36) 0.002 (0.53) −0.002∗∗∗ (−5.79) 0.008 (1.50) 0.003 (1.18) 0.102 0.094 13.51∗∗∗

Panel C Transpa Svolatility IdioRisk− daily Breturn1 Crating Maturity Beta Coupon Treasury

0.303∗∗∗ (5.27) 0.292∗∗∗ (4.86) 83.076∗∗∗ (13.98) 1.303∗∗∗ (4.50) – −0.277∗∗∗ (−5.84) −0.717∗∗∗ (−7.62) 0.095 (1.14) 0.063 (0.64)

0.001∗∗∗ (2.81) 0.001∗∗∗ (5.35) – 0.003∗∗∗ (2.68) 0.001∗∗∗ (14.06) – 0.007∗∗∗ (19.43) – –

0.007 (1.35) 0.011∗∗ (2.05) 1.478∗∗∗ (2.63) – 0.010∗∗∗ (4.50) 0.001 (0.31) −0.013 (−1.53) −0.030∗∗∗ (−4.04) −0.053∗∗∗ (−6.03) (Continued)

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Table 6. Continued ROE/100 Lev Age Diver Mb R2 Adjusted R2 F-value

2.403∗∗∗ 0.324∗∗∗ −0.047∗∗∗ −0.249∗∗∗ −0.175∗∗∗ 0.513 0.509 120.07∗∗∗

(13.88) (7.05) (−10.09) (−4.27) (−5.87)

−0.006∗∗∗ (−8.72) −0.001∗∗∗ (−2.59) −0.000∗∗∗ (−5.34) −0.000 (−0.28) 0.001∗∗∗ (6.95) 0.464 0.460 128.25∗∗∗

0.028∗ (1.69) 0.003 (0.82) −0.002∗∗∗ (−5.49) 0.006 (1.11) 0.003 (1.01) 0.100 0.092 12.69∗∗∗

Notes: t-Values are given within parentheses. Total number of samples is 1637. Variable definitions: Breturn1 denotes the estimated bond return with the conversion feature stripped out; Transpa denotes the level of a company’s information transparency; Maturity is the remaining life of a bond; IdioRisk denotes idiosyncratic risk; Crating denotes a proxy for bond’s credit ranking; Svolatility denotes the volatility of stock return; Coupon denotes treasury bill’s interest; Treasury denotes interest spread between short- and long-term treasury bonds; Beta denotes Beta risk; ROE is a ratio of net income to equity; Lev denotes leverage; Diver is a dummy variable for whether a company adopted a diversified strategy; Age denotes the seniority of a company; Mb is a ratio as a proxy for a company’s growth opportunity. Idiosyncratic risk IdioRisk is estimated using the Fama–French three-factor model. We adopt monthly, weekly, and daily bases in Panels A, B, and C, respectively. ∗ Significant at 10% level. ∗∗ Significant at 5% level. ∗∗∗ Significant at 1% level.

Table 7. Sensitivity analysis for idiosyncratic risk and returns on convertible bonds.

IdioRisk Breturn IdioRisk Breturn IdioRisk Breturn

Diversification samples

Un-diversification samples

Mean Median 0.067 0.058 0.085 0.089 High Mb samples Mean Median 0.071 0.065 0.108 0.099 High-transparency samples Mean Median 0.065 0.056 0.106 0.093

Mean Median 0.077 0.070 0.099 0.094 Low Mb samples Mean Median 0.059 0.055 0.095 0.079 Low-transparency samples Mean Median 0.073 0.066 0.096 0.086

Z test 4.263∗∗∗ 1.554 Z test 3.521∗∗∗ 4.221∗∗∗ Z test −4.378∗∗∗ 2.251∗∗

Notes: Total number of observations is 1637. Variable definitions: Breturn denotes return on convertible bonds; IdioRisk denotes idiosyncratic risk. We divide the sample into high and low groups based on diversification (Diver), growth opportunity (Mb) and information transparency (Transpa) to analyse the properties of idiosyncratic risk and returns on convertible bonds using the Z test of the median of high and low sample groups. ∗∗ Significant at 5% level. ∗∗∗ Significant at 1% level.

bonds. The results are shown in Table 7. From the Z test of the median of high and low sample groups, we find that there are significant differences between the high and low groups in both idiosyncratic risk and returns on convertible bonds. The firms with more diversification and greater growth potential have less idiosyncratic risk and better returns on convertible bonds, suggesting that diversification of firm operations can also diversify idiosyncratic risk; thus, they are popular with investors. Moreover, both the mean and the median of idiosyncratic risk are higher for HT

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Return Change

20% 10% 0%

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

2

4

6

8 10 12 14 16 18 20

-10% -20% -30%

Period

Figure 3. Stock returns 20 days before and after the issue of the convertible bonds.

Stock Return 50% 40% 30% 20%

Return Change

10% 0% -40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0 -10%

4

8 12 16 20 24 28 32 36 40

-20% -30% -40% Period

Figure 4. Stock returns 40 days before and after the issue of the convertible bonds.

firms than for LT firms. This result is consistent with Hypothesis 2. We also find that the HT group has higher returns on convertible bonds, which is consistent with Hypothesis 1. 4.6

Beta and stock returns for 20 and 40 days before and after the issuance of convertible bonds

The issuance of convertible bonds complicates a company’s asset portfolio. Theoretically, the presence of a call option in the convertible bond should have an impact on company share prices. Hence, we further analyse how the beta and the returns correspondingly change before and after the issuance of convertible bonds. Figures 3 and 4 show the changes in stock returns 20 days and 40 days before and after the issue of the convertible bonds, respectively. Figures 5 and 6 show the changes in beta 20 days and 40 days before and after the issue of the convertible bonds, respectively. From Figures 3 and 4, we find that the returns on convertible bonds significantly

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99

Beta

-8

-5

-2

1

4

7

10

13

16

19

Period

Figure 5. Betas 20 days before and after the issue of the convertible bonds. 0.960 0.940 0.920 Beta

0.900 0.880 0.860 0.840 0.820 -40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0

4

8 12 16 20 24 28 32 36 40

Period

Figure 6. Betas 40 days before and after the issue of the convertible bonds.

increase during 5 continuous days before the issuance and during 7 continuous days after the issuance of convertible bonds, implying that the issuance of convertible bonds is good news for investors. Moreover, Figures 5 and 6 show that the betas ascend from the 20th to the 8th day before the issuance of convertible bonds and significantly descend from the 25th day after the issuance of the convertible bonds, implying that the issuance of convertible bonds conveys better future prospects for investors and, thus, that systematic risk decreases. 5. Conclusions This paper contributes to the literature by examining how idiosyncratic risk affects the returns on convertible bonds and the credit rating of the issuer and how transparency is associated with idiosyncratic risk and returns on convertible bonds. Furthermore, we also isolate the bond price whilst excluding its conversion value to obtain the estimated bond return with the conversion feature stripped out. According to a simultaneous equations model analysis, the study finds that both transparent disclosures and idiosyncratic risk positively impact a company’s credit rating, indicating that

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when a company discloses more relevant information, idiosyncratic variance will increase along with its credit rating. Returns on convertible bonds are positively affected by the volatility of stock returns and a company’s credit rating. This finding is consistent with the notion that volatile stock returns contribute to the value of an option, whereas a better credit rating contributes to the value of pure bonds. Additionally, information transparency appears to be unrelated to returns on convertible bonds even when idiosyncratic risk is measured on monthly, weekly, and daily bases. Because the essence of convertible bonds is the debt contract, the value of convertible bonds is primarily driven by the market interest rate. Therefore, returns on convertible bonds are less influenced by what a company discloses to investors. We also suggest that the age of a company and the market-to-book ratio also influence returns on convertible bonds. Moreover, we further measure the returns on convertible bonds excluding conversion value, that is, replacing the returns on convertible bonds with the estimated pure bond returns. The evidence indicates that returns on convertible bonds without conversion value are positively related to information transparency under monthly-based idiosyncratic risk but are unrelated to transparency under weekly- and daily-based idiosyncratic risks. A company’s idiosyncratic risk is positively associated with information transparency, credit rating, and returns on convertible bonds without conversion features, and returns on convertible bonds without conversion features are positively related to the volatility of stock returns and a company’s credit rating. Finally, we perform another analysis to observe the impact of issuing convertible bonds on stock returns and beta. We conclude that the issuance of convertible bonds appears to be good news for investors, with stock returns significantly increasing for 5 continuous days before and for 7 continuous days after the issuance of the convertible bonds. Meanwhile, the beta, which ascends from the 20th to the 8th day before the issuance of the convertible bonds and significantly descends from the 25th day after their issuance, implies that a company issuing convertible bonds signals better future prospects for investors and that its systematic risk is lower. We experienced a few obstacles while collecting information. For example, the relative bond information in the Taiwan bond market was incomplete, transactions in the Taiwan bond market are not highly active, and information about updated prices of convertible bonds is not easy to obtain. These challenges create limitations in our research data. In addition, when evaluating the disclosure transparency of companies, we evaluated the disclosed data without confirming its accuracy, which created another limitation. Future research may try to conquer these limitations to validate our findings. Acknowledgement This work is supported by the Ministry of Education, Taiwan under the ATU plan.

Notes 1. Taiwan’s bond market currently imposes no trading tax. 2. Since 2001, the US SEC has required the National Association of Securities Dealers (NASD) to report OTC trading records for the bond market to improve information transparency. The information transparency of bond markets was further improved with the development of the Trade Reporting and Compliance Engine (TRACE), which also helped the US bond market become more active. 3. Idiosyncratic volatility is also called idiosyncratic risk. 4. Jensen (1986) believes that for companies with higher free cash flow, debt financing can hold managers’ overinvestment or ineffective disbursement and decrease managers’ chances and incentives in manipulating earnings. Due to shareholders’ claims of falling behind creditors, if a company uses too much debt financing to deal in high-risk investment, shareholders would gain the profit, with creditors only receiving interest when the investment brings profit.

The European Journal of Finance

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6.

7.

8.

9. 10.

11.

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Once the company faces collapse, shareholders would have only limited liability, but creditors would experience great losses. Mayers (1998) indicates that convertible bonds play an important role in long-term debt-financing plans for companies, which can adjust cash flow in accordance with investment expenditures. Issuance of convertible bonds is also the most economical method to raise capital for future investment opportunities; it can simultaneously decrease issue costs and agency costs of issuance. Before 1985, simply holding more than 20 types of stocks could decrease the excess standard deviation by 5%. However, from 1986 to 1997, to decrease the excess standard deviation by 5%, investors needed to hold at least 50 types of stocks for risk diversification. Malkiel and Xu (2003) find that the volatility of individual stocks appears to have increased over time. This trend is not solely attributed to the increasing prominence of the NASDAQ market, and they go on to suggest that the idiosyncratic volatility of individual stocks is associated with the degree to which their shares are owned by financial institutions. Two significant credit-risk evaluating models exist: the Credit Metrics model used by J.P. Morgan and the KMV model used by KMV Corporation. The Credit Metrics model is primarily concerned with a company’s credit rating, changes in credit rating, yield, and recovery rate if the creditors bankrupted; the analysis is dependent on a company’s historical credit rating records, which is a kind of back method. The KMV model uses expected default frequency as the tool to measure the credit risk of a company, which analyses simultaneous chances of stock prices. The stock market reflects not only past and current situations, but also includes investors’ expectations about the companies’ future development. Therefore, it is a type of forward method. Giesecke et al. (2011) find that stock returns, stock return volatility, and changes in GDP are strong predictors of default rates. We also adopt the market model, Ri = ai + bi Rm + ei , to estimate regression coefficients α and β for the past 36 months, where Ri is the actual monthly return of firm i, and Rm is the actual monthly market return. If the estimated coefficients α and β are plugged back into the market model to calculate the value of α + βRmt , where Rmt is the actual monthly market return of month t, we obtain the expected monthly return of firm i in month t. Using the actual monthly return of firm i in month t minus the expected monthly return of firm i in month t, we can obtain the residual (or error term) of firm i in month t. We take the standard deviation of residuals of 12 months to obtain the annual data for the idiosyncratic risk of firm i. Due to space limitations, we do not show the results for idiosyncratic risk calculated by the market model. These results are similar to those for the Fama–French three-factor model. For example, Ingersoll (1976) and Smith (1984) use the volatility of assets value as one of the parameters to establish their convertible bond pricing model.

References Alexander, G.J., R.D. Stover, and D.B. Kuhnau. 1979. Market timing strategies in convertible debt financing. Journal of Finance 34: 143–55. Alexander, G.J., A. Edwards, and M. Ferri. 2000. The determinants of trading volume of high-yield corporate bonds. Journal of Financial Markets 3: 177–204. Ammann, M., M. Fehr, and R. Seiz. 2006. New evidence on the announcement effect of convertible and exchangeable bonds. Journal of Multinational Financial Management 16: 43–63. Ang, A., R. Hodrick, Y. Xing, and X. Zhang. 2006. The cross-section of volatility and expected returns. Journal of Finance 61: 259–99. Ayache, E., P.A. Forsyth, and K.R. Vetzal. 2003. Valuation of convertible bonds with credit risk. Journal of Derivatives 11: 9–29. Bali, T.G., N. Cakici, X. Yan, and Z. Zhang. 2005. Does idiosyncratic risk really matter? Journal of Finance 60: 905–29. Bello, Z. 2005. Idiosyncratic risk and mutual fund return. Journal of Business and Economic Studies 11: 62–73. Bessembinder, H., W. Maxwell, and K. Venkataraman. 2006. Optimal market transparency: Evidence from the initiation of trade reporting in corporate bonds. Journal of Financial Economics 82: 251–88. Billingsley, R.S., and D.M. Smith. 1996. Why do firms issue convertible debt? Financial Management 25: 93–9. Billingsley, R.S., R. Lamy, and D.M. Smith. 1990. Units of debt with warrants: Evidence of the penalty-free. Journal of Financial Research 8: 187–99. Brennan, M.J., and E.S. Schwartz. 1988. Time-invariant portfolio insurance strategies. Journal of Finance 43: 283–99. Campbell, J., M. Lettau, B. Malkiel, and Y. Xu. 2001. Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk. Journal of Finance 49: 1–43.

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Quantitative Finance, Vol. 10, No. 10, December 2010, 1137–1151

Up and down credit risk TOMASZ R. BIELECKI*y, STE´PHANE CRE´PEYz and MONIQUE JEANBLANCzx yDepartment of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA zDe´partement de Mathe´matiques, Universite´ d’E´vry Val d’Essonne, 91025 E´vry Cedex, France xEuroplace Institute of Finance (Received 16 October 2008; in final form 1 October 2009) This paper discusses the main modeling approaches that have been developed for handling portfolio credit derivatives, with a focus on the question of hedging. In particular, the so-called top, top down and bottom up approaches are considered. We give some mathematical insights regarding the fact that information, namely the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multi-name derivative by single-name derivatives. We then illustrate by means of numerical simulations (semi-static hedging experiments) why and when the portfolio loss process may not be a ‘sufficient statistic’ for the purpose of valuation and hedging of portfolio credit risk. Keywords: Credit risk; Computational finance; Financial mathematics; Model calibration

1. Introduction Presently, most if not all credit portfolio derivatives have cash flows that are determined solely by the evolution of the cumulative loss process generated by the underlying portfolio. Thus, as of today, credit portfolio derivatives can be considered as derivatives of the cumulative loss process L. The consequence of this is that most of the models of portfolio credit risk, and related derivatives, focus on modeling the dynamics of the process L, or directly on modeling the dynamics of the related conditional probabilities, such as ProbðL takes some values at future time(s)j given present informationÞ: In this paper we shall study the various methodologies that have been developed for this purpose, particularly the so-called top, top-down and bottom-up approaches. In addition, we shall discuss the issue of hedging of loss process derivatives, and we shall argue that loss processes may not provide a sufficient basis for this, in the sense described later in the paper. In fact, we engage in some in-depth study of the role of information with regard to valuation and hedging of derivatives written on the loss process. The paper is organized as follows. In section 2 we provide an overview of the main modeling approaches that *Corresponding author. Email: [email protected]

have been developed for handling portfolio credit derivatives. In section 3 we revisit the notion of thinning that was recently advocated for the purpose of hedging a multi-name credit derivative by single-name credit derivatives, such as CDS contracts. In section 4 we illustrate by means of numerical simulations why and when the portfolio loss process may not be a sufficient statistic for the purpose of the valuation and hedging of portfolio credit risk. Conclusions and perspectives are drawn in section 5. Finally, an appendix gathers definitions and results from the theory of the processes that we use repeatedly in this paper, such as, for instance, the definition of the compensator of a non-decreasing adapted process.

2. Top, top-down and bottom-up approaches: an overview This section provides an overview and a discussion about the so-called top, top-down and bottom-up approaches in portfolio credit risk modeling. Some related discussion can also be found in Inglis et al. (2009). Let us first introduce some standing notation. . If X is a given process, we denote by FX its natural filtration satisfying the usual conditions (perhaps after completion and augmentation).

Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2010 Taylor & Francis http://www.informaworld.com DOI: 10.1080/14697680903382776

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T.R. Bielecki et al. . By the F-compensator of an F-stopping time , where F is a given filtration, we mean the F-compensator of the (non-decreasing) one point process It (see section A.2). . For every d, k 2 N, we denote Nk ¼ {0, . . . , k}, Nk ¼ f1, . . . , kg and Ndk ¼ f0, . . . , kgd .

From now on, t will denote the present time, and T4t will denote some future time. Suppose that  represents a future payment at time T, which will be derived from the evolution of the loss process L on a credit portfolio, and representing a specific (stylizedy) credit portfolio derivative claim. There may be two tasks at hand: . to compute the time-t price of the claim, given the information that we may have available and we are willing to use at time t; and . to hedge the claim at time t. By this, we mean computing hedging sensitivities of the claim with respect to hedging instruments that are available and that we may want to use.

ordered sequence, that is  (1)5 (2)5  5 (n). We denote Hit ¼ 11i t . P Accordingly, set HðiÞ t ¼ 11ðiÞ t . Therefore, Pn we ðiÞ n i obviously, i¼1 H ¼ i¼1 H , and the representation n X Hi ð2Þ L¼ i¼1

holds if and only if

ti ¼ ðiÞ ,

i 2 Nn

ð3Þ

(in which case  (i) are F-stopping times). From now on we assume that (2) is satisfied. The random times  i can thus be interpreted as the default times of the pool names, and Hi as the default indicator process of name i. We stress that for any i the random time  i may or may not be an F-stopping time, and thus the process Hi may or may not be F-adapted, although all  (i) are F-stopping times in this W case. i We denote Hi ¼ FH , H ¼ i2Nn Hi . 2.1. Information is it!

For simplicity, we shall assume that we use a spot martingale measure, say P, for pricing, and that the interest rate is zero. Thus, denoting by F ¼ (F t)t2[0,T ] a filtration that represents the flow of information we use for pricing, and by E the expectation relative to P, the pricing task amounts to computation of the conditional expectation E( j F t) ( being assumed F T -measurable and P-integrable). More specifically, on a standard stochastic basis ( , F , F, P), we consider a (strictly) increasing sequence of stopping times ti, for i 2 Nn , representing the ordered default times of the names in the credit pool, and we define the (F-adapted) portfolio loss process, for t  0, L by Lt ¼

n X i¼1

11ti t

ð1Þ

(assuming, for simplicity, zero recoveries). Therefore, L is a non-decreasing ca`dla`g process stopped at time tn, taking its values in Nn, with jumps of size one (L is, in particular, a point process; see, e.g., Bre´maud 1981 and Last and Brandt 1995). We shall then consider (stylized) portfolio loss derivatives with payoff  ¼ (LT), where () is an appropriate integrable function. Throughout this paper we work under the standing assumption that ti are totally inaccessible F-stopping times, which is tantamount to assuming that their compensators i are continuous processes and are therefore P stopped at ti (see section A.2). The compensator  ¼ ni¼1 i of L is therefore, in turn, continuous and stopped at tn. Let  i, i 2 Nn , denote an arbitrary collection of (mutually avoiding) random (not necessarily stopping) times on ( , F , P), and let  (i), i 2 Nn , denote the corresponding

Various approaches to the valuation of derivatives written on credit portfolios differ depending on the content of the model filtration F. Thus, loosely speaking, these approaches differ depending on what they presume to be sufficient information to price, and, consequently, to hedge, credit portfolio derivatives. The choice of a filtration is of course a crucial modeling issue. In particular, the compensator  of an adapted non-decreasing (and bounded, say) process K, defined as the predictable non-decreasing Doob–Meyer component of K (see section A.2), is an information- (i.e. filtration-) dependent quantity. So is, therefore, the intensity process (time-derivative of , assumed to exist) of K. Let thus K denote an F-adapted non-decreasing process and G be a filtration larger than F (so K is, of course, G-adapted). Let  and denote the F-compensator and the G-compensator of K, respectively. The following general result, which is proved in section A.3 (see also sections A.1 and A.2 for the various notions of the projections involved), establishes the relation between  and and the related F- and G-intensity processes  and  (whenever they exist, for the latter). Proposition 2.1: (i)  is the dual predictable projection of on F. (ii) Moreover, in the case where  and are time-differentiable with related F- and G-intensity processes  and , then  is the optional projection of  on F. Figure 1 provides an illustration of the dependence of the intensities on information in a simple model with n ¼ 3 stopping times.z The figure shows a trajectory over the

y Of course, most credit products are swapped and therefore involve coupon streams, so, in general, we need to consider a cumulative ex-dividend cash flow t on the time interval (t, T ]. z We thank Behnaz Zargari from the Mathematics Departments at the University of Evry, France, and Sharif University of Technology, Tehran, Iran, for these simulations.

Up and down credit risk 2.2. Top and top-down approaches

(a) 0.70 lambda1 lambda2 lambda3

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.0 (b) 0.8

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2.0

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Figure 1. Simulated sample path of the pre-default intensity of  1 with respect to H1 _ FZ, H1 _ H2 _ FZ and H1 _ H2 _ H3 _ FZ, with Z trivial (constant) on the left versus Z ¼ W on the right.

time interval [0, 5] years of the pre-default intensity of  1 with respect to H1 _ FZ, H1 _ H2 _ FZ and H1 _ H2 _ H3 _ FZ (curves respectively labeled lambda1, lambda2 and lambda3 in figure 1), where Hi correspond to the filtrations generated by Hi, and . the reference filtration FZ is trivial on the left side, . it is given as a (scalar) Brownian filtration FZ ¼ FW on the right side. We refer the reader to Zargari (2009) for more details on these simulations. In this example we have  2 ¼ 1.354,  3 ¼ 0.669 in the case where FZ is trivial and  2 ¼ 1.3305,  3 ¼ 0.676 in the case where FZ ¼ FW (the same random numbers were used in the two experiments). Observe that: . lambda2 and lambda3 jump at  2 (¼1.354 in the left graph and 1.3305 in the right graph), . only lambda3 jumps at  3 (¼0.669 in the left graph and 0.676 in the right graph), and . lambda1 does not jump at all. The facts that lambda2 does not jump at  3 and lambda1 does not jump at all are of course consistent with the definitions of lambda1 and lambda2 as the pre-default intensities of  1 with respect to H1 _ FZ and H1 _ H2 _ FZ, respectively. Also note the effect of adding a reference filtration (noisy pre-default intensities on the right side, versus pre-default intensities ‘deterministic between default times’ on the left side).

The approach that we dub the pure top approach takes as F the filtration generated by the loss process alone. Thus, in the pure top approach we have that F ¼ FL. Examples of this approach are Laurent et al. (2007), (most of) Herbertsson (2007), and Cont and Minca (2008). The approach that we dub the top approach takes as F the filtration generated by the loss process and by some additional relevant (preferably low-dimensional) auxiliary factor process, say Y. Thus, in this case, F ¼ FL _ FY. Examples of this approach are Bennani (2006), Ehlers and Scho¨nbucher (2006), Scho¨nbucher (2006), Sidenius et al. (2006), and Arnsdorf and Halperin (2007). The so-called top-down approach starts from the top, that is it starts by modeling the evolution of the portfolio loss process subject to information structure F. It then attempts to decompose the dynamics of the portfolio loss process down into the individual constituent names of the portfolio, so as to deduce the dynamics of processes Hi (for the purpose typically of hedging of credit portfolio derivatives by vanilla individual contracts such as default swaps). This decomposition is done by a method of random thinning formalized by Giesecke and Goldberg (2007) (see also Halperin and Tomecek 2008), which will be discussed in detail in section 3.

2.3. Bottom-up approaches The approach that we dub the pure bottom-up approach takes as F the filtration generated by the state of the pool process H ¼ (H1, . . . , Hn), i.e. F ¼ FH ¼ H (see, for instance, Herbertsson 2008). The approach that we dub the bottom-up approach takes as F the filtration generated by process H and by an auxiliary factor process Z. Thus, in this case, F ¼ FH _ FZ. Examples of this approach are Duffi and Garleanu (2001), Frey and Backhaus (2007, 2008), and Bielecki et al. (2007, 2008).

2.4. Discussion The pure top approach is undoubtedly the best suited for the fast valuation of portfolio loss derivatives, as it only refers to a single driver—the loss process itself. However, this approach may produce incorrect pricing results, as it is rather unlikely that the financial market evaluates derivatives of the loss process based only on the history of the evolution of the loss process alone. Note, in particular, that the loss process is not a traded instrument. Thus, it appears to be advisable to work with more information than that carried by filtration FL alone. This is quite likely the reason why several versions of the top approach have been developed. Enlarging the filtration from FL to FL _ FY may lead to increased computational complexity, but at the same time it is quite likely to increase the accuracy of the calculation of important quantities, such as CDO tranche spreads and/or CDO prices.

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From the hedging perspective, both the pure top approach and the top approach may not be adequate. Indeed, operating on the top level prohibits computing sensitivities of a loss process derivative with respect to constituents of the credit portfolio. Therefore, for example, when operating just on the top level, one cannot compute sensitivities of CDO tranche prices with respect to prices of the CDS contracts underlying the portfolio. In these approaches, it is only possible to hedge one loss derivative by another (e.g., hedging a CDO tranche using iTraxx). However, as we shall see in section 4, in certain circumstances this kind of hedging may not be quite precise, or even not possible at all. This is, of course, the problem that led to the idea of the top-down approach, that is the idea of thinning. But, as we shall now see, it seems to us that thinning cannot really help in developing a consistent approach to hedging credit loss derivatives by single-name credit derivatives.

stopping times, one will seek a martingale representation in the form n Z t m Z t X X i i s dMs þ ð4Þ sj dNsj , Eð j F t Þ ¼ E þ i¼1

Note that processes Hi and H (i) are sub-martingales, and can therefore be compensated, with respect to any filtration for which they are adapted, as non-decreasing processes (see section A.2). Thinning refers to the recovery of individual compensators of H (i) and Hi, starting from the loss compensator  as input data. Since the compensator is an information- (filtration-) dependent quantity, thinning of course depends on the filtration under consideration. A preliminary question regarding thinning is why would one wish to know the individual compensators. Suppose that all one wants to do is pricing, in other words computing the expectation E( j F t) for 0  t5T, where the integrable random variable  ¼ (LT) represents the stylized payoff of a portfolio loss derivative. Under Markovian assumptions, or conditionally Markovian assumptions (assuming further factors Y ), about process L with respect to the filtration F, in principle the expectation E( j F t) can be computed (at least numerically). For computation of E( j F t), one does not really need to know the individual compensators of  i (which do not even need to be assumed to be F-stopping times in this regard). Therefore, with regard to the problem of pricing of derivatives of the loss process, a top model may be fairly adequate. In particular, the filtration F may not necessarily containPthe pool filtration H. Also, the representation L ¼ ni¼1 Hi (see (2)) need not be considered at all in this context. But computing the price E( j F t) is just one task of interest. Another key task is hedging. From a mathematical point of view, hedging relies on the derivation of a martingale representation of E( j F t), which is useful in the context of computing sensitivities of the price of  with respect to changes in the prices of liquid instruments, such as credit indices and/or CDS contracts, corresponding to the credit names composing the credit pool underlying the loss process L. Typically, assuming here that  i are F

j¼1

0

where Mi are some fundamental martingales associated with the non-decreasing processes Hi, and N j are some fundamental martingales associated with all relevant auxiliary factors included in the model. The coefficients i and  j can, in principle, be computed given a particular model specification; now, for the practical computation of i and  j, but also for the very definition of Mi and N j, one will typically need to know the compensators i.

3.1. Thinning of the ordered default times Let (i) denote the F-compensator of  (i) (recall that  (i) are F-stopping times). Proposition 3.1:

3. Thinning revisited

0

We have, for t  0, ðiÞ t ¼ t^ðiÞ  t^ði1Þ :

ð5Þ

Therefore, in particular, (i) ¼ 0 on the set t   (i1). Proof: Note first that Lt^ðiÞ  t^ðiÞ

ð6Þ

is a F-martingale, as it is equal to the F-martingale L   (see equation (A1) in the appendix) stopped at the F-stopping time  (i). Taking the difference between the expression in (6) for i and i  1 yields that  ðiÞ  ðiÞ HðiÞ t  t , with t defined as the RHS of (5), is an F-martingale (starting at  (i1) and stopped at  (i)). Hence, (5) follows, due to the uniqueness of compensators (recall  ðiÞ is continuous, hence that  is continuous, so  predictable). œ Formula (5) represents the ‘ordered thinning’ of . Note that proposition 3.1 is true regardless of whether  i are F-stopping times or not. This reflects the fact that modeling the loss process L is the same as modeling the ordered sequence of  (i), no matter what the informational context of the model.

3.2. Thinning of the default times Let us first denote by i the F-compensator of  i, assumed to be an F-stopping time. We of course have that ¼

n X

i :

i¼1

ð7Þ

Moreover, the following is true. Proposition 3.2: There exists F-predictable non-negative processes Zi, i 2 Nn , such that Z1 þ Z2 þ    þ Zn ¼ 1 and Z i ¼ Zit dt , i 2 Nn : ð8Þ 0

Up and down credit risk Proof: In view of (7), the existence of Zi ¼ di/d follows from theorem VI68, p. 130, of Dellacherie and Meyer (1982) (see also Giesecke and Goldberg 2007). œ In the special case where the random times  i constitute an ordered sequence, therefore  i ¼  (i), then the ordered thinning formula (5) yields that Zit ¼ 11i1 5ti . Proposition 3.2 tells us that, if one starts building a model from the top, that is if one starts building the model by first modeling the F-compensator  of the loss process L, then the only way to go down relative to the information carried by F, i.e. to obtain F-compensators i, is to do thinning in the sense of equation (8). We shall refer to this as F-thinning of . 3.2.1. Thinning with respect to a sub-filtration. Now suppose that Fi is some sub-filtration of F and that  i is an Fi-stopping time. We want to compute the Fi-compensator bi of  i, starting with . The first step is to do the  F-thinning of , that is to obtain the F-compensator i of  i (see (8)). The second step is to obtain the Fibi of  i from i. The following result compensator  follows by application of proposition 2.1.

bi is the dual predictable projection of i Proposition 3.3:  i b i and i are on F . Moreover, in the case where  i time-differentiable with related F - and F-intensity processes bi and i, then b i is the optional projection of i on Fi.  bi is also the dual predictable Remark 1: Note that  i i projection of H on F (see section A.2).

Proposition 3.3 is important regarding the issue of the calibration of a portfolio credit model to marginal data, one of the key issues in relation to hedging a credit loss derivative by single-name credit instruments. For example, one may want to calibrate the credit portfolio model to spreads on individual CDS contracts. If the spread on the ith CDS contract is computed using conditioning with bi of  i will typically be respect to Fi, then the Fi-intensity  used as input data in the calibration (for determining an F-adapted process i with Fi-optional projection b i given in the market). 3.3. The case when si are not stopping times In the case where  i is not an F-stopping time, Giesecke and Goldberg (2007) introduce the notion of (we call it the top-down) the intensity of  i, defined as the time-derivative, assumed to exist, of the dual predictable projection of Hi on F. In view of remark 1, this is indeed a generalization of the usual notion of intensity to the case where  i is not an F-stopping time. However, our opinion is that such a top-down intensity does not make much sense. Indeed the market intensity of name i (intensity of name i as extracted from the marginal market data on name i, typically the CDS curve on i) corresponds to an intensity in a filtration adapted to  i,

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which, in particular, vanishes after  i (contrarily to a top-down intensity, unless  i is an F-stopping time). A top-down intensity is thus not represented in the market, and it can therefore not be calibrated (unless, again,  i is an F-stopping time). 3.4. Limitations of thinning In view of the above observations, one must, in our opinion, restrict consideration of thinning to the case where all  i are F-stopping times, that is, to the case of thinning in the sense of section 3.2. Observe, though, that thinning in this sense is equivalent to building the model from the bottom up. This is because modeling of processes  and Zi, proposition 3.2, is equivalent to modeling the processes i. The relevance of the top-down construction of a model by thinning with respect to a filtration containing all Hi (therefore, a bottom-up model, ultimately) thus seems questionable. In defense of such an approach, one might say that, since this approach starts from the top, then it gives the modeler better control over designing the dynamics of the portfolio loss process L so as to tailor-design this process through a ‘nice’ and simple dynamics. But the point is precisely that in a model with a ‘nice’ and simple top portfolio loss process L, there is no need of use single-name instruments for hedging. In fact, typically, a small number of other loss derivatives will be able to do the hedging job (see, for instance, Laurent et al. (2007)). Models with a ‘too simple’ loss process L are actually not a good family for considering the issue of hedging credit loss derivatives by single-name instruments, because single-name instruments are, in principle, not required for hedging in such a model.

4. Sufficient statistics For credit derivatives with a stylized payoff given by  ¼ (LT) at maturity time T, it is tempting to adopt a Black–Scholes-like approach, modeling L as a Markov process and performing factor hedging of one derivative by another, balancing the related sensitivities computed by the Itoˆ–Markov formula (see, for instance, Laurent et al. 2007). However, since the loss process L may be far from Markovian in the market, there may be circumstances under which L is not a ‘sufficient statistic’ for the purpose of valuation and hedging of portfolio credit risk. In other words, ignoring the potentially non-Markovian dynamics of L for pricing and/or hedging may cause significant model risk, even though the payoffs of the products at hand are given as functions of LT. In this section we want to illustrate this point by means of numerical hedging simulations (see also Cont and Kan 2008 for an extensive empirical study of the real-life hedging performances of a variety of top models on preas well as post-crisis data sets). For these numerical experiments we introduce a non-zero recovery R, taken as a constant R ¼ 40%. We thus need to distinguish the

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T.R. Bielecki et al.

P cumulative default process Nt ¼ ni¼1 Hit and the cumulative loss process Lt ¼ (1  R)Nt. We shall consider the benchmark problem of pricing and hedging a stylized loss derivative. Specifically, for simplicity, we only consider protection legs of equity tranches (respectively super-senior tranches) (i.e. detachment of 100%), with stylized payoffs  þ LT LT ^ k, respectively k ðNT Þ ¼ n n at maturity time T. The ‘strike’ (detachment, respectively attachment point) k belongs to [0, 1]. In this formalism the stylized credit index corresponds to the equity tranche with k ¼ 100% (or senior tranche with k ¼ 0). With a slight abuse of terminology, we shall refer to our stylized loss derivatives as tranches and index, respectively. We shall now consider the problem of hedging the tranches with the index, using a simplified market model of credit risk. 4.1. Homogeneous groups model We consider a Markov chain model of credit risk (Frey and Backhaus 2008) (see also Bielecki et al. 2007). Namely, the n names of a pool are grouped into d classes of  1 ¼ n/d homogeneous obligors (assuming n/d an integer). The cumulative default processes Nl, l 2 Nd , of different groups are jointly modeled as a d-variate Markov point process N , with a FN -intensity of Nl given as el ðt, N t Þ, lt ¼ ð  1  Nlt Þ

ð9Þ

for some pre-default individual intensity functions el ðt, {Þ, where { ¼ ði1 , . . . , id Þ 2 Nd 1 . The related el ¼   infinitesimal generator at time t may then be written in (very sparse) matrix, say At. the form of a d-dimensional P l Also note that N ¼ N. For d ¼ 1 we recover the well-known local intensity model (N modeled as a Markov birth point process stopped at level n) of Laurent et al. (2007) or Cont and Minca (2008). At the other extreme, for d ¼ n, we are in effect modeling the vector of the default indicator processes of the pool names. As d varies between 1 and n, we thus obtain a variety of models of credit risk, ranging from pure top models for d ¼ 1 to pure bottom-up models for d ¼ n. Remark 1: Observe that, in the homogeneous case where P b¼ b b el ðt, {Þ ¼ ðt, i Þ for some function  ðt, iÞ (inde j j pendent of l ), the model effectively reduces to a local intensity model (with d ¼ 1 and pre-default individual b iÞ therein). Further specifying the model to intensity ðt, b be  independent of i corresponds to the situation of homogeneous and independent obligors. In general, introducing parsimonious parameterizations of the intensities allows one to account for inhomogeneity between groups and/or default contagion. It is also possible to extend this setup to more general credit migrations models, or to generic bottom-up models of credit migrations influenced by macro-economic factors

(see Frey and Backhaus 2004 and Bielecki et al. 2007, 2008).

4.1.1. Pricing in the homogeneous groups model. Since N is a Markov process and Nt is a function of N t, the related tranche price process is given by, for t 2 [0, T ] (assuming (NT) integrable) t ¼ EððNT Þ j F N t Þ ¼ uðt, N t Þ,

ð10Þ

where u(t, ) or u (t) for t 2 [0, T ] and { 2 Nd 1 is the pricing function (system of time-functions u ). Using the Itoˆ formula in conjunction with the martingale property of , the pricing function can then be characterized as the solution to the following pricing equation (system of ODEs): ð@t þ At Þu ¼ 0 on ½0, T Þ,

ð11Þ

with terminal condition u (T ) ¼ ( ), for { 2 Nd 1 . In particular, in the case of a time-homogeneous generator A (independent of t), one has the semi-closed matrix exponentiation formula uðtÞ ¼ eðTtÞA :

ð12Þ

Pricing in this model can be achieved by various means, such as numerical resolution of the ODE system (11), numerical matrix exponentiation based on (12) (in the time-homogeneous case) or Monte Carlo simulation. However resolution of (11) or computation of (12) by deterministic numerical schemes is typically precluded by the curse of dimensionality for d greater than a few units (depending on ). Therefore, for high d, simulation methods appear to be the only viable computational alternative. Appropriate variance reduction methods may help in this regard (see, for instance, Carmona and Cre´pey 2009). The distribution of the vector of time-t losses (for each group), that is q (t) ¼ P(N t ¼ ) for t 2 [0, T ] and { 2 Nd 1 , and the portfolio cumulative loss distribution, p ¼ pi(t) ¼ P(Nt ¼ i) for t 2 [0, T ] and i 2 Nn, can be computed likewise by numerical solution of the associated forward Kolmogorov equations (for more details, see, e.g., Carmona and Cre´pey 2009).

4.1.2. Hedging in the homogeneous groups model. In general, in the Markovian model described above, it is possible to replicate dynamically in continuous time any payoff provided that d non-redundant hedging instruments are available (see Frey and Backhaus 2007 or Bielecki et al. 2008; see also Laurent et al. 2007 for results in the special case where d ¼ 1). From the mathematical side, this corresponds to the fact that, in general, this model is of multiplicity d (model with d fundamental martingales; see, e.g., Davis and Varaiya 1974). Therefore, in general, it is not possible to replicate a payoff, such as a tranche, by the index alone in this model, unless the model dimension d is equal to 1 (or reducible to 1; see remark 1). Now our point is that this potential lack

Up and down credit risk of replicability is not purely speculative, but can be very significant in practice. Since delta-hedging in continuous time is expensive in terms of transaction costs, and because the main changes occur at default times in this model (in fact, default times are the only events in this model, if not for time flow and the induced time-decay effects), we shall focus on semi-static hedging in what follows, only updating at default times the composition of the hedging portfolio. More specifically, denoting by t1 the first default time of a reference obligor, we shall examine the result at t1 of a static hedging strategy on the random time interval [0, t1]. Let  and  denote the tranche and index model price processes, respectively. Using a constant hedge ratio b 0 over the time interval [0, t1], the tracking error or profit-and-loss of a delta-hedged tranche at t1 is given by 0 ðt1  0 Þ: et1 ¼ ðt1  0 Þ  b

ð13Þ

The question we want to consider is whether it is possible to make this quantity ‘small’, in terms, say, of variance, relative to the variance of t1  0 (which corresponds to the risk without hedging), by a suitable choice of b 0 . It is expected that this should depend on the following.

. First, on the characteristics of the tranche, and, in particular, on the value of the strike k: A high strike equity tranche or a low strike senior tranche is quite close to the index in terms of cash flows, and should therefore exhibit a higher degree of correlation and be easier to hedge with the index than a low strike equity tranche or a high strike senior tranche. . Second, on the ‘degree of Markovianity’ of the loss process L, which in the case of the homogeneous groups model depends both on the model nominal dimension d and on the specification of the intensities (see, e.g., remark 1).

Moreover, it is intuitively clear that, for too large values of t1, time-decay effects matter and the hedge should be re-balanced at some intermediate points of the time interval [0, t1] (even though no default has yet occurred). To keep it as simple as possible we shall merely apply a cutoff and restrict our attention to the random set {! : t1(!)5T1} for some fixed T1 2 [0, T ]. 4.2. Numerical results We work with the above model for d ¼ 2 and ¼ 5. We thus consider a two-dimensional model of a stylized credit portfolio of n ¼ 8 obligors. The model generator is a d  d-(sparse) matrix with 2d ¼ 54 ¼ 625. Recall that the computation time for exact pricing using a matrix exponentiation based on (12) in such a model increases as 2d, which motivated the previous modest choices for d and . Moreover, we take e l given by (see (9)) 2ð1 þ i1 Þ e  1 ðt, {Þ ¼ , 9n

e2 ðt, {Þ ¼ 16ð1 þ i2 Þ :  9n

ð14Þ

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Therefore, in this case, which is an admittedly extreme case of inhomogeneity between two independent groups of obligors, the individual intensities of the obligors of groups 1 and 2 are given by (1 þ i1)/36 and [8(1 þ i2)]/36, where i1 and i2 represent the number of currently defaulted obligors in groups 1 and 2, respectively. For instance, at time 0 with N 0 ¼ (0, 0), the individual intensities of the obligors of groups 1 and 2 are equal to 1/36 and 8/36, respectively; the average individual intensity at time 0 is thus equal to 1/8 ¼ 0.125 ¼ 1/n. We set the maturity T equal to 5 years and the cutoff T1 equal to 1 year. We thus focus on the random set of trajectories for which t151, meaning that a default occurred during the first year of hedging. In this toy model the simulation takes the following very simple form. Compute 0 for the tranche and 0 for the index by numerical matrix exponentiation based on (12). Then, for every j ¼ 1, . . . , m: . draw a pair ðe t1j , b t1j Þ of independent exponential random variables with parameter (see (9) and (14))     1 8 1 8 1 2 , ¼ , ; ð15Þ ð0 , 0 Þ ¼ 4  36 36 9 9 . set t1j ¼ minðe t1j Þ and N t1 ¼ (1, 0) or (0, 1) t1j , b t1j or b t1j ; depending on whether t1j ¼ e . compute t j for the tranche and t j for the index 1 1 using (12). Doing this for m ¼ 104, we obtain 9930 draws with t15T ¼ 5 years, of which 6299 are with t15T1 ¼ 1 years, subdividing themselves into 699 defaults in the first group of obligors and 5600 defaults in the second group.

4.2.1. Pricing results. We consider two T ¼ 5 year tranches in the above model: an ‘equity tranche’ with k ¼ 30%, corresponding to a payoff   ð1  RÞNT 60NT ^k¼ ^ 30 % n 8 (of a unit nominal amount), and a ‘senior tranche’ defined simply as the complement of the equity tranche to the index, thus with payoff  þ  þ ð1  RÞNT 60NT k ¼  30 %: n 8 We computed the portfolio loss distribution at maturity by numerical matrix exponentiation corresponding to explicit solution of the associated forward Kolmogorov equations (see, e.g., Carmona and Cre´pey 2009). Note that there is virtually no error involved in the previous computations, in the sense that our simulation is exact (without simulation bias), and the prices and loss probabilities are computed by numerical quasi-exact matrix exponentiation. The left side of figure 2 represents the histogram of the loss distribution at the time horizon T; we indicate by a vertical line the loss level x beyond which the equity tranche is wiped out, and the senior tranche starts being

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T.R. Bielecki et al.

Figure 2. (Left) Portfolio loss distribution at maturity T ¼ 5 years. (Right) Tranche prices at t1 for t15T ¼ 5 (equity tranche (þ), senior tranche () and index (*)). In this and the following figures, blue and red points correspond to defaults in the first and second group of obligors, respectively.

hit (therefore, [(1  R)x]/n ¼ k, i.e. x ¼ 4). The right side of figure 2 displays the equity (labeled þ), senior () and index (*) tranche prices at t1 (in ordinate) versus t1 (on the abscissa), for all the points in the simulated data with t155 (9930 points). Blue and red points correspond to defaults in the first (N t1 ¼ (1, 0)) and in the second group of obligors, respectively. (N t1 ¼ (0, 1)) We also present in black the points ð0, 0 Þ (for the equity tranche and the senior tranche) and (0, 0) (for the index). Note that in the case of the senior tranche and of the index, there is a clear difference between prices at t1 depending on whether t1 corresponds to a default in the first or in the second group of obligors, whereas in the case of the equity tranche there seems to be little difference in this regard. In view of the portfolio loss distribution on the left side, this can be explained by the fact that, in the case of the equity tranche, the probability

Figure 3. Equity tranche versus index price changes between 0 and t1 (left: t15T ¼ 5; right: zoom on t15T1 ¼ 1).

conditional on t1 that the tranche will be wiped out at maturity is important unless t1 is rather large. Therefore, the equity tranche price at t1 is close to k ¼ 30% for t1 close to 0. Moreover, for t1 close to T the intrinsic value of the tranche at t1 constitutes the major part of the equity tranche price at t1, for the tranche has low time-value close to maturity. In conclusion, the state of N at t1 has a small impact on t1, unless t1 is in the middle of the time-domain. On the other hand, in the case of the senior tranche or in the case of the index, the state of N at t1 has a large impact on the corresponding price, unless t1 is close to T (in which case intrinsic value effects are dominant). This explains the ‘two-track’ pictures seen for the senior tranche and for the index on the right side of figure 2, except close to T (whereas the two tracks are superimposed close to 0 and T in the case of the equity tranche). Looking at these results in terms of price changes 0  t1 of a tranche versus the corresponding index price changes 0  t1, we obtain the graphs of figure 3 for

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Up and down credit risk

Table 1. Time t ¼ 0—prices, spreads and instantaneous deltas in the semi-homogeneous model. 0 or 0

0 or S0

10

20

0

Eq 0.2821814 1881.209 0.1396623 0.7157741 0.2951399 Sen 0.03817907 254.5271 0.8603377 0.2842259 0.7048601

4.2.2. Hedging results. We then computed the empirical variance of t1  0 and of the profit-and-loss et1 in (13) on the subset t15T1 ¼ 1 of the trajectories, using for b 0 the empirical regression delta of the tranche with respect to the index at time 0. Therefore, b 0 ¼

d Covð t1  0 , t1  0 Þ : d t  0 Þ Varð 1

ð16Þ

Moreover, we also performed these computations restricting further attention to the subsets of t151 corresponding to defaults in the first and in the second group of obligors (blue and red points in the figures), respectively. The latter results are to be understood as giving proxies of the situation that would prevail in a one-dimensional complete model of credit risk (‘local intensity model’ for N (see section 4.2.3). The results are displayed in tables 1 and 2. In table 1 we denote by:

Figure 4. Senior tranche versus index price changes between 0 and t1 (left: t15T ¼ 5; right: zoom on t15T1 ¼ 1).

the equity tranche and figure 4 for the senior tranche. We consider all points with t15T on the left side and focus on the points with t15T1 in the right side. We use the same blue/red color code as above, and we further highlight in green on the left side the points with t151, which are focused upon on the right side. Figure 3 gives a further graphical illustration of the low level of correlation between price changes of the equity tranche and of the index. Indeed, the cloud of points on the right side is obviously ‘far from a straight line’, due to the partitioning of points between blue points/defaults in group 1 on one segment versus red points/defaults in group 2 on a different segment. In contrast (figure 4), at least for t1 not too far from 0 (see the close up view on the points for which t151 on the right side), there is evidence of a linear correlation between price changes of the senior tranche and of the index, since, in this case, the blue and the red segments are not far from being on a common line.

. 0 ¼ (104/kT)0 or [104/(1  R  k)T ]0 (for the equity or senior tranche) or S0 ¼ [104/(1  R)T ]0 (for the index), stylized ‘bp spreads’ corresponding to the time zero prices 0 and 0 of the equity or senior tranche and of the index; . 10 , 20 and 0, the functions 1u/ 1v, 2u/ 2v and the continuous time min-variance delta function (as it follows easily by application of a bilinear regression formula)  1  1 ð 1 uÞð 1 vÞ þ 2 ð 2 uÞð 2 vÞ 1 ð 1 vÞ2 u ¼ 2 2 2 2 1 v 1 1 2 2 1 1 2 2  ð vÞ þ  ð vÞ  ð vÞ þ  ð vÞ  2  2 ð 2 vÞ2 u þ , 2 2 2 v 1 1 2 2  ð vÞ þ  ð vÞ evaluated at t ¼ 0 and ¼ N 0 ¼ (0, 0), therefore u1,0  u0,0 u0,1  u0,0 ð0Þ, 20 ¼ ð0Þ, 10 ¼ v1,0  v0,0 v0,1  v0,0 0 ¼

ð17Þ

10 ðu1,0  u0,0 Þðv1,0  v0,0 Þ þ 20 ðu0,1  u0,0 Þðv0,1  v0,0 Þ 10 ðv1,0  v0,0 Þ2 þ 20 ðv0,1  v0,0 Þ2

where we recall from (15) that ð10 , 20 Þ ¼ ð1=9, 8=9Þ.

,

ð18Þ

The three deltas 10 , 20 and 0 were thus computed by matrix exponentiation based on (12) for the various terms u, v (0) involved in formulas (17) and (18). Note that the prices and deltas of the equity and senior tranche of the same strike k sum to  and to 1, by construction (see also table 2 for b 0 ). Therefore, the results for the senior tranche could be deduced from those for the equity tranche, and conversely. However, we present detailed results for the equity and senior tranche for the reader’s convenience.

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T.R. Bielecki et al. Table 2. Hedging tranches by the index in the semi-homogeneous model.

Eq Eq1 Eq2 Sen Sen1 Sen2

b 0

0.00275974 0.2269367 0.3391563 1.002760 0.7730633 0.6608437



R2

Dev

RedVar

0.03099014 0.9980242 0.997375 0.9960836 0.9998293 0.9993066

0.0009603885 0.9960522 0.994757 0.9921825 0.9996586 0.9986137

0.006612626 0.007576104 0.006134385 0.07475331 0.02576152 0.01192970

1.000961 253.306 190.7276 127.9176 2928.847 721.3244

Remark 2: The instantaneous min-variance delta 0 (which is a suitably weighted average of 10 and 20 ) can be considered as a measure of the distance to the index of a tranche: a far-from-the-index low strike equity tranche or a high strike senior tranche with 0 less than 0.5, versus a close-to-the-index high strike equity tranche or low strike senior tranche with 0 greater than 0.5. The further from the index a tranche and/or ‘the less Markovian’ a porfolio loss process L, the poorer the hedge by the index (see end of section 4.1.2). In table 2 (see also (16)): .  in column 2 is the empirical correlation of the tranche price increments t1  0 versus the index price increments t1  0; d t  0 Þ in column 3 is d t Þ=Varð . R2 ¼ 1  Varðe 1 1 the coefficient of determination of the regression, which, in the present setup of a simple linear regression, coincides with 2; d . Dev in column 4 stands for Stdevð t1  0 Þ=0 ; . the hedging variance reduction factor d t Þ in the last d t  0 Þ=Varðe RedVar ¼ Varð 1 1 column is equal to 1/(1  R2) ¼ 1/(1  2).

Remark 3: It is expected that b 0 should converge to 0 in the limit where the cutoff T1 would tend to zero, provided that the number of simulations m jointly goes to infinity. For T1 ¼ 1 year and m ¼ 104 simulations, however, we shall see below that there is a clear discrepancy between 0 and b 0 , and all the more so that we are in a non-homogeneous model with low correlation between the tranche and index price changes between times 0 and t1. The reason for this is that the coefficient of determination of the linear regression with slope b 0 is given by R2 ¼ 2. In the case where  is small, R2 is even smaller, and the significance of the estimator (for low T1) b 0 of 0 is also low. In other words, in the case where  is small, we recover mainly noise through b 0 . This, however, does not weaken our statements below regarding the ability or not to hedge the tranche by the index, since the variance d t  0 Þ=Varðe d t Þ is reduction factor RedVar ¼ Varð 1 1 2 equal to 1/(1   ), which, for  small, is close to one, whatever the noisy value of b 0 may be.

Recall that, qualitatively, the senior tranche’s dynamics is rather close to that of the index (at least for t1 close to 0, see section 4.2.1, the right side of figure 4). Accordingly, we find that hedging the senior tranche with the index is possible (variance reduction factor of about 128 (entry in

Table 3. Replicating the equity tranche by the index and the senior tranche in the semi-homogeneous model. b ind 0 1

b sen 0 1

RedVar 2.56eþ29

bold/italic type in the last column)). This case thus seems to be supportive of the claim according to which one could use the index for hedging a loss derivative, even in a non-Markovian model of portfolio loss process L. However, in the case of the equity tranche we obtain the opposite message: the index is useless for hedging the equity tranche (the variance reduction factor is essentially equal to 1 (shown in bold type) in the table, therefore there is no variance reduction in this case). Moreover, the equity tranche variance reduction factors conditional on defaults in the first and second groups of obligors (italic type in the table) amount to 253 and 190, respectively. This supports the interpretation that the unhedgeability of the equity tranche by the index really comes from the fact that the full model dynamics is not represented in the loss process L. Incidentally, this also means that hedging the senior tranche by the equity tranche, or vice versa, is not possible either. We conclude that, in general, at least for certain ranges of the model parameters and tranche characteristics (strongly non-Markovian loss process L and/or far-from-the-index tranche), hedging tranches with the index may not be possible. Since the equity and the senior tranche sum up to the index, a perfect static replication of the equity tranche is provided by a long position in the index and a short position in the senior tranche. As a reality check of this statement, we performed a bilinear regression of the equity price increments versus the index and the senior tranche price increments in order to minimize over bsen ðb ind 0 , 0 Þ the (risk-neutral) variance of eq sen bind bsen sen e et1 ¼ ðeq t1  0 Þ  0 ðt1  0 Þ  0 ðt1  0 Þ: ð19Þ

The results are displayed in table 3. We recover numerically the perfect two-instruments replication strategy which was expected theoretically, whereas a single-instrument hedge using only the index was essentially useless in this case (see entry in bold type in table 2).

4.2.3. Fully homogeneous case. For confirmation of the previous analysis and interpretation of the results, we re-performed the computations using the same values as

Up and down credit risk

Figure 5. (Left) Portfolio loss distribution at maturity T ¼ 5 years. (Right) Tranche prices at t1 (for t15T ).

before for all the models, products and simulation parameters, except for the fact that the following pre-default individual intensities were used, for l ¼ 1, 2: ! P X i‘ 1 1‘d l e ð{Þ ¼ þ b  ¼:  i‘ : ð20Þ nd n 1‘d

For instance, at time 0 with N 0 ¼ 0, the individual intensities of the obligors are all equal to 1/8 ¼ 0.125 ¼ 1/n. We are thus in a case of homogeneous obligors, reducible to a local intensity model (with d ¼ 1 and pre-default individual intensity b ðiÞ therein; see remark 1). Therefore, in this case we expect that hedging tranches by the index should work, including in the case of the far-from-the-index equity tranche. This is what happens numerically, as is evident from figures 5–7 and tables 4 and 5 (which are the analogs of those in previous sections, using the same notation everywhere). Note that all red and blue curves are now superimposed, which is consistent with the fact that the

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Figure 6. Equity tranche versus index price decrements between 0 and t1 (left: t15T ¼ 5; right: zoom on t15T1 ¼ 1).

group of a defaulted name has no bearing in this case, given the present specification of the identities. Of the new 104 draws using the intensities given in (20), we obtained 9922 draws with t155, of which 6267 were with t151, subdivided into 3186 defaults in the first group of obligors and 3081 defaults in the second group. Looking at table 5, we find, as in the semi-homogeneous case, that hedging the senior tranche with the index works very well, and even better than before: the variance reduction factor of 11,645 in bold/ italic type in the last column. Yet these even better results may be partly due to an effect of distance on the index and not only to the fact that we are now in a fully homogeneous case: the senior tranche is now closer to the index than before, with a senior tranche 0 of about 0.7 in table 1 versus 0.8 in table 4. But, as opposed to the situation in the semi-homogeneous case, hedging the equity tranche with the index now also works very well (variance reduction factor of about 123 (entries in italic type in the last column)), and this holds even though the

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T.R. Bielecki et al. equity tranche is further from the index now than it was before, with an equity tranche 0 of about 0.3 in table 1 versus 0.2 in table 4 (see remark 2). Therefore, the degradation of the hedge when we pass from the homogeneous model to the semi-homogeneous model is really due to the non-Markovianity of L, and not to an effect of distance on the index (see the end of section 4.1.2). Moreover the unconditional variance reduction factor and variance reduction factor conditional on defaults in the first and second groups of obligors are now essentially the same (for the equity tranche as for the senior tranche). This also means that hedging the equity tranche by the senior tranche, or vice versa, is quite effective in this case. These results support our previous analysis that the impossibility of hedging the equity tranche by the index in the semi-homogeneous model was due to the non-Markovianity of the loss process L. Note, incidentally, that b 0 and 0 are closer now (in tables 4 and 5) than they were previously (in tables 1 and 2). This is consistent with the fact that R2 is now larger than before (b 0 and 0 would be even closer if the cutoff T1 was less than 1 year, provided, of course, that the number of simulations m is large enough; see remark 3).

5. Conclusions and perspectives

Figure 7. Senior tranche versus index price decrements between 0 and t1 (left: t15T ¼ 5; right: zoom on t15T1 ¼ 1).

Table 4. Time t ¼ 0—prices, spreads and instantaneous deltas in the fully homogeneous model. 0 or 0

0 or S0

10

20

0

Eq 0.2850154 1900.103 0.2011043 0.2011043 0.2011043 Sen 0.1587075 1058.050 0.7988957 0.7988957 0.7988957

In the case of a non-Markovian portfolio process L, factor hedging of a loss derivative by another one may not work, and hedging by single-name credit derivatives, such as CDS contracts, may then be necessary. Models with filtration that is at least as large as the filtration H, that is the filtration of the indicator processes of all the default times in the pool, are the only ones which are able to deal with this issue in a theoretically sound way. Such models can, arguably, be constructed in a top-down way by thinning, starting from a top model with ‘nice’ dynamics for the portfolio process L. But focusing on having a model with a ‘nice’ dynamics for the top process L is misguided for dealing with a situation in which the index does not do the job in terms of hedging, for such a situation precisely means that the market dynamics of the top process L is not nice, and, as illustrated in section 4, insisting on using a simplistic model for L in a complex world may lead to a highly ineffective hedge. It is thus our opinion that bottom-up models are the only ones that are really suited to deal in a self-consistent way with the issue of hedging credit loss derivatives by single-name derivatives.

Table 5. Hedging tranches by the index in the fully homogeneous model.

Eq Eq1 Eq2 Sen Sen1 Sen2

b 0

0.0929529 0.09307564 0.09282067 0.9070471 0.9069244 0.9071793



R2

Dev

RedVar

0.9959361 0.995929 0.995946 0.999957 0.9999569 0.9999573

0.9918887 0.9918745 0.9919084 0.9999141 0.9999137 0.9999146

0.004754811 0.004794916 0.004713333 0.04621152 0.04653322 0.0458808

123.2852 123.0695 123.5853 11,645.15 11,590.83 11,710.42

Up and down credit risk 5.1. A tractable bottom-up model of portfolio credit risk A common objection to the use of a bottom-up model is made with regard to the issue of the so-called curse of dimensionality. In this regard we wish to stress that suitable developments in bottom-up modeling enable one to efficiently cope with this curse of dimensionality (see, for instance, Elouerkhaoui 2007 or Bielecki et al. 2008). It is thus possible to specify bottom-up Markovian models of portfolio credit risk with automatically calibrated model marginals (to the individual CDS curves, say). Much like in the standard static copula models, but in a dynamized setup, this effectively reduces the main computational effort, i.e. the effort related to model calibration, to the calibration of only a few dependence parameters in the model at hand. Thus, model calibration can be achieved in a very reasonable time, also by pure simulation procedures if need be (without using any closed pricing formulae, if there are none available for the model under consideration). To illustrate the previous statements let us briefly present a simple model (see Bielecki et al. 2008 for more general theory and models). We postulate that, for i 2 Nn , the individual default indicator process Hi is a Markov process admitting the following generator, for u ¼ ue(t) with e 2 {0, 1}: Ait ue ðtÞ ¼ i ðtÞðu1 ðtÞ  ue ðtÞÞ, for a pre-default intensity function i of name i given by i(t) ¼ ai þ bit. For constant and given interest rate r  0 and recovery rate Ri 2 [0, 1], the individual time-t ¼ 0 spread i0 ðTÞ is then given by a standard explicit formula in terms of r, Ri, ai and bi (see Bielecki et al. 2008 for details). The (non-negative) parameters ai and bi are then fitted so as to match the five and 10 year (say) spreads of the related credit index constituents. Next, in order to couple together the model marginals Hi, we define a certain number of groups of obligors susceptible to default simultaneously. Setting n ¼ 125, for l 2 L ¼ {10, 20, 40, 125}, we thus define Il as the set containing the indices of the l riskiest obligors, as measured by the spread of the corresponding five year CDS. In particular, we have I10 I20 I40 I125 ¼ Nn ¼ f1, 2, . . . , 125g: Let I ¼ {Il}l2L. We then construct the generator of process H ¼ (H1, . . . , Hn) as, for u ¼ u(t) with  ¼ (e1, . . . , en) 2 {0, 1}n, At u ðtÞ ¼

125 X X l ðtÞÞðui ðtÞ  u ðtÞÞ ði ðtÞ  i¼1

þ

X

I2I ;I3i

I ðtÞðuI ðtÞ  u ðtÞÞ,

I2I

where, for i 2 Nn and I 2 I , . i (respectively I) denotes the vectors obtained from  by replacing the component ei (respectively the components ej for j 2 I) by 1, and

1149

. I ðtÞ ¼ a I þ bI t, with a I ¼ I min{i2I} ai, bI ¼ I minfi2Ig bi , for some [0, 1]-valued model dependence parameters i. In other words, the form of the above generator implies that, at every time instant, either each alive obligor can default individually, or all the surviving names whose indices are in one of the sets I 2 I can default simultaneously. Observe that the martingale dimension (or multiplicity, see section 4.1.2) of the model is 125 þ 4 ¼ 129. This makes the simulation of process H very fast, as we essentially need to draw 125 þ 4 i.i.d. exponential random variables in order to recover a set of default times ði Þi2Nn . Pricing CDO tranches in this model can thus be effectively done by simulation. Moreover, we only need to calibrate four parameters, namely I with I 2 I (since the marginal model parameters aI and bI were calibrated in a previous stage). Finally, since this is a bottom-up Markov model, dynamic delta hedging by multi- and single-name derivatives can be considered in a model-consistent way, in the sense of replication if there are enough hedging instruments at hand, or in a min-variance sense in any case. Of course, the grouping of the names in the above model is rather arbitrary. Also, it would be better to use exact or approximate analytics for the CDO tranches, rather than relying only on simulation as proposed above. There is thus much room for improvement. However, we refer the reader to Bielecki et al. (2008) for numerical results on real market data demonstrating that this simple approach does a very good job in practice in terms of calibration to CDS and CDO data.

Acknowledgements The research of T.R. Bielecki was supported by NSF grant 0604789. The research of S. Cre´pey and M. Jeanblanc benefited from the support of the ‘Chaire Risque de cre´dit’, Fe´de´ration Bancaire Franc¸aise, and the Europlace Institute of Finance.

References Arnsdorf, M. and Halperin, I., BSLP: Markovian bivariate spread-loss model for portfolio credit. Working Paper, 2007. Bennani, N., The forward loss model: a dynamic term structure pricing of portfolio credit derivatives. Working Paper, 2006. Bielecki, T.R., Cre´pey, S., Jeanblanc, M. and Rutkowski, M., Valuation of basket credit derivatives in the credit migrations environment. In Handbook of Financial Engineering, edited by J. Birge and V. Linetsky, 2007 (Elsevier: Amsterdam). Bielecki, T.R., Vidozzi, A. and Vidozzi, L., A Markov copulae approach to pricing and hedging of credit index derivatives and ratings triggered step-up bonds. J. Credit Risk, 2008, 4, 47–76. Bre´maud, P., Point Processes and Queues, Martingale Dynamics, 1981 (Springer: Berlin). Bre´maud, P. and Yor, M., Changes of filtrations and of probability measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1978, 45, 269–295.

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Carmona, R. and Cre´pey, S., Importance sampling and interacting particle systems for the estimation of Markovian credit portfolios loss distribution. IJTAF, 2009, forthcoming. Cont, R. and Kan, Y.H., Dynamic hedging of portfolio credit derivatives. Working Paper, 2008. Cont, R. and Minca, A., Recovering portfolio default intensities implied by CDO quotes. Working Paper, 2008. Davis, M.H. and Varaiya, P., The multiplicity of an increasing family of -fields. Ann. Probab., 1974, 2, 958–963. Dellacherie, C. and Meyer, P.-A., Probabilities and Potential, 1982 (North-Holland: Amsterdam). Duffie, D. and Garleanu, Risk and valuation of collateralized debt olbligations. Financial Anal. J., 2001, 7, 41–59. Ehlers, P. and Scho¨nbucher, P., Background filtrations and canonical loss processes for top-down models of portfolio credit risk. Working Paper, 2006. Elouerkhaoui, Y., Pricing and hedging in a dynamic credit model. Int. J. Theor. Appl. Finance, 2007, 10, 703–731. Frey, R. and Backhaus, J., Portfolio credit risk models with interacting default intensities: a Markovian approach. Working Paper, 2004. Frey, R. and Backhaus, J., Dynamic hedging of synthetic CDO tranches with spread risk and default contagion. Working Paper, 2007. Frey, R. and Backhaus, J., Pricing and hedging of portfolio credit derivatives with interacting default intensities. IJTAF, 2008, 11, 611–634. Gaspar, R. and Schmidt, T., Quadratic models for portfolio credit risk with shot-noise effects. Working Paper, 2008. Giesecke, K. and Goldberg, L., A top down approach to multiname credit. Working Paper, 2007.

Halperin, I. and Tomecek, P., Climbing down from the top: single name dynamics in credit top down models. Working Paper, Quantitative Research, JP Morgan, September 2008. Herbertsson, A., Pricing portfolio credit derivatives. PhD thesis, Go¨teborg University, 2007. Herbertsson, A., Pricing kth-to-default swaps under default contagion: the matrix-analytic approach. J. Comput. Finance, 2008, 12, 1–30. Inglis, S., Lipton, A., Savescu, I. and Sepp, A., Dynamic credit models. Statist. Interface, 2009, forthcoming. Jacod, J., Calcul Stochastique et Proble´mes de Martingales (Lecture Notes in Mathematics, Vol. 714), 1979 (Springer: Berlin). Last, G. and Brandt, A., Marked Point Processes on the Real Line: The Dynamical Approach, 1995 (Springer: Berlin). Laurent, J.-P., Cousin, A. and Fermanian, J.-D., Hedging default risks of CDOs in Markovian contagion models. Working Paper, 2007. Protter, P.E., Stochastic Integration and Differential Equations, 2nd ed., version 2.1, 2005 (Springer: Berlin). Scho¨nbucher, P., Portfolio losses and the term structure of loss transition rates: a new methodology for the pricing of portfolio credit derivatives. Working Paper, 2006. Sidenius, J., Piterbarg, V. and Andersen, L., A new framework for dynamic credit portfolio loss modelling. Working Paper, 2006. Zargari, B., Dependence modelling in multi-name credit risk. PhD thesis, University of Evry–University of Sharif, 2009 (in preparation).

Appendix A: A glimpse at the general theory

non-decreasing process ðKpt Þt0 , called the dual predictable projection of K on F, such that, for any positive predictable process H, Z 1  Z 1  Hs dKps : Hs dKs ¼ E E

For the convenience of the reader, in this appendix we recall definitions and results from the theory of the processes that we used in the paper. We refer to, for example, Dellacherie and Meyer (1982) for a comprehensive exposition. Let us be given a standard stochastic basis ( , F , F, P). The probability measure P will be fixed throughout. By default, all the filtration-dependent notions like adapted, stopping time, compensator, intensity, (local) martingale, etc., implicitly refer to the filtration F (as opposed to, for instance, the larger filtration G which appears in section A.3). A.1. Optional projections Let X be an integrable process, not necessarily (F-) adapted. Then there exists a unique adapted process (oXt)t0, called the optional projection of X on F, such that, for any stopping time , EðX 11f5þ1g jF  Þ ¼ o X 11f5þ1g : In the case where X is non-decreasing, then sub-martingale.

o

X is a

0

0

In the case where K is adapted, it is a sub-martingale, and it admits as such a unique Doob–Meyer decomposition Kt ¼ Mt þ t ,

ðA1Þ

where M is a uniformly integrable martingale (since K is bounded) and the compensator  of K is a predictable finite variation process. Therefore, K p ¼ , by identification in the Doob–Meyer decomposition. If, moreover, K is stopped at some stopping time , and if K p ¼  is continuous, then K p ¼  is also stopped at , by uniqueness of the Doob–Meyer decomposition of K ¼ KR6 . In the case where  is time-differentiable,   ¼ 0 t dt for some intensity process  of K (also called the intensity of  when K ¼ It for some stopping time ), and the intensity process  vanishes after . Remark A1: If K is a point process (like a marginal or cumulative default process Hi or L in this paper), the continuity of  is equivalent to the ordered jump times of K being totally inaccessible stopping times (see, e.g., Dellacherie and Meyer 1982).

A.2. Dual predictable projections and compensators Let K be a non-decreasing and bounded process, not necessarily adapted (typically in the context of this paper, K corresponds to marginal or portfolio loss processes Hi or L). Then there exists a unique predictable

A.3. Proof of proposition 2.1 We recall that, in the context of proposition 2.1, K denotes an F-adapted non-decreasing process with F G,

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Up and down credit risk while  and denote the F-compensator and the G-compensator of K, respectively. Let  denote the F-predictable non-decreasing component of the F-sub-martingale o , the optional projection of on F (see section A.1). The tower property of iterated conditional expectations yields  Z T dKu  d  u F t E t  Z T ¼E dKu  dðo Þu F t t  Z T ¼E dKu  d u F t t    Z T ¼E E dKu  d u Gt F t ¼ 0,

that Z

is an G-martingale. This proves that  ¼ :

ðA2Þ

Moreover, one has  ¼ p , the dual predictable projection of on F (see, e.g., proposition 3 of Bre´maud and Yor 1978), hence ¼

p

,

as stated in words in proposition 2.1(i). Now, in the case where  and are time-differentiable with related intensity processes  and , (A2) indicates

0

 Z t s ds  E s ds F t 0

ðA3Þ

is an F-martingale. Moreover it is immediate to check, using the tower property of iterated conditional expectations, that E

Z

t

 Zt Eðs j F s Þ ds s ds F t  0

0

ðA4Þ

is also an F-martingale. By addition between (A3) and (A4), Z

t

since K 

t

0

t

s ds 

Z

t 0

Eðs j F s Þ ds

is in turn an F-martingale. Since it is also a predictable (continuous) finite variation process, it is thus in fact identically equal to 0, therefore for t  0, t ¼ Eðt j F t Þ, and therefore  ¼ o , which is the statement of proposition 2.1(ii).

China Journal of Accounting Studies, 2013 Vol. 1, Nos. 3–4, 236–259, http://dx.doi.org/10.1080/21697221.2013.867401

Why do state-owned enterprises over-invest? Government intervention or managerial entrenchment Jun Baia* and Lishuai Lianb a

School of Economics and Management, Shihezi University, Xinjiang 832000, China; bSchool of Management, Fudan University, Shanghai, 200433, China In a transition economy, corporate investment decisions are affected not only by managerial discretion, but also by government intervention. Using the data of publicly listed state-owned enterprises (SOEs) in China, we investigate how government intervention and corporate managerial entrenchment affect over-investment. The results show that both the policy burden from government intervention and rentseeking due to managerial entrenchment can lead to over-investments, and these two effects appear to be complementary to each other. With a weak government intervention, managerial discretion is greater and management behavior tends toward opportunism. Keywords: government intervention; managerial entrenchment; over-investment

1. Introduction China’s economy has been growing rapidly since the start of its economic reform in 1978. Its GDP reached US$6.01 trillion in 2010 (National Bureau of Statistics of China, 2011) and it is now the world’s second largest economy. The China’s growth pattern can be described by the investment-growth model (Zhang, 2003). The investment-to-GDP ratio in China, with continuous rapid growth from 36% to 48% in the period of 2001 and 2010, is high relative to the world average (23.7% in 2010), and it is also higher than most other East Asian countries, which are also known for relying on capital accumulation for their growth (27% in 2010).1 Despite this, a high investment-to-GDP ratio does not necessarily correspond to optimal investment as firms may over-invest, e.g. firms’ engagement in redundant constructions (Tang, Zhou, & Ma, 2007, 2010). This problem is particularly severe among state-owned enterprises (SOEs), where investment efficiency is significantly lower than that of domestic private or foreign-owned firms (Dollar & Wei, 2007).2 Traditional investment theory suggests that over-investment is rooted in efficiency loss for an internal agency chain. In modern firms, where ownership and control are separate, managers, acting as agents, hold the significant control rights over firms’ asset allocation decisions (Shleifer & Vishny, 1997). Managers may pursue their own private goals that might be in conflict with those of outside shareholders. Managers can take wasteful, negative net present value investment projects as a result of their desire to derive personal benefits from controlling more assets, which leads to over-investment (e.g., Jensen, 1986, 1993; Stulz, 1990). *Corresponding author. Email: [email protected] Paper accepted by Tong Yu. © 2013 Accounting Society of China

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In a transition economy, the government plays a vital role in the business activities of firms, and more profoundly impacts SOEs’ investment decisions (Chen, Sun, Tang, & Wu, 2011). The enhanced government control over SOEs, due to fiscal decentralization reform (Qian & Weingast, 1997) and the reform of political promotion mechanisms (Blanchard & Shleifer, 2001; Li & Zhou, 2005), allows the government to impose economic growth and social stability targets on firms by intervening in their investment decisions. This makes the investment decisions deviate from maximizing the benefits of shareholders, thereby promoting SOEs’ over-investment. Furthermore, during gradual decentralization in the SOE reformation, the role of corporate managers becomes increasingly important. Managers may maximize their rent-seeking through investment expansion, which yields SOEs’ over-investment (Wei & Liu, 2007; Xin, Lin, & Wang, 2007; Zhang, Wu, & Wang, 2010). Government intervention and managerial entrenchment have become the primary forces driving SOEs’ over-investment, and these two forces potentially interact with each other. Therefore, to understand the influences of government intervention and managerial entrenchment on over-investment, the paper examines the following. (1) What are the impacts of these two forces on SOEs’ investment decisions? (2) How do they interact with each other during the process? Differently put, are they supplementary or complementary to each other? (3) Which one is the dominant force? Is there a difference in managerial entrenchment under heterogeneous levels of government interventions? To address the above questions, the paper investigates the impacts of policy burden from government intervention and rent-seeking behavior that originated from managerial entrenchment on firms’ over-investment. We find that in SOEs, both government intervention and managerial entrenchment can yield over-investment, but the two have a complementary relationship in affecting over-investment. More powerful government interventions suppress managerial entrenchment, and management behavior tends toward opportunism in a weak government intervention environment. The paper explores the interactions between the government and firm management from an investment perspective and contributes to the literature in three ways. First, we provide microscopic evidence on how government behavior affects economic growth, which enables us to better understand the mechanism underlying the impacts of government behavior on microscopic financial decisions, thereby enriching the extant literature on the relationship between government and firms. China’s economy is in a transition stage, where various government policies are undergoing reforms. The relationship between government behavior and economic growth has been the focus of academic research, but the micro-channels and mechanisms through which government affects economic growth have not been well-understood. Herein, by incorporating government behavior into the firm micro-financial decision-making framework, this paper investigates the impacts of mutual influences between government intervention and management behavior on firms’ investment decisions, providing a new theoretical perspective to better understand current government behavior in China and its economic consequence. Second, by measuring the ‘policy burden’ on firms, this paper evaluates government behavior from a microscopic perspective, bridging studies in macro-public and micro-corporate governance. Following La Porta, Lopez-de-Silanes, Shleifer, and Vishny, (1998), studies have recognized the importance of the impacts of government behavior and institutional environment on firm’s financial decisions. However, little has been done to quantify government intervention. Based on the ratio and structure of government ownership and focusing on the Chinese financial market, researchers have attempted to quantify government behavior mainly from two aspects. One is by

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assessing government equity (Sun & Tong, 2003; Tang et al., 2010; Wu, 2009; Xia & Chen, 2007). The other is using pyramid structure as the substitution variable of government decentralization (Fan, Wong, & Zhang, 2012) or intervention (Cheng, Xia, & Yu, 2008; Zhong, Ran, & Wen, 2010). In line with the view that the policy burden reflects the direct economic consequence of government intervention in SOEs (Lin & Li, 2008), we employ policy burden as a proxy for government intervention, which is a new way to quantify government behavior at the micro level. Finally, our findings shed new light on the literature that examines distortional investment behavior due to the traditional agency conflicts between shareholders and managers as well as the conflicts between shareholders and debt holders in mature markets. The traditional investment theory suggests that corporate investment is independent of the government regulatory environment. Studies focus on the impacts of the internal stockholder, manager and external investor agency relationships on firm financial decision-making (Jensen & Meckling, 1976; Shleifer & Vishny, 1986, 1989). We show that governments in transition economies like China can expropriate the firms by intervening in their investment decisions. Meanwhile government activities also restrict the managerial entrenchment. By considering the government intervention and managerial entrenchment in the transitional economy of China, the findings facilitate our understanding of these institutional features’ impacts on SOEs’ investment activities. The remainder of the paper proceeds as follows. Section 2 discusses the institutional background and the theoretical framework of the study. Section 3 presents the research design, introduces the samples, data and models adopted in the paper. Section 4 reports the empirical findings and Section 5 concludes. 2. Institutional background and theoretical analysis 2.1. Inducements of over-investment Investment decisions are not only vital for the survival and growth of firms at the micro level, but also they serve an economic engine at the macro-economic level. In a perfect capital market, a firm’s investment policy is solely dependent on its investment opportunities (Fazzari, Hubbard, & Petersen, 1988; Modigliani & Miller, 1958). However, in reality, firms would deviate from the optimal investment level for various reasons. Investment decisions reflect not only the agency relationship within the firms (Bushman & Smith, 2001; Stein, 2003) but also government behavior and the relationship between government and firms (Macroeconomic Study Group, CCER, Peking University, 2004). At the firm level, agency problems would lead managers to over-invest when they pursue their own interests (Jensen, 1986). In the government and firm relationship, to accomplish social and political goals, the government has strong incentives to intervene in firms’ decision-making, which may also lead to over-investment. The problem is particularly severe for SOEs (Chen, Sun, Tang, & Wu, 2011). 2.1.1. Managerial entrenchment and over-investment At the firm level, the agency problem stems from the separation of ownership and control. When managers end up with significant control rights over how to allocate investors’ funds, they may use their discretion for rent-seeking. This incurs great costs to the firms, and thereby reduces shareholders’ wealth. The agency theory suggests that managers tend to expand investments and are motivated to invest free cash in non-profit projects for their personal benefits. In other words, managers have incentives

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to expand investment beyond the optimal level. To interpret over-investment caused by the agency problem, researchers have proposed various theoretical frameworks, such as the free cash-flow agency (Jensen, 1986), empire-building (Stulz, 1990), entrenchment (Shleifer & Vishny, 1989) and overconfidence (Roll, 1986) hypotheses. Generally, three principal explanations have been presented for why managers over-invest. First, Jensen (1986) argues that managers have incentives to over-invest because payouts of free cash to shareholders reduce the resources under their control, thereby reducing managers’ power, and making it more likely that the shareholders will incur the monitoring of the capital markets when a firm needs to raise fund. Second, by making manager-specific investment that often beyond its optimal size, managers can reduce the probability of being replaced (Shleifer & Vishny, 1989). Third, Stulz (1990) shows that over-confident managers tend to invest to the maximum extent when managers have discretion over firms’ decision-making. 2.1.2. Government intervention and over-investment One cannot ignore the effect of government behavior on corporate investment decisions. Government may internalize non-economic efficiency goals into firms’ investment activities by intervening in management decisions, with both effects of ‘grabbing hands’ and ‘helping hands’. For the former, government intervenes in firms’ investment activities to increase fiscal revenue, to improve social welfare, to maintain social stability and to achieve other social and political goals, which yields deviations in investment decisions from market efficiency and encroaches on and reduces shareholders’ wealth. The latter manifests itself such that government is intimately involved in promoting economic activity, and supports firms with close ties to it (Frye & Shleifer, 1997). Meanwhile, in a market economy, a firm may be short-sighted and this can be corrected by government intervention. In particular, when an internal management agency problem is serious, government control and supervision can inhibit managerial entrenchment, thereby protecting external investors’ interests from being abused. When a government takes actions to improve social welfare by expropriating firms, and firm managers also expropriate firms to maximizing their own benefits, government agency and management agency problems will occur simultaneously3 (Stulz, 2005). These two agency problems are neither irrelevant nor independent, but intertwined. Government behavior affects or restricts the personal benefits of managers, while managers are motivated to prevent such predatory government behavior to protect their personal interests. Therefore, firm investment decisions are a balanced outcome from interactions between government and management. 2.2. Why do China’s SOEs over-invest? Theoretical analysis 2.2.1. Economic reform and government intervention in China China has experienced the devolution of fiscal authority from central to local governments since 1978. The major objectives of fiscal reform were to make localities fiscally self-sufficient, to reduce the central state’s own fiscal burden including subsidizing inefficient firms, to maintain the social stability, and to provide incentives for local authorities to promote economic development. During the reform, localities became independent in terms of fiscal entities that had both responsibility for local expenditures and the unprecedented right to use the revenue that they retained (Oi, 1992). Local governments gained powers such as financial discretion and management

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of social public economy and also assumed social goals such as economic development, employment, social pension and social stability. To adapt to economic reform, in the early 1980s, China introduced the government official incentive system, where the Chinese central government was ready to reward or punish local officials on the basis of their economic performance in order to motivate them to promote the local economy (Blanchard & Shleifer, 2001; Li & Zhou, 2005). The central government has the power to appoint and dismiss local officials, and has exercised such power both to support the local officials whose regions have performed well economically and to discipline local officials who have followed anti-growth policies. In this context, local government officials are motivated to promote the local economy by generating fiscal revenue, GDP growth and social stability, among other goals, to gain political advancement opportunities. Firms’ investment not only boosts local economic growth but also increases employment opportunities, maintains social stability and facilitates government (officials’) performance evaluation goals, which facilitate political objectives. Therefore, investment may be the best option for local governments under competitive pressure to enhance their fiscal share and performance (Macroeconomic Study Group, CCER, Peking University, 2004). Meanwhile, inherent in gradual reforms, the government maintains ‘super control’ over SOEs. In order to promote economic growth and maintain social stability, the government (officials) has strong incentives to intervene in firms under the jurisdiction, especially SOEs,4 leading to the distortions of investment such as over-investment (Chen, Sun, Tang, & Wu, 2011), and SOEs bear the policy burden from multiple government objectives. Government intervention affects firms’ investment decisions in two different ways. One is referred to as a ‘protective effect’ on companies in a particular jurisdiction. On one hand, the bureaucrats’ political promotions rely on the performance of the SOEs under their jurisdiction. Therefore, they have responsibilities to support SOEs that undertake more policy burdens through providing resources and investment opportunities, and subsidizing the SOEs when they are in trouble. In addition, when the legal systems are not well developed, the government can provide guarantees to the SOEs’ partners by using its own creditworthiness for implementing firm contracts, and help the SOEs be in a favorable position in the market competition. On the other hand, by monitoring managers and providing incentive plans for them, the government can efficiently mitigate the agency problems between the managers and outside investors. When the formal mechanisms, such as the legal mechanisms, are not efficient enough for the outside investors to protect themselves from expropriation by managers, the monitoring role played by the government can be viewed as an informal alternative (Zhong et al., 2010). Second, high government intervention in SOEs may distort their investment decisions when government (officials) has incentives to expropriate resources for its own goals, which yields a government-to-business ‘predatory effect’. The government (officials) may impose its own goals such as economic growth, social stability due to either fiscal reform or the previously mentioned political promotion, on SOEs, which leads these firms to over-invest simply because over-investment can create more jobs and promote GDP growth (Tang et al., 2010; Zhang & Wang, 2010). 2.2.2. SOE reformation and managerial entrenchment During the economic reforms in the 1980s, the Chinese government launched a program that decentralized the managerial decision rights of SOEs from the central government down to the local firm level. The owner of an SOE, a governmental

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bureaucrat, typically faces decision-making constraints due to insufficient expertise and information, and thus allocates some decision rights to SOE managers. The SOE reformation went through four stages. The first stage ran from 1979 to 1983 with the major goals of administrative decentralization and profit retention, and in the second stage, SOEs were required to pay taxes instead of turning in profits, while the funding for SOEs’ capital investment, instead of being allocated directly from government financial reserves, had to come through bank loans (Sun & Tong, 2003). One of the major goals of the first two stages was to make the SOE mangers take responsibilities for their losses, and in this process, the SOE managers began to obtain some control rights from the government. In the third stage (1987–1992) with a Contractual Management System, the government gave managers a free hand to run their operations, that is, the separation of government ownership from control of SOEs’ operations (Sun & Tong, 2003). In the fourth stage of corporatization, the government granted significant controlling rights to SOE managers, except for the decision rights concerning M&A and the disposal of shares and assets, as well as the decision rights on the appointment of CEOs (Fan, Wong, & Zhang, 2007). However, empowered managers can expropriate substantial gains from the SOEs, resulting in severe agency costs. This is because, unlike a private firm, an SOE does not have a ‘true’ owner looking after the firm’s interests (Fan et al., 2012). Although the State-owned Assets Supervision and Administration Commission (SASAC) of the State Council and local governments were, since 2003, to take responsibility for managing the remaining SOEs, including appointing top executives and approving any mergers or sales of stock or assets and so on, it cannot mitigate the problem of ‘the absence of ownership’ in SOEs completely, and it is even possible that the officials of SASAC could make some concession to the managers in SOEs. Therefore, when empowered managers try to pursue their own benefits through investment decisions, over-investment would be their rational option (Wei & Liu, 2007; Xin et al., 2007; Zhang et al., 2010). When government intervenes in corporate investment decisions, managers’ responses to government actions potentially result in two types of relationship. One is a ‘complementary relationship’ between management and government. Predatory effects from government actions not only harm outside investor interests but also encroach on managers’ vested interests. Thus, to protect their own interests, managers are motivated to prevent government intervention, which yields a reciprocal substitutive relationship between the two entities. The other is a ‘supplementary relationship’ between management and government. From the above, under institutional fiscal decentralization, the government (officials) is interdependent with the firms under its jurisdiction. To get resources and investment opportunities from the government, managers may intentionally meet government (officials’) economic development needs through investments; thus, SOEs become indispensable to local economic development and employment stability (Chen & Li, 2012). Therefore, the firms’ investment behavior results from both managerial entrenchment and government intervention. 2.2.3. Interaction between government intervention and managerial entrenchment Under the economic transition, fiscal decentralization and gradual reform of SOEs yielded that local governments were deeply involved in local economic development. Furthermore, local protectionism and regional market segmentation exists commonly, leading to significant variances in government impacts on SOEs. Within heterogeneous levels of the government intervention environment, government behavior and its

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impacts on managerial entrenchment evolve in different ways. In an environment where there is more severe government intervention, there will be less discretion for managers to make investment decisions due to government constraints and supervision, and thereby limiting their over-investment motivation. Hence, there are two options for corporate managers. First, they take actions to decrease expropriation by the government. Second, managers may cater to the needs of the government. Owing to the interconnected and interdependent relationships between government (officials) and SOEs under its jurisdiction, managers of SOEs have to cater to the needs of the government (officials), so as to acquire the resources and investment opportunities provided by the government. In an environment with a relatively low government intervention, reduced government intervention is associated with an increased discretion in decision-making for the management; consequently, managers’ capacity to engage within government also increases. In this case, managers tend to behave in two different ways. One is opportunistic. Owing to the weaker government intervention and managerial entrenchment when internal governance fails, while an effective market discipline mechanism is unavailable, unfettered managers tend to over-invest. The other behavioral tendency is market efficiency. If a firm is in a favorable market environment, management behavior is restricted by market forces, and the manager will follow market efficiency standards for investment decision-making. In summary, the economic transition in China yields government ‘super control’ over SOEs, and the government intervenes in SOEs’ investment decision-making to promote its political objectives. Such control over investment activities leads to deviations from the objective to maximize shareholders’ wealth, resulting in over-investment through government intervention. Furthermore, SOEs’ managers have the benefit of investment expansion, which yields over-investment through management discretion. Therefore, considering both government intervention and managerial entrenchment of SOEs, it would be more meaningful to understand the current SOE reformation in China by analyzing the inducements of over-investment. Based on the above discussion, we pose three questions as follows. (1) What are the impacts of these two forces on SOEs’ investment decisions? (2) How do they interact during the process? Differently put, are they supplementary or complementary to each other? (3) Which one is the dominant force? Is there a difference in managerial entrenchment under heterogeneous levels of government interventions? 3. Research design 3.1. Samples and data The samples used in this study are drawn from A-share firms listed on Shanghai Stock Exchange and Shenzhen Stock Exchange in China during 2003–2010. Our final samples are unbalanced panel data consisting of 7997 firm-year observations after excluding: (1) firms in the financial sector; (2) observations with missing value; (3) cases without sufficient information on the nature of firm ownership and industry; (4) observations in their IPO years, and (5) observations in 2003 because our regression model (1) and (2) uses variables lagged by one period. We winsorize all continuous variables at the 1% and 99% levels. The data on the nature of firm ownership are from the China Center for Economic Research (CCER) database, and the data on managerial power are hand-collected from finance. sina.com.cn. Other firm-specific information is collected from the China Stock Market and Accounting Research (CSMAR) database.

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3.2. Variable definitions and regression models 3.2.1. Proxy for over-investment Consistent with prior research (e.g., Chen, Hope, Li, & Wang, 2011), we measure over-investment by a positive deviation from expected investment using the model based on Richardson (2006). We estimate optimal investment according to the following regression: Invt ¼ a0 þ a1 Growtht1 þ a2 Levt1 þ a3 Casht1 þ a4 Aget1 þ a5 Sizet1 X X þ a6 Rett1 þ a7 Invt1 þ Industry þ Year þ e

(1)

where the dependent variable, Invt, is the capital investment in year t. Growtht–1 represents firm growth opportunities. We use the sale growth rate in year t–1 in the main tests, and Tobin’s Q at the end of year t–1 for robustness tests. Levt–1, Casht–1, Aget–1, Sizet–1, Rett–1 and Invt–1 represent the financial leverage, cash holdings, age of being listed, firm size, stock return and capital investment at the end of year t–1, respectively. In addition, Industry and Year are dummy variables used to control for the industry and year effects. The residual ε represents deviations from expected investment, and positive deviations are considered as over-investments (OverInv). 3.2.2. Proxy for government intervention Providing the fact that the policy burden that Chinese enterprises bear reflects the direct economic consequences of government intervention in SOEs (Lin & Li, 2008), the policy burden is used as a proxy for government intervention. To implement a comparative-advantage defying (CAD) strategy, the Chinese government needs to impose its policy burdens on firms, known as the strategic policy burden and the social policy burden. The former stems from the fact that firms are forced to enter CAD industries or adopt CAD technologies, which were transferred to encourage local economic growth (Qian & Roland, 1998). The latter refers to keeping redundant workers. Since the capital-intensive industries cannot provide enough job opportunities, in order to solve employment problems and maintain social stability, Chinese government expects SOEs to retain excess workers, resulting in a social policy burden (Lin & Li, 2008). Either the strategic policy burden or the social policy burden is a result of the government strategy of CAD. Under the government intervention, firms invest in projects that should be financed by the government or provide jobs that should be removed. In the trading game between government and firms, stronger government interventions yield heavier policy burdens for firms. Lin, Liu and Zhang (2004) employed the square of the deviation for a firm’s optimal capital–labor ratio determined by economic factor endowments, as a proxy for policy burden. Motivated by their research design, we measure policy burden by using an expanded model based on Zeng and Chen (2006) and Liu, Zhang, Wang, and Wu (2010). We estimate the optimal capital–labor ratio based on the following regression specification. Cit ¼ b0 þ b1 Sizet1 þ b2 Levt1 þ b3 Roat1 þ b4 Growtht1 þ b5 Tangiblet1 X X X þ Zone þ Industry þ Yearþd

(2)

where the dependent variable, Cit, is capital-intensity, defined as capital–labor ratio, which is measured as the ratio of a firm’s net value of Property, plant and equipment (PPE) to the number of employees at the end of year t. Sizet–1, Levt–1, Roat–1 and

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Tangiblet–1 represent the firm size, financial leverage, return on assets and tangible assets at the end of year t–1, respectively. Zone is the regional dummy variable. In addition, we control for industry and year effects. The residual δ represents the deviation from the optimal capital-intensity determined by economic factor endowments. A positive residual indicates that a firm’s capital-intensity surpasses its optimal capitalintensity, primarily driven by the strategic policy burden. A negative residual indicates that a firm’s capital-intensity is lower than its optimal capital-intensity, primarily driven by social policy burden. We use the absolute value for residual δ as the policy burden (Ovci). 3.2.3. Proxy for managerial entrenchment The agency problem stems from the separation of ownership and control. A manager has discretion over firms’ decisions because he or she ends up with significant control rights of the firm (Shleifer & Vishny, 1997). Consistent with Lu, Wei, and Li (2008) and Quan, Wu, and Wen (2010), we measure managerial entrenchment by using managerial power to reflect managers’ ability to expropriate shareholders or to misallocate firms’ funds. Managers with a stronger managerial power have more discretion over firms’ decisions, and thus are more likely to serve the interests of themselves rather than those of shareholders. Managerial power depends on two aspects: first, how do managers acquire and reinforce their power, which comes from the balance of power between managers and the board members. Second, how are they constrained or monitored, which mainly comes from the balance of power between the large and minority shareholders. Therefore, we measure the managerial power from the above two aspects: managerial power structure and ownership structure. The former reflects the position power, which will be strengthened if a manager, in particular, the CEO, is also a member of the board, or even the chairman of the board. The latter reflects the balance of power of ownership. When control rights are concentrated in the hands of a smaller number of shareholders, the managers may have less discretion over the firm, and thus they have less power. 3.3. Baseline regression: government intervention, managerial entrenchment and over-investment Next, in model (3), we examine how government intervention and managerial entrenchment affect over-investment. X X Control t þ Year þ l OverInvt ¼ c0 þ c1 Ovcit þ c2 Powert þ c3 Ovcit Powert þ (3)

where the dependent variable, OverInv, represents over-investment, policy burden (Ovci) is a proxy for government intervention, and managerial power (Power) is a proxy for managerial entrenchment. The coefficients γ1 and γ2 denote the correlations of government intervention and managerial entrenchment with over-investment, respectively; the coefficient on the interaction term (Ovci × Power) is γ3, which captures the relationship between government intervention and managerial entrenchment and its influence on over-investment. If γ1 and γ2 are significantly positive, while γ3 is significantly negative, both government intervention and managerial entrenchment potentially lead to overinvestment but have a complementary relationship. In comparison, if γ3 is significantly

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positive, the government and management potentially have a supplementary relationship, wherein the two conspire to further encroach on external investor interests. The Control comprises a set of control variables that could affect over-investment, including the management expense (Adm), free cash flow (Fcf), cash occupied by related parties (Orecta), executive compensation (Comp), the proportion of independent directors (Indep) and the marketization index (Market). Ang, Cole, and Lin (2000) argue that the management expense ratio is an economic consequence of conflicts between managers and shareholders, which may also result in over-investment; thus, it is unlikely to predict the relation between these two variables. Jensen (1986) argues that empire-building preferences will cause managers to spend essentially all free cash flow on investment projects. The considerations lead to the prediction that over-investment will be increasing in free cash flow (Fcf). Xin et al. (2007) argue that ‘other receivables’ used as a vehicle for large shareholder tunneling will make firms more financially constrained, and thus they have to cut the capital investment, which finally make them less likely to over-invest. Therefore, we expect cash occupied by related parties (Orecta) to have a negative impact on a firm’s over-investment. Xin et al. (2007) also find that the existence of regulation on executive compensation in Chinese SOEs leads to the execution failure of the compensation contraction between the executives and the governments, making the executives more likely to over-invest. We therefore predict that executive compensation (Comp) would have a negative impact on over-investment. Zhi and Tong (2005) find that independent directors can effectively identify the managers’ earnings management behavior, and Ye, Lu and Zhang (2007) find that independent directors can reduce the funds embezzled by large shareholders. Tang, Luo, and Wang (2005) and Gao, He, and Huang (2006), however, fail to find a significant and negative relation between the proportion of independent directors on board and the extent of tunneling in a firm. Inconsistencies in empirical evidence and the conclusions of prior research make it hard to predict the relation between the proportion of independent directors on board (Indep) and overinvestment (OverInv). Tang et al. (2010) argue that, firms located in places where governments have strong incentives to influence their operations, are more likely to over-invest. Therefore, we expect that the marketization index (Market) would have a negative coefficient. We adjust all continuous variables by subtracting the industry median for the current year, except over-investment (OverInv), policy burden (Ovci) and the marketization index (Market).5 Detailed variable definitions are provided in Table 1.

4. Descriptive statistics and empirical results 4.1. Variable estimates and descriptive statistics Table 2 reports the regression results based on Model (1). Consistent with prior research (e.g., Xin et al. 2007), the coefficients on Growtht–1, Casht–1, Sizet–1, Rett–1 and Invt–1 are all positive and significant at a 1% level, and Levt–1 and Aget–1 are negatively and significantly associated with Invt. According to the regression results of model (1), we obtain 3177 observations in the over-investment sample, including 2108 SOE observations and 1069 non-SOE observations. Table 3 reports the regression results based on model (2). Consistent with prior studies (e.g., Liu et al., 2010), the coefficients on Sizet–1 and Tangiblet–1 are positive and significant at a 1% level, and Levt–1 is negatively and significantly associated with

246 Table 1.

Bai and Lian Definitions of the variables.

Variables Definition Inv OverInv Ovci State Structure Disp Power Adm Growth

Lev Cash Age Size Ret Ci Tangible Roa Fcf Orecta Comp Indep Market Zone Industry

Year

Capital investment is measured as cash payments for fixed assets, intangible assets, and other long-term assets from the cash flow statement minus receipts from selling these assets, scaled by beginning total assets. Over-investment is measured as the positive residuals of model (1). Policy burden is measured as the absolute value of the residuals of model (2). A dummy variable equals one if the firm is an SOE and zero if the firm is a non-SOE. Managerial power structure is measured as an ordinal dummy variable, which equals one if the CEO is not a member of the board, two if the CEO is also the member of the board, and three if the CEO is also the chairman of the board. Ownership structure is measured as the ratio of the sum of the percentage points of shareholding by the 2nd to the 10th largest shareholders to the percentage points of shareholding by the 1st largest shareholders. Managerial power is measured as the sum of Structure and Disp after being standardized.6 Management expense is measured as the ratio of administrative expenses to total revenue. Firm growth opportunities is measured as Tobin’s Q or Dsales. Tobin’s Q is measured as the sum of market value of tradable shares, book value of non-tradable shares and liabilities divided by book value to total assets. Dsales is measured as the proportion of change in sales. Financial leverage is measured as the ratio of total liabilities to total assets. Cash holdings is measured as the ratio of cash to total assets. Age of being listed is measured as the number of years a firm being publicly listed. Firm size is measured as the natural logarithm of a firm’s total asset. Stock return is measured as the annual market stock return of a firm. Capital-intensity is measured as the ratio of net value of Property, plant and equipment (PPE) (in million yuan) to the number of employees. Tangible assets is measured as the ratio of net value of Property, plant and equipment (PPE) to total assets. Return on assets is measured as the ratio of net income to total assets. Free cash flow is measured as the ratio of the operating cash flow of a firm to total assets. Cash occupied by related parties is measured as the ratio of other receivables to total assets. Executive compensation is measured as the natural logarithm of cash compensation of the sum of the three highest paid executives. The proportion of independent directors on board. A comprehensive index measuring the development of the regional market in which a firm is registered (Fan & Wang, 2010), where higher values indicate greater regional market development.7 A region-level dummy variable to proxy for the degree of development of China’s regional institutions. Zone1 equals one if a firm is registered in coastal areas, and zero otherwise. Zone2 equals one if a firm is registered in central areas, and zero otherwise. The classification of industry follows the CSRC document, Guidance on Listed Firms’ Industries, issued on April, 2001. There are altogether 13 industries coded from A, to M, and 10 subindustries under C. We classify all the listed firms into 22 industries as we treat the 10 subindustries under manufacturing as distinct industries. A dummy variable equals one if the firm went public during that year, and zero otherwise.

China Journal of Accounting Studies Table 2.

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Regression results of investment equation.

Variables

Invt

Intercept

–0.0359*** (–3.12) 0.0026*** (2.74) –0.0171*** (–7.34) 0.0328*** (6.11) –0.0007*** (–4.65) 0.0032*** (6.04) 0.0027*** (3.35) 0.5062*** (40.12) Yes 0.407 7997

Growtht–1 Levt–1 Casht–1 Aget–1 Sizet–1 Rett–1 Invt–1 Year and Industry Adj-R2 N *

, **, and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively, for a two-tailed test. T-statistics are reported in parentheses, and are based on Huber-White’s robust standard errors.

Cit. The coefficient on Roat–1 is negative, while the coefficient on Growtht–1 is positive, but none of them are statistically significant. Table 4 provides the descriptive statistics for main variables. As reported in the table, both the number of over-investment observations (n) and the mean or median over-investment (OverInv) are significantly larger for the SOE group than the corresponding numbers for the non-SOE group. Furthermore, in the SOE group, the policy burden (Ovci) is significantly higher than that in the non-SOE group, while Table 3.

Regression results of policy burden equation.

Variables

Cit

Intercept

–4.6127*** (–11.23) 0.2339*** (12.36) –0.1885** (–2.54) –0.3293 (–1.39) 0.0268 (0.64) 1.4645*** (10.91) Yes 0.184 7997

Sizet–1 Levt–1 Roat–1 Growtht–1 Tangiblet–1 Year and Industry Adj-R2 N

* ** , , and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively, for a two-tailed test. T-statistics are reported in parentheses, and are based on Huber-White’s robust standard errors.

0.0242 (3177) 0.3061 –0.3559 0.0715 0.0515 0.0154 13.5222 0.3333

0.0388 (3177) 0.6289 0.0000 0.1062 0.0527 0.0406 13.4890 0.3574

0.0397 (2108) 0.6827 –0.1952 0.0933 0.0553 0.0355 13.5110 0.3550

Mean 0.0246 (2108) 0.3291 –0.5553 0.0684 0.0531 0.0142 13.5515 0.3333

Median

SOEs: State=1

0.0370 (1069) 0.5232 0.3762 0.1309 0.0479 0.0504 13.4465 0.3620

Mean

0.0236 (1069) 0.2657 0.0853 0.0773 0.0475 0.0183 13.4568 0.3333

Median

Non-SOEs: State=0

0.0027 0.1595*** –0.5714*** –0.0376*** 0.0074*** –0.0149*** 0.0645*** –0.007***

*

t value

0.0010* 0.0634*** –0.6406*** –0.0089*** 0.0056*** –0.0041*** 0.0947*** 0.0000***

z value

Test for difference in mean and median

, , and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively. The test for mean difference is student’s test, and the test for median difference is Wilcoxon test. Figures in parentheses denote number of the observation for firms with overinvestment.

* **

Median

Mean

Full sample

Descriptive statistics for main variables.

OverInv (n) Ovci Power Adm Fcf Orecta Comp Indep

Variables

Table 4.

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managerial power (Power) is significantly less than that in the non-SOE group. In addition, the management expense (Adm) and cash occupied by related parties (Orecta) are significantly lower for the SOE group than those for the non-SOE group, while free cash flow (Fcf) in the SOE group is significantly higher than that in the non-SOE group. The results from univariate tests indicate that, relative to non-SOEs, SOEs assume more policy burdens due to government intervention, acquire more funds due to government support and have a lower agency cost due to government supervision and restriction of managerial power. From the investment efficiency perspective, government intervention may cause over-investment due to its social responsibility goals, while also limiting the manager’s over-investment behavior by constraining managerial power. Overall, in China, wherein market competition and corporate governance mechanisms are not well developed, both the government’s protective and predatory side may coexist in SOEs. On one hand, in order to encourage economic growth and maintain social stability, the government imposes more policy burdens on SOEs, making their decisions deviate from the goal of maximizing shareholders’ interests and instead, reducing shareholders’ wealth. That’s the grabbing hand of government. On the other hand, by strengthening supervision over managers, the government can limit managerial entrenchment, and therefore protect shareholders’ interests from expropriation by managers, and that’s the helping hand of the government. Table 5 reports Pearson and Spearman correlations for the main variables. Both policy burden (Ovci) and managerial power (Power) are significantly and positively correlated with over-investment (OverInv) whether the Spearman or Pearson correlation coefficient is used, indicating that both government intervention and managerial entrenchment can lead to over-investment. The government may be highly motivated to request firms expand investment to achieve social goals, such as promoting local economic growth and maintaining social stability, which results in over-investment. Managers with greater managerial power have greater motivation and capacities for over-investment. Policy burden (Ovci) and managerial power (Power) are significantly and negatively correlated, suggesting a complementary relationship between these two variables. For given interest constraints, government and managers are involved in competitive games in expropriating the outside shareholders. In addition, managerial power (Power) is also significantly and positively correlated with management expenses (Adm) and cash occupied by related parties (Orecta), indicating that the greater the managerial power, the greater tendency to opportunism managers would have. Conversely, the policy burden (Ovci) is significantly and negatively correlated with cash occupied by related parties (Orecta) and management expenses (Adm), implying that government intervention can restrict management opportunistic behavior. Consequently, the government, as a regulatory authority, has a restrictive effect on managerial entrenchment. Meanwhile, policy burden (Ovci) is significantly and positively correlated with free cash flow (Fcf), indicating that government intervention can help firms acquire more fund as well. 4.2. Effects of government intervention and managerial entrenchment on over-investment Table 6 reports the regression results for government intervention, managerial entrenchment and over-investment in SOE and non-SOE groups. In SOEs, both the coefficient on policy burdens (Ovci) and that of managerial power (Power) are positive

0.0732*** 0.0407** –0.0237 0.1347*** –0.1639*** 0.0074 –0.0104

OverInv ***

–0.0401** –0.0578*** 0.0616*** –0.0902*** 0.1098*** 0.0209

0.0776

Ovci

Correlation matrix for main variables.

0.0974** 0.0452*** 0.0620*** 0.0938 *** 0.0162

0.0322 –0.0497***

*

Power ***

–0.0708*** 0.2662*** –0.1160*** 0.0079

–0.0462 –0.0120 0.1103***

Adm ***

–0.1339*** 0.1343*** –0.0515***

0.1202 0.0402*** 0.0089 –0.1453***

Fcf

–0.1996*** 0.0039

–0.0960 –0.0256** 0.0653*** 0.4102*** –0.1454***

***

Orecta

0.0316*

–0.0088 0.0611*** 0.1047*** –0.1876*** 0.1079*** –0.2376***

Comp

–0.0347* 0.0034 0.0062 0.0002 –0.0512*** –0.0293** 0.0612***

Indep

* ** , , and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively. This table reports Pearson (above the diagonal) and Spearman (below the diagonal) correlation matrix of the main variables.

OverInv Ovci Power Adm Fcf Orecta Comp Indep

Table 5.

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China Journal of Accounting Studies Table 6.

Impacts of government intervention and managerial entrenchment on over-investment. SOEs: State=1

Variables Intercept Ovci

(1)

(2) ***

0.0482 (11.57) 0.0031*** (3.59)

Power

(3) ***

0.0489 (11.73)

0.0017** (2.29)

Ovci×Power Adm Fcf Orecta Comp Indep Market Year and Industry Adj-R2 N

0.0201* (1.85) 0.0463*** (3.43) –0.1113*** (–7.87) –0.0020 (–1.38) –0.0076 (–0.40) –0.0012** (–2.41) Yes 0.032 2108

0.0176 (1.60) 0.0460*** (3.40) –0.1122*** (–8.02) –0.0024 (–1.64) –0.0110 (–0.56) –0.0010** (–2.12) Yes 0.025 2108

Non-SOEs: State=0 (4) ***

0.0471 (11.25) 0.0030*** (3.62) 0.0028*** (3.57) –0.0014*** (–2.61) 0.0175 (1.61) 0.0457*** (3.40) –0.1142*** (–8.07) –0.0027* (–1.91) –0.0063 (–0.33) –0.0011** (–2.22) Yes 0.038 2108

0.0490 (7.91) 0.0013 (1.08)

(5) ***

(6) ***

0.0488 (7.89) 0.0012 (1.55)

–0.0076 (–1.14) 0.0575*** (3.51) –0.0118 (–0.61) 0.0011 (0.66) –0.0212 (–0.79) –0.0014** (–2.10) Yes 0.017 1069

–0.0089 (–1.32) 0.0555*** (3.40) –0.0123 (–0.63) 0.0007 (0.44) –0.0194 (–0.73) –0.0014** (–2.11) Yes 0.018 1069

0.0487*** (7.80) 0.0012 (1.05) 0.0007 (0.89) 0.0010** (2.35) –0.0086 (–1.27) 0.0558*** (3.42) –0.0112 (–0.58) 0.0009 (0.55) –0.0221 (–0.83) –0.0014** (–2.16) Yes 0.019 1069

The table provides the regression results of government intervention and managerial entrenchment on overinvestment on SOEs and non-SOEs. The dependent variable OverInv is calculated according to Richardson (2006). *,**, and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively, for a two-tailed test. Tstatistics are reported in parentheses, and are based on Huber-White’s robust standard errors.

and statistically significant, while the coefficient on the interaction term (Ovci × Power) is significantly negative. The results indicate that both government intervention and managerial entrenchment can lead the SOEs to over-invest. Meanwhile, these two are complementary to each other in affecting over-investment. However, in non-SOEs, neither the coefficient on policy burden (Ovci) nor that of managerial power (Power) is significant and, in particular, the coefficient on the interaction term (Ovci × Power) is significantly positive. Such a difference suggests that government intervention, managerial entrenchment and their interaction effects can be found mainly in SOEs. The government can impose policy burdens such as economic growth and social stability on SOEs, resulting in over-investment in SOEs. Meanwhile, as agents, managers may also expropriate outside shareholders through over-investment to pursue their own benefits. However, the counterbalance between government and managers can suppress overinvestment. The interaction effects between government intervention and managerial entrenchment on over-investment in SOEs can be explained as follows. On one hand, through supervising the manager and optimizing the protective mechanism for minority shareholders, the government increases the costs of expropriating outside shareholders by managers, thereby restricting such expropriation. On the other hand, managers may resist the predatory behavior of government to pursue their own benefits.

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4.3. Further analysis The above analyses show that in SOEs, both government intervention and managerial entrenchment can lead to over-investment, while these two effects appear to be complementary to each other. One may expect to see which is the dominant force, government intervention or managerial entrenchment? If government intervention dominates managerial entrenchment, then how do heterogeneous levels of government intervention affect managerial entrenchment? Differently put, is there a difference of managerial entrenchment under heterogeneous levels of government interventions? To answer these questions, we examine the interaction effects between government intervention and managerial entrenchment on SOEs’ over-investment under heterogeneous levels of government interventions in this section. 4.3.1. Is government intervention the dominant force leading to over-investment? In this section, we divide the sample firms into groups with high or low government intervention by using the median value of policy burden (Ovci), and then examine the interaction effects between government intervention and managerial entrenchment on overinvestment between SOEs with high government intervention and SOEs with low. We create a dummy variable, Ctrllevel, which is coded as one if the policy burden (Ovci) is greater than the industry median value in SOEs in that year, and zero otherwise. Descriptive statistics are reported in Table 7. It shows that both the mean and median of over-investment (OverInv) are significantly higher for the SOEs with high government intervention than for those with low, while there is no statistical difference for managerial power (Power) between these two groups (SOEs with high versus low government intervention). The results of univariate tests indicate that SOEs under heterogeneous levels of government interventions have different over-investment expenditures when the managerial power is relatively constant. Hence, government intervention may be the dominant force that affects SOEs’ over-investment. For the SOEs with high government intervention, the government imposes more policy burdens on SOEs, which leads to over-investment. 4.3.2. Different management behavior under heterogeneous levels of government interventions The further univariate results reported in Table 7 suggest that government intervention may be the dominant force affecting SOEs’ over-investment. We subsequently examine Table 7.

Descriptive statistics of over-investment and managerial power. High government intervention （Ctrllevel=1）

Variables

Mean 0.0431 (1053) -0.1121

OverInv (n) Power ***

Median 0.0269 (1053) -0.4708

Low government intervention （Ctrllevel=0） Mean 0.0364 (1055) -0.1916

Median 0.0229 (1055) -0.5354

Test for difference in mean and median t value 0.0067 0.0795

***

z value 0.0040*** 0.0646

*, **, and indicate the 0.1, 0.05, and 0.01 levels of significance, respectively. The test for mean difference is student’s test, and the test for median difference is Wilcoxon test. Figures in parentheses denote number of the observation for firms with overinvestment.

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the interaction effects between government intervention and managerial entrenchment on SOEs’ over-investment and the behavioral characteristics of managers in response to heterogeneous levels of government interventions. The empirical results are reported in Table 8. The results indicate that for SOEs with high government intervention, the coefficient on policy burden (Ovci) is significantly positive at the 1% level and the coefficient on managerial power (Power) is positive at the 10% level, while the coefficient on the interaction term (Ovci×Power) is significantly negative at the 5% level, suggesting that for SOEs within a strong government intervention environment, government supervision on a manager is greater when the government has stronger incentives to intervene in SOEs’ decisions. A strong control by the government over SOEs constrains managers’ opportunistic behavior, thereby weakening the managers’ discretion over firms. However, managers may resist government intervention as long as they have incentives to maximize their personal benefits of control. For SOEs with low government intervention, policy burden (Ovci) is significantly positive at the 10% level and managerial power (Power) is significantly positive at the 1% level correlation with over-investment, while the interaction term Table 8.

Determinants of overinvestment: SOEs with high or low government intervention. SOEs (State=1) High government intervention （Ctrllevel=1）

Variables Intercept Ovci

（1）

（2） ***

0.0518 (8.25) 0.0025*** (2.64)

（3） ***

0.0542 (8.73) 0.0004 (0.34)

Power Ovci×Power Adm Fcf Orecta Comp Indep Market Year and Industry Adj-R2 N

0.0105 (0.53) 0.0456** (2.09) –0.1063*** (–4.75) –0.0040* (–1.83) –0.0306 (–1.06) –0.0012* (–1.73) Yes 0.029 1053

0.0105 (0.53) 0.0450** (2.06) –0.1027*** (–4.75) –0.0041* (–1.88) –0.0371 (–1.28) –0.0012 (–1.63) Yes 0.019 1053

Low government intervention （Ctrllevel=0） （4） ***

（5） ***

0.0512 0.0430 (8.11) (7.15) 0.0023*** 0.0211* (2.62) (1.67) 0.0022* (1.82) –0.0013** (–2.34) 0.0114 0.0301*** (0.57) (2.69) 0.0458** 0.0460*** (2.10) (2.78) –0.1057*** –0.1128*** (–4.76) (–6.66) –0.0047** 0.0000 (–2.18) (0.03) –0.0292 0.0113 (–1.01) (0.44) –0.0012* –0.0011* (–1.65) (–1.71) Yes Yes 0.032 1053

0.025 1055

（6） ***

0.0446 (7.82)

0.0032*** (3.20) 0.0228** (2.00) 0.0442*** (2.67) –0.1183*** (–6.96) –0.0005 (–0.28) 0.0119 (0.46) –0.0010 (–1.44) Yes 0.033 1055

0.0415*** (6.84) 0.0230* (1.84) 0.0051*** (2.78) –0.0115 (–1.22) 0.0231** (1.99) 0.0450*** (2.74) –0.1191*** (–6.98) –0.0004 (–0.24) 0.0139 (0.54) –0.0010 (–1.53) Yes 0.035 1055

The table provides the regression results of interaction effects between government intervention and managerial entrenchment on over-investment with high or low government intervention. The dependent variable OverInv is calculated according to Richardson (2006). *,**, and ***indicate the 0.1, 0.05, and 0.01 levels of significance, respectively, for a two-tailed test. T-statistics are reported in parentheses, and are based on Huber-White’s robust standard errors.

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(Ovci × Power) is negative but not statistically significant, suggesting that the managers have more discretion over investment decisions in SOEs within a weak government intervention environment. With less supervision from the government, managers are more likely to over-invest and tend toward opportunism. Using China as our research setting, our study complements and extends existing research on the economic consequences of ‘the twin agency problems’ proposed by Stulz (2005). In his model, Stulz argues that all the investors risk expropriation by the government and outside investors additionally risk expropriation by those who control firms, whom he calls corporate insiders, since they are sometimes managers and at other times controlling shareholders. In the transitional economy of China, government (officials) has stronger incentives to intervene in SOEs’ decision-making due to the fiscal decentralization and the reform of political promotion mechanisms, while the managers of SOEs end up with the de facto operating decision rights during the process of the Chinese government launching a program that decentralized the managerial decision rights of SOEs from the central government down to the local firm-level. This is the ‘two agency problems’ described by Stulz (2005). However, different from his model, the Chinese government also plays a protective role, by constraining managerial opportunism, in addition to its expropriating role. Consequently, managerial entrenchment in SOEs is determined by the institutional environment of government reform. It grants large controlling rights to the managers in SOEs, however, when internal governance fails, it can also be a monitor to prevent the managers from making over-investment decisions. 4.4. Robustness tests We conducted a battery of additional tests to check the robustness of our results. First, we used Tobin’s Q as an alternative measure for growth opportunities, instead of sales growth as used before. Xin et al. (2007) argue that Chinese stock markets are less developed relative to other markets, such as the US markets, which makes the Tobin’s Q an inappropriate measure of growth opportunities. To examine whether our results are robust to using Tobin’s Q, we re-do the analysis by using this alternative measure of growth opportunities. Second, following Xin et al. (2007), we categorize firms into three groups based on the residuals derived from model (1) and remove the middle group because these firms, whose unexpected investments are closest to 0 among all firms, are more likely to be affected by measurement error in the investment model (Chen, Hope, Li, & Wang, 2011). Firms in the group with the largest residuals are regarded as firms with overinvestment. Third, the panel data sets we use here contain observations on multiple firms in multiple years, so we correct the standard error for clustering of observations by firms as suggested by Petersen (2009). Fourth, we tackle endogeneity. One may ask whether the relationship between government intervention, managerial entrenchment and over-investment is endogenously determined. Following Lin and Li’s (2008) theoretical analysis, we use the ratio of secondary industry production to gross production, the ratio of population employed by secondary industry to total employed persons, as well as GDP per capita,8 as instrumental variables of policy burden. Fifth, we use negative residuals derived from model (2) as a proxy for policy burden (Ovci). In the above analysis, we measure the policy burden by the absolute value

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of residual from model (2). As Lin and Li (2008) point out, policy burdens consist of both strategic policy burdens and social policy burdens. A strategic policy burden stems from the fact that the firms are forced to enter CAD industries or adopt CAD technologies. Since the capital-intensive industries cannot provide enough job opportunities, in order to solve employment problems and maintain social stability, government needs to retain excess workers, which results in a social policy burden. Therefore, the social policy burden is a more direct economic consequence of government intervention, and it may be a more appropriate measure for the incentives of the government to intervene in SOEs. Sixth, the levels of government intervention and the interaction between government intervention and managerial entrenchment are examined both cross-sectionally and over time. In the cross-sectional analyses, we divide the SOE samples into high or low government intervention groups in two different ways. One, following Fan et al. (2012),9 uses the unemployment rate of the region under the jurisdiction of the local government to measure the regional policy burden, thereby classifying the levels of government intervention environment. Given that social stability is a major policy burden of government, achieving higher unemployment provides government with the stronger incentives to intervene in corporate investments and results in policy burdens at the regional level. We create a dummy variable, Unemrate_dum, which equals one if the unemployment rate of the region under the jurisdiction of the local government is greater than the median value in China in that year, and zero otherwise. The other way is by using the median value of the marketization index by which we divide the SOE samples into high or low government intervention groups. SOEs with high government intervention are firms located in places where the marketization index is less than the median value in that year, while SOEs with low government intervention are firms located in places where the marketization index is greater than the median value in that year. In the time series analysis, we divide the SOE samples into high or low government intervention groups in a particular period. During the recent financial crisis, the Chinese government was forced to unleash a plan of government stimulus and credit to keep growth growing, including a 4-trillion Yuan fiscal stimulus and a 10-trillion Yuan bank loans stimulus. The stimulus programs were implemented through direct instructions from the government to banks for loan issuance and project financing to maintain economic growth through capital expenditure expansion. In the process, the government directly intervened in bank loan allocations through commands (Bai & Lian, 2012), imposed its objectives on SOEs, and forced SOEs to expand their investment for political objectives instead of maximizing the value of corporate shareholders. Therefore, we predict that the level of the government intervention in 2008–2010 should be greater than that in 2004–2007. In summary, our results are robust to alternative measures of growth opportunities, over-investment, policy burden, government intervention and the levels of government interventions, in order to tackle endogeneity, and to correct the standard error for clustering of observations by firms. For the sake of brevity, we do not tabulate the results for the additional tests. 5. Conclusions During economic restructuring, fiscal decentralization and SOE reformation result in excessive control of local governments over SOEs and provide incentives to corporate managers to pursue greater authority and controls. Government intervention and managerial entrenchment have become two major drivers for SOEs’ over-investments.

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Government may force firms to carry out policy burdens, such as economic development and employment, leading to SOEs’ over-investment. Alternatively, empowered managers can also expropriate substantial gains from SOEs, resulting in over-investment for their own personal benefits. Moreover, the Chinese government also plays a protective role (‘helping hand’) by constraining managerial opportunism in SOEs, in addition to its role of the ‘grabbing hand’, and in turn managers will resist the monitoring from the government.10 Our empirical results reveal that, in SOEs, both the policy burden from government intervention and managerial rent-seeking due to managerial entrenchment can lead to over-investment, and these two forces are complementary to each other. More powerful government intervention can suppress managerial entrenchment, and in turn managers have more discretion over firms’ decisions, whose behavior tends toward opportunism where government intervention is weak. Our findings raise an interesting question for further study. Should the Chinese government only play the role as a predator to expropriate resources from these firms, given the government’s incentives to intervene in SOEs for policy burdens and political promotion? If this conjecture is confirmed, one would have to attribute the remarkable economic growth in China to non-SOEs. Our results show that in a transitional economy such as China, where corporate governance and market economy mechanisms are not well developed, government can be conducive to reducing management agency cost through monitoring the managers, ensuring that contracts are fulfilled and partially protecting outside investor interests. Thus, during the market-oriented reform process in China, where the investor protection systems are significantly less developed than most of the countries in the world (Allen, Qian, & Qian, 2005), a gradual reform strategy of deregulation should be persistently implemented as it can retain the alternative systems of informal investor protection that the government brings. Acknowledgements The authors acknowledge financial support from the National Natural Science Foundation of China (No. 71262007), the National Social Science Foundation of China (No. 11XGL002), the Humanistic and Social Science Foundation of the Ministry of Education of China (No. 10YJC630002; No. 09YJC630160), the Social Science Foundation of Xinjiang Production and Construction Corps (No. 10BTYB12) and the Key Research Center for Social Sciences of Committee of Education of Xinjiang (No. XJEDU020112C03). They also thank the Talents Boosting Program for the New Century of the Ministry of Education of China and the National Accounting Talents (Reserve) Program of the Ministry of Finance of China. The authors would like to offer their most sincere thanks to three anonymous reviewers and the editors, Jason Xiao and Tong Yu, for their insightful comments. In addition, the authors specially thank Dr Liang Han, at University of Surrey, UK, for his valuable advice and proofreading. The authors would take full responsibility for the paper.

Notes 1.

2. 3.

The international data of gross capital formation (% of GDP) is collected from the World Bank (http://data.worldbank.org/indicator/NE.GDI.TOTL.ZS). Other East Asian countries and areas are Cambodia, Hong Kong (PRC), Indonesia, Japan, Lao PDR, Malaysia, Mongolia, South Korea, Singapore, Thailand, and Vietnam, where the data of gross capital formation (% of GDP) are available. Consistent with their findings, Zhang (2003) finds that the investment was efficiently and largely reaped through the rural industrialization and proliferation of small firms in non-state sectors. According to the hypothesis of ‘grabbing hands’ of government, Stulz (2005) came up with the twin agency problems. The core part of his point is that the owners of a firm bear the

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risks of state expropriation, and outside investors take the risk of expropriation by the insiders, who often refer to controlling shareholders and managers. Specifically, in order to maximize their own interests, insiders would acquire private benefits by expropriating outside investors, which can produce ‘the agency problem of insider discretion’. At the same time, government rulers expropriate investors by virtue of the authority of government for their own sake, and then ‘the agency problem of state ruler discretion’ arises. 4. Government may also intervene in non-SOEs, but compared with the SOEs, both the possibility and the intensity of the intervention may be lower (Tian, 2005). 5. Because the estimation models of over-investment (OverInv) and policy burden (Ovci) have controlled for the industry and year effects, it can cause repetitive control if we still control for these effects. The marketization index is not affected by industry. Therefore, all continuous variables except for OverInv, Ovci and Market are adjusted by using their industry median value. We thank Professor Xin Zhang, at the School of Management, Fudan University, for his helpful comments. 6. The standardized equation for variable x is: (x–m/sd, and m stands for the mean of variable x, while sd is the standard deviation. 7. The most updated Marketization Index covers all of the provinces from 1997–2007, so we match the cases after 2008 with the index of 2007. 8. The data are collected from The Statistical Data of New China for Fifty Years. We use statistical data in 1985, because data under some jurisdictions are not available until 1985. In addition, we exclude firms registered in Chongqing, as it was established as a municipality in 1997. 9. The unemployment rate data is obtained from www.drcnet.com.cn. 10. Shleifer and Vishny (1994) argue that when the government (officials) controls firms, managers may even use bribes to convince them not to push firms to pursue political objectives, which can be regarded as a means deployed by the managers to resist the intervention by the government (officials).

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